TOWARDS A NON-LINEAR CALDERÓN-ZYGMUND THEORY
GIUSEPPE MINGIONE
To Tadeusz Iwaniec on his 60th birthday
Contents
1. Road map
2. The linear world
3. Iwaniec opens the non-linear world
4. Review on measure data problems
5. Nonlinear Adams theorems
6. Beyond gradient integrability
References
1
2
8
20
27
36
47
1. Road map
Calderón-Zygmund theory is a classical topic in the analysis of partial differential
equations, and deals with determining, possibly in a sharp way, the integrability
and differentiability properties of solutions to elliptic and parabolic equations, and
especially of their highest order derivatives, once an initial, analogous information is
known on the given datum involved. Now it happens that while a linear theory has
been developed in a quite satisfactory way, a complete theory for gradient estimates
for solutions to quasilinear equations of the type
(1.1)
div a(x, Du) = “suitable right hand side”
is not yet developed, at least up to that complete extent one could wish for.
The reason for such a difference lies of course in that linear structures allow,
via certain explicit representation formulas, for applying rather abstract tools from
Harmonic Analysis, semigroup theory, abstract Functional Analysis, interpolation
theory and so forth. The use of all such tools is clearly ruled out in the case
of non-linear structures as in (1.1), basically because representation formulas are
not available, and more in general because no linear or sub-linear operator can be
associated to the non-linear problem in question.
The purpose of this paper is now to collect all those pieces - that is the available theorems - that put together should form what we may call a non-linear
Calderón-Zygmund theory. In fact we shall review some recent and less recent
results concerning the integrability and (weak) differentiability properties of solutions to non-homogeneous equations involving operators of the type in (1.1), with
a final emphasis on the content of a couple of recent papers we wrote [105, 106].
We would like to remark that a very deep, fully non-linear Calderón-Zygmund
theory for fully non linear problems of the type
(1.2)
F (x, D2 u) = f
is available, being a fundamental contribution of Caffarelli - see [32, 31]. For obvious reasons the phenomena and the techniques involved for the case (1.2), where
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solutions are intended in the viscosity sense and equations cannot be differentiated,
are quite different from the divergence form/variational case (1.1), where a notion
of weak solution in the integral sense is adopted. For this reason we will not touch
the theory available for operators of the type (1.2), instead we refer the reader to
the excellent monograph [33]. On the other hand, while the theory for (1.2) is
essentially scalar, due to the fact that a vectorial analog of viscosity solutions has
not been developed yet, we shall see that in the quasilinear case (1.1) systems of
PDE can be nevertheless dealt with, at least up to certain extent.
A bridge between the viscosity methods and quasilinear structures has been
anyway built in [34], a paper who eventually inspired the proof of many results for
divergence form operators.
General notation. From now on, and for the rest of the paper, we will use CZ
as an acronym for Calderón-Zygmund. By Ω we denote a bounded open domain of
Rn , with n ≥ 2. We shall denote by BR (x0 ) the open ball in Rn of radius R and
center x0 , that is
BR (x0 ) = {x ∈ Rn : |x − x0 | < R} .
When the center will be unimportant we shall simply denote BR (x0 ) ≡ BR . With
BR ⊂ Rn being a a ball with positive and finite radius, if g : BR → Rk is an
integrable map, the average of g over BR is
Z
Z
1
g(x) dx .
(g)BR := − g(x) dx :=
|BR | BR
BR
When considering a function space X(Ω, Rk ) of possibly vector valued measurable
maps defined on an open set Ω ⊂ Rn , with k ∈ N, e.g.: Lp (Ω, Rk ), W β,p (Ω, Rk ), we
shall define in a canonic way the local variant Xloc (Ω, Rk ) as that space of maps
f : Ω → Rk such that f ∈ X(Ω0 , Rk ), for every Ω0 ⊂⊂ Ω. Moreover, also in the
case f is vector valued, that is k > 1, we shall also use the short hand notation
X(Ω, Rk ) ≡ X(Ω), or even X(Ω, Rk ) ≡ X when the domain is not important, and
we want to emphasize a qualitative property.
Finally, several times, in order to simplify the exposition, we shall not specify
the domain of integration when stating certain results; in such cases we shall mean
that the domain is not important, or that the result in question holds in a local
way, and then also up to the boundary provided suitable boundary conditions are
made. Other times, the domain considered will be simply the whole space Rn .
Acknowledgments. This research is supported by the ERC grant 207573 “Vectorial Problems”.
2. The linear world
The material in this section is classical, and we are just giving a short survey of
results in order to settle a background of linear results to later present in a more
efficient way the forthcoming non-linear ones. We shall prefer here a more informal
presentation, not giving all the details but rather aiming to give a general overview.
References for the next results and to more details can be for instance [53, 64, 116].
2.1. Lebesgue spaces. The basic example is at this stage the Poisson equation
(2.1)
4u = f ,
which for simplicity we shall initially consider in the whole Rn , for n ≥ 2; here f is
the datum. The previous equation, as all the other ones considered in this section
is satisfied in the distributional sense, while all solutions are supposed to be at least
of class W 1,1 .
TOWARDS A NON-LINEAR CALDERÓN-ZYGMUND THEORY
3
A classical result, going back to the fundamental work of Calderón & Zygmund
asserts the solvability in the right function spaces
(2.2)
f ∈ Lγ =⇒ D2 u ∈ Lγ
for every γ > 1 .
Of course the last result comes along together with an a priori estimate
kD2 ukLγ . kf kLγ
(2.3)
for every γ > 1 .
We remark that well-known counterexamples show that the same implication is
false for γ = 1, even locally. As a consequence of (2.2) and of Sobolev embedding
theorem we have also
(2.4)
nγ
f ∈ Lγ =⇒ Du ∈ L n−γ
for every γ ∈ (1, n) .
The classical proof of (2.2)-(2.4) goes via a representation formula involving the so
called fundamental solution, that is the Green’s function, say for n ≥ 2
Z
(2.5)
u(x) ≈ G(x, y)f (y) dy
where
(
(2.6)
|x − y|2−n
G(x, y) ≈
if
n≥3
log |x − y| if n = 2 .
Then, after differentiating twice (2.5) one arrives at a new representation formula
Z
(2.7)
D2 u(x) ≈ K(x − y)f (y) dy
where now K(·, ·) is a so called CZ kernel, that is
kK̂kL∞ ≤ B ,
(2.8)
where K̂ denotes the Fourier transform of K(·), and moreover the following Hörmander
cancelation condition holds:
Z
(2.9)
|K(x − y) − K(x)| dx ≤ B
for every y ∈ Rn .
|x|≥2|y|
A this point the standard CZ theory of singular integrals comes into the play: the
linear operator
f 7→ I0 (f )
where, in fact
Z
(2.10)
I0 (f )(x) := K(x − y)f (y) dy ,
is bounded from Lγ to Lγ , for every γ ∈ (1, ∞) and therefore (2.2) follows from
(2.7). Related a priori estimates for solutions to (2.1) follow from the a priori
bounds on I0 in the various function spaces involved. The crucial point in the
Calderón-Zygmund theory of singular integrals is that although the kernel K(·) is
singular - i.e. not integrable - in the sense that
1
,
|K(x)| .
|x|n
condition (2.9) encodes enough cancelations to ensure the convergence of I0 (f ) in
Lγ for γ > 1.
Related to the equation (2.1) is the following one:
4u = div F ,
(2.11)
and since this can be re-written as
div Du = div F ,
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GIUSEPPE MINGIONE
we expect, for homogeneity reasons - that is Du scales as F - that Du enjoys the
same integrability of F . This is the case, indeed it holds that
(2.12)
kDukLγ . kF kLγ
for every γ > 1 ,
as desired, and this is again achievable via the use of singular integrals [68, Proporition 1].
As mentioned above, (2.4) follows from (2.2) applying Sobolev embedding theorem, but there is another, more direct way to get it, without appealing to the
the theory of singular integrals, but rather relying on a lighter one: that of Riesz
potentials, also called fractional integrals. In fact, once again starting from (2.5),
but differentiating it once we gain yet another representation formula
Z
(2.13)
Du(x) ≈ K1 (x, y)f (y) dx ,
where, accordingly
|K1 (x, y)| .
(2.14)
1
.
|x − y|n−1
This motivates the introduction of so called fractional integrals.
Definition 2.1. Let β ∈ [0, n); the linear operator, acting on measurable functions
and defined by
Z
f (y)
Iβ (f )(x) :=
dy ,
|x
−
y|n−β
Rn
is called the β-Riesz potential of f .
Needless to say it is possible to define the action of Riesz potentials over measures
with finite total mass as follows as
Z
dµ(y)
Iβ (µ)(x) :=
.
|x
− y|n−β
n
R
As a matter of fact the following theorem holds:
Theorem 2.1 ([66]). Let β ∈ [0, n); for every γ > 1 such that βγ < n we have
(2.15)
kIβ (f )k
nγ
L n−βγ (Rn )
≤ c(n, β, γ)kf kLγ (Rn ) .
See also [99, Theorem 1.33] for a proof. At this point the derivation of (2.4) is
straightforward from (2.13), (2.14) and (2.15). This is not a surprise since a less
general version of Theorem 2.1 was used - and actually re-derived - by Sobolev in
order to prove his celebrated embedding theorem for the case t > 1 - the one for
the case t = 1 actually necessitates other tools, that is the celebrated GagliardoNirenberg inequalities.
Remark 2.1. There is a basic difference between fractional and singular integrals:
in the theory of fractional integrals one does not use cancelation properties of the
kernel as the one in (2.9). Indeed estimate (2.15) only uses the size of |x − y|β−n ,
and the constant c(n, β, γ) blows-up for β → 0, if using the technique leading to
Theorem 2.1. In other words the crucial fact is that for β > 0 the function
1
|x|n−β
is locally integrable in Rn .
TOWARDS A NON-LINEAR CALDERÓN-ZYGMUND THEORY
5
2.2. The borderline case γ = 1. A natural question raises now: What happens
in the borderline case γ = 1? The results presented up to now fail, but an answer
can be nevertheless obtained by considering suitable functions spaces. Let me
start by the so called Marcinkiewic spaces Mt (A, Rk ), also called Lorentz spaces
and denoted by Lt,∞ (A), or by Ltw (A), when they are called “weak-Lt ” spaces, or
Lorentz spaces - see Definition 5.4 below.
Definition 2.2. Let t ≥ 1. A measurable map w : Ω → Rk belongs to Mt (Ω, Rk ) ≡
Mt (Ω) iff
(2.16)
sup λt |{x ∈ A : |w| > λ}| =: kwktMt (Ω) < ∞ .
λ≥0
It turns out that linear CZ integral operators send L1 into M1 ; therefore, in the
borderline case γ = 1 estimates (2.3) and (2.12) turn to
(2.17)
kD2 ukM1 . kf kL1 ,
and
(2.18)
kDukM1 . kF kL1 ,
respectively, while (2.15) turns to
(2.19)
kIβ (f )k
n
M n−β
≤ c(n, β)kf kL1 ,
so that for the Poisson equation (2.1) it holds that
(2.20)
n
f ∈ L1 =⇒ Du ∈ M n−1 .
For the limiting embedding property of Riesz potential see the classical paper of
Adams [6]; this paper contains results we shall examine in greater detail later.
Remark 2.2. There is a problem in (2.17)-(2.18); these have to be thought as a
priori estimates, since the derivatives involved there are not the distributional ones.
To avoid complications, we shall consider (2.17)-(2.18) when f , F are smooth, and
therefore u is also smooth, and we shall retain (2.17)-(2.18) only in such a qualitative
form.
The importance of the space Mt clearly lies in the fact that it serves to describe in
a sharp way certain limiting integrability situations, very often occurring in modern
non-linear analysis, as those given by potential functions. In fact, the prototype of
Mt functions is given by the potential |x|−n/t ; note that
1
t
t
(2.21)
for every t ≥ 1 .
n ∈ M (B1 ) \ L (B1 ) ,
|x| t
In general the following inclusions hold:
(2.22)
Lt $ Mt & Lt−ε
for every ε > 0 .
As for the first one, observe that
Z
|{|w| > λ}| =
dx
{|w|>λ}
Z
≤
{|w|>λ}
(2.23)
≤
|w|t
dx
λt
kwkLt
λt
so that
kwkMt ≤ kwkLt
holds, and in fact the estimation in (2.23) motivates the definition of Marcinkiewicz
spaces. Marcinkiewicz spaces are nowadays of crucial importance in the analysis of
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GIUSEPPE MINGIONE
problems with critical non-linearities as those involving harmonic, p-harmonic, and
bi-harmonic maps, Euler equations and other pdes from fluid-dynamics.
Inclusions (2.22) tell us that Marcinkiewicz spaces interpolate Lebesgue spaces;
we shall see later a more refined way of interpolating Lebesgue spaces, when Lorentz
spaces will be introduced, extending both Marcinkiewicz and Lebesgue spaces,
and introducing finer scales for measuring the size of a function. For more on
Marcinkiewicz spaces we refer to Section 5.3 below.
The second natural question is now to find a condition for which in estimates
(2.17)-(2.18) we can obtain the full Lebesgue integrability scale instead of the
Marcinkiewicz one. The question is the same that finding a function space “slightly
smaller” than L1 , which is mapped into L1 by singular integral operators. For the
developments we are concerned with the answer is actually doublefold.
In order to replace M1 by L1 in the left hand side of (2.17) one has to consider
a slightly larger space for the right hand datum f , called Hardy space and denoted
by H1 . The elements of such space are functions enjoying enough cancelation properties to match with those by singular integrals and finally yielding convergence
in L1 . The story of the Hardy spaces is essentially a complex function theory one
until the fifties, when a completely real function theory characterization of such
spaces was settle down mainly by authors like Stein and Weiss; see the classical
paper of Fefferman & Stein [58]. A central concept is the one of atomic decomposition, initially due to Coifman [38], see also [92], allowing to give a particularly
simple definition which we adopt here. We recall that an atom a over a cube Q is
a function such that supp a(·) ⊆ Q and moreover
Z
1
,
a(x) dx = 0
kakL∞ ≤
|Q|
Q
hold. The atom a(·) can actually be thought as a bump function exhibiting cancelations. Then a function f belongs to the Hardy H1 (Rn ) iff there exists a sequence
of atoms {ak } such that the following atomic decomposition holds:
X
X
(2.24)
w(x) =
λk ak (x) and
|λk | < ∞ .
k∈N
k∈N
P
The inf of the sums
|λk | over all possible such decompositions naturally defines
the Hardy space norm of w. We won’t any longer deal with Hardy spaces here;
this in another story, too much unrelated to the non-linear setting we are going
to switch to in the next sections. As a matter of fact the atomic decompositions
of Hardy functions allows a perfect match with the cancelations properties of the
CZ kernel. In fact, using both the cancelation properties of CZ kernels and the
zero-average property of a over the cube Q, it is easy to see that kI0 (a)kL1 ≤ c,
where the constant c ultimately depends on the constant B occurring in (2.8)-(2.9),
but is actually independent of the atom considered a. Using this fact, and the very
definition of Hardy spaces, that is the decomposition in (2.24), the boundedness in
H1 follows:
kI0 (f )kL1 . kf kH1 .
Such a proof cannot be apparently extended to the case of non-linear operators,
in that such a kind of delicate cancelation phenomena are apparently destroyed
by general non-linearities, in other words, non-linear operators seems not to read
cancelation properties of the Hardy functions.
It remains to try with additional size conditions. The space L log L(Ω), with
Ω ⊆ Rn being a bounded domain, is therefore defined as those of the functions f
satisfying
Z
|w| log(e + |w|) dx < ∞ .
Ω
TOWARDS A NON-LINEAR CALDERÓN-ZYGMUND THEORY
7
Such space, a particularly important instance of what are called Orlicz spaces,
becomes a Banach space when equipped with the following Luxemburg norm:
Z w w
(2.25)
kwkL log L(Ω) := inf λ > 0 : − log e + dx ≤ 1 < ∞ .
λ
Ω λ
Note that for homogeneity reasons that will be clear later we have incorporated in
the above definition a dependence on the measure Ω, by considering an averaged
integral in (2.25). The following equivalence due to T. Iwaniec [73]:
Z
w
R
(2.26)
kwkL log L(Ω) ≈ − |w| log e +
dx =: |w|L log L(Ω) ,
−A |w(y)| dy
Ω
and the striking fact is that the last quantity actually defines a true norm in
L log L(Ω), which is therefore equivalent to the usual Luxemburg one (2.25) via
a constant independent of the domain Ω. An obvious consequence of the definitions above is the following inclusion:
Z
L log L(Ω) $ L1 (Ω) and − |w| dx ≤ kwkL log L(Ω) .
Ω
As a matter of fact the space L log L is sent into L1 by singular integrals operators
kI0 (f )kL1 . kf kL log L .
as a consequence we have limiting L1 -estimates in (2.3) and (2.12): these turn to
(2.27)
kD2 ukL1 . kf kL log L ,
and
(2.28)
kDukL1 . kF kL log L ,
respectively, while (2.15) turns to
(2.29)
kIβ (f )k
n
L n−β
≤ ckf kL log L ,
so that for the Poisson equation (2.1) it holds that
(2.30)
n
f ∈ L log L =⇒ Du ∈ L n−1 .
For the last two results we again refer to Adams’ paper [6].
2.3. Perturbations of the linear theory. The results on the differentiability
properties explained in Section 2.1 extend to linear elliptic equations of the type
(2.31)
AD2 u = f
where A is a constant coefficients matrix such that
ν|λ|2 ≤ hAλ, λi ≤ L|λ|2
holds for any λ ∈ Rn . In fact, up to changing coordinates, the equation (2.31)
behaves as the usual Poisson equation. The same results hold for equations with
2
variable, continuous coefficients, that is A ≡ A(x) ∈ C 0 (Rn , Rn ); in fact in this
case, due to the continuity of the coefficients, the equation can be considered as
a local perturbation of the Laplace operator, and CZ estimates follow by mean of
local perturbation and fixed point arguments; see for instance [62, Paragraph 10.4].
It is now obvious that the same results cannot hold when the coefficient matrix
A(x) is just measurable, due to well-known counterexamples. Anyway certain types
of mild discontinuities for the matrix A(·) can still be allowed. This is the case when
the entries of A(x) are assumed to be VMO functions - see Section 3 below. These
are functions which are basically continuous up to an averaging process.
The first results in this direction have been obtained by Chiarenza & Frasca &
Longo [39] by using a few deep theorems from Harmonic Analysis on the boundedness of commutator operators involving BMO coefficients. This strategy, applied
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GIUSEPPE MINGIONE
to a variety of problems with different boundary conditions, heavily relies on the
linearity of the problem considered, in that it is based on the use of representation
formulas via fundamental solutions. This approach, relying on a series of sophisticated tools, can be actually completely bypassed, via the use of suitable maximal
function operators, as we shall see more in detail in Section 3.3 below; moreover
various type of boundary value problems with very weak regularity assumptions on
the boundary are treatable, see for instance [28] and related references.
2.4. “Extremals” of the linear theory. In the last years the problem of determining a CZ for linear problem not involving treatable kernels - for instance kernels
which are not regular enough to satisfy Hörmander condition (2.9) - has been very
often dealt with. This are problems in which the linearity of the equations considered still allows to apply - in suitably tailored sophisticated forms - abstract
methods from operator theory, interpolation theory, and Harmonic Analysis. Very
often, for instance, the linearity of the equations involved suffices to associate to
them a suitable sub-linear operator, so that interpolation methods and techniques
are applicable. For such developments we refer the reader to the interesting papers
of Auscher & Martell [13, 14, 15], and their related references.
Plan for the next sections. At the end of this introductory part we would
like to conclude with a few remarks, and a plan for the next sections. The very
basic survey of results in this section settles the mood for the rest of the paper. The
purpose here is to describe some extension of the integrability properties in (2.2)
and (2.4) to solutions to non-linear elliptic problems. We would like to emphasize
the doublefold character of the issue: basic integrability of the gradient in (2.4),
and second order differentiability in (2.2), which is the maximal regularity result
since equation (2.1) is a second order one. Of course, in general (2.2) implies (2.4)
via Sobolev embedding theorem, but there are anyway cases in which (2.4) holds,
but (2.2) does not, unless using additional assumptions. The case of equations
with low regularity coefficients is an instance. Section 3 will be dedicated to results
concerning the integrability of the gradient for large exponents, and we shall deal
with classical weak or energy solutions. Sections 4 and 5 will be still dedicated
to integrability issues, this time for small exponents. We shale therefore deal with
so-called very weak solutions, i.e. solutions not belonging to the natural energy
space. In particular, in Section 4 we shall give a rapid introduction to measure
data problems, including basic regularity results. In Section 5 we shall present
more delicate integrability results in different scales of spaces as Morrey spaces,
and in finer scales as Lorentz spaces. Finally in Section 6 we shall concentrate on
the higher differentiability of solutions.
3. Iwaniec opens the non-linear world
3.1. The notion of solution. The general setting we are going to examine concerns non-linear equations and systems which in the most general form look like
(3.1)
−div a(x, u, Du) = H
in Ω ,
where a : Ω×RN ×RN n → RN n is a measurable vector field, with n ≥ 2 and N ≥ 1,
which is continuous in the last two arguments, and initially satisfies the following
p-growth assumption:
(3.2)
|a(x, u, z)| ≤ L(1 + |z|2 )
p−1
2
for p > 1 .
When N = 1 (3.1) reduces to an equation and we are in the scalar case. On the
right hand side of (3.1) we initially assume that
H ∈ D0 (Ω, RN ) .
TOWARDS A NON-LINEAR CALDERÓN-ZYGMUND THEORY
9
The notion of distributional solution prescribes that a map u ∈ W 1,1 (Ω, RN ) is a
weak solution to (3.1) iff u is such that a(x, u, Du) ∈ L1 (Ω, RN ) and satisfies
Z
(3.3)
a(x, u, Du)Dϕ dx = hH, ϕi
for every ϕ ∈ Cc∞ (Ω, RN ) .
Ω
As it will be clear later, this definition is too general. Therefore we recall the
following:
Definition 3.1. An energy solution to (3.1), under the assumption (3.2), is a
distributional solution in the sense of (3.3) enjoining the additional property
u ∈ W01,p (Ω, RN ) .
In turn, when dealing with energy solutions, the natural conditions to impose
0
on H is of course H ∈ W −1,p (Ω); this allow to test (3.3) with W01,p (Ω) functions
- this follows by simple density arguments - and in particular with multiples of
0
the solution itself. In turn, when having H ∈ W −1,p (Ω) and a suitable monotonicity assumption on a(·), the existence of an energy solution follows by classical
monotonicity methods. For this we refer to the classical [98].
In the rest of this section, unless otherwise stated, we shall deal with the notion of
energy solution; non-energy solutions will appear later, when dealing with measure
data problems.
3.2. A starting point. The starting point here is the following natural p-Laplacean
analog of equation (2.11):
div (|Du|p−2 Du) = div (|F |p−2 F )
(3.4)
for p > 1 ,
which indeed reduces to (2.11) for p = 2. Note that the right hand side of (3.4)
is written in the peculiar form div (|F |p−2 F ) in order to facilitate a more elegant
presentation of the results, and also because such form naturally arises in the study
of certain projections problems motivated by multi-dimensional quasi-conformal
geometry [68]. Anyway, one could immediately consider a right hand side of the
type div G by on obvious change of the vector field
G ≡ |F |p−1
1
F
G
⇐⇒ F ≡ |G| p−1
.
|F |
|G|
The following fundamental result in essentially due to Tadeusz Iwaniec, who, in the
paper [68] established the foundations of the non-linear CZ theory.
Theorem 3.1 ([68]). Let u ∈ W 1,p (Rn ) be a weak solution to the equation (3.4)
in Rn . Then
F ∈ Lγ (Rn , Rn ) =⇒ Du ∈ Lγ (Rn , Rn )
for every γ ≥ p .
The local version of this result is
Theorem 3.2. Let u ∈ W 1,p (Ω) be a weak solution to the equation (3.4) in Ω,
where Ω is a bounded domain in Rn . Then
(3.5)
F ∈ Lγloc (Ω, Rn ) =⇒ Du ∈ Lγloc (Ω, Rn )
for every γ ≥ p .
Moreover, there exists a constant c ≡ c(n, p, γ) such that for every ball BR b Ω it
holds that
! γ1
Z
p1
Z
γ1
Z
p
γ
γ
≤ c − |Du| dx
+ c − |F | dx
.
(3.6)
−
|Du| dx
BR/2
BR
BR
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A proof can be adapted from [3], for instance. From now on, for ease of presentation we shall confine ourselves to treat local regularity results.
The non-trivial extension to the case when (3.4) is a system has been obtained
by DiBenedetto & Manfredi, who caught a borderline case too.
Theorem 3.3 ([49]). Let u ∈ W 1,p (Ω, RN ) be a weak solution to the system (3.4),
where Ω is a bounded domain in Rn , and N ≥ 1. Then (3.5) holds. Moreover
(3.7)
F ∈ BMOloc (Ω, RN n ) =⇒ Du ∈ BMOloc (Ω, RN n ) .
3.3. BMO and VMO functions. There appears a new function space in (3.7),
the space of functions with bounded mean oscillations, introduced by John & Nirenberg [74]. In order to introduce BMO functions let us introduce the quantity
Z
(3.8)
[w]R0 ≡ [w]R0 ,Ω :=
sup
− |w(x) − (w)BR | dx .
BR ⊂Ω,R≤R0 BR
Then a measurable map w belongs to BMO(Ω) iff [w]R0 < ∞, for every R0 < ∞.
It turns out that BM O ⊂ Lγ for every γ < ∞, while a deep and celebrated result
of John & Nirenberg tells that every BMO function actually belongs to a suitable
weak Orlicz space generated by an N-function with exponential growth [74], and
depending on the BMO norm of w. Specifically, we have
c2 λ
(3.9)
|{x ∈ QR : |w(x) − (w)QR | > λ}| ≤ c1 (n) exp −
[w]2R,QR
where QR is a cube whose sidelength equals R, and c1 , c2 are absolute constants.
Anyway BMO functions can be unbounded, as shown by log(1/|x|). For the proof
of (3.9) a good reference is for instance [53, Theorem 6.11].
Related to BMO functions are functions with vanishing means oscillations. These
have been originally defined by Sarason [110] as those BMO functions w, such that
lim [w]R,Ω = 0 .
R→0
In this way one prescribes a way to allow only mild discontinuities, since the oscillations of w are measured in an integral, averaged way.
As outlined in Section 2.3, the linear CZ theory can be extended to those problems/operators involving VMO coefficients. This happens also in the non-linear
case, as proved by Kinnunen & Zhou who considered a class of degenerate equations whose model is given by
(3.10)
div (c(x)|Du|p−2 Du) = div (|F |p−2 F )
for p > 1 ,
where the coefficient c(·) is a VMO function satisfying
(3.11)
c(·) ∈ VMO(Ω)
and
0 < ν ≤ c(x) ≤ L < ∞ .
The outcome is now
Theorem 3.4 ([84]). Let u ∈ W 1,p (Ω) be a weak solution to the equation (3.10) in
Ω, where Ω is a bounded domain in Rn , and the function c(·) satisfies (3.11). Then
assertions (3.5)-(3.6) hold for u, and the constant in estimate (3.6) also depends
on the coefficient function c(·).
Before going on, we will comment on the strategy adopted in order to prove
Theorems 3.1-3.4; in turn, up to non-trivial complications due to the increasing
level of generality, this goes back to the original Iwaniec’s paper [70]. The idea of
Iwaniec is very natural; in the linear case the estimates are obtained using exactly
two ingredients: global representation formulas as in (2.5), and then the use of CZ
TOWARDS A NON-LINEAR CALDERÓN-ZYGMUND THEORY
11
theory of singular integrals (2.10). Then Iwaniec replaces the first ingredient using
a local comparison argument with solutions to the homogeneous p-Laplace equation
div (|Dv|p−2 Dv) = 0 .
(3.12)
The pointwise regularity estimates of Uhlenbeck-Ural’tseva [121, 122] are then applied in order to provide an analog of the local representation formula in the linear
case. Then finally Iwaniec shows that it is possible to pointwise estimate the sharp
maximal function of Du with the maximal function of the datum F , and the conclusion follows by applying the well-known maximal theorems of Hardy-Littlewood,
and Fefferman-Stein, which, at this stage, play the role of the boundedness of singular integrals.
3.4. More general operators. We will now turn to more general equations of
the type
div a(x, Du) = div (|F |p−2 F )
(3.13)
for p > 1 ,
where a : Ω × Rn → Rn is a continuous vector field satisfying the following strong
p-monotonicity and growth assumptions:

2
2 p−1
2
2 1

 |a(x, z)| + (s + |z| ) 2 |Da(x, z)| ≤ L(s + |z| ) 2


p−2
(3.14)
ν(s2 + |z|2 ) 2 |λ|2 ≤ hDa(x, z)λ, λi



p−1

|a(x, z) − a(x0 , z)| ≤ Lω (|x − x0 |) (s2 + |z|2 ) 2 ,
whenever x, x0 ∈ Ω, z, λ ∈ Rn . Here we take p > 1, s ∈ [0, 1], while ω(·) is a
modulus of continuity, that is a non-decreasing function defined on [0, ∞] such that
(3.15)
lim ω(t) = 0 .
t→0+
We have seen from the previous section that the regularity of solutions to homogeneous related homogeneous equations
(3.16)
div a(x, Dv) = 0 ,
is an important ingredient in the proof of the gradient estimates, in that the regularity estimates for solutions to (3.16) are then used in a comparison scheme to get
proper size estimates for the gradient of solutions to (3.13). Therefore in order to
state a theorem of the type 3.2 one has to consider operators such that solutions
v to (3.16) enjoy the maximal regularity, which in our case is Dv ∈ Lγ for every
γ < ∞. On the other hand, an obvious a posteriori arguments is that if an analog
of Theorem 3.2 would hold for equation (3.13), then applying it with the choice
F ≡ 0 would in fact yield Dv ∈ Lγ for every γ < ∞.
This is the case for solutions to (3.16) under assumptions (3.14). Therefore it
holds the following:
Theorem 3.5. Let u ∈ W 1,p (Ω) be a weak solution to the equation (3.13), where
Ω is a bounded domain in Rn , and such that assumptions (3.14) are satisfied. Then
(3.5) holds for u and moreover
! γ1
Z
1
Z
1
Z
(3.17)
−
|Du|γ dx
BR/2
≤ c − (|Du|p + sp ) dx
BR
p
+c −
|F |γ dx
γ
,
BR
where c ≡ c(n, p, ν, L, γ).
See for instance [3], from which a proof of the previous result can be adapted.
There is a number of possible variants of the previous result; we shall outline a
couple of them.
12
GIUSEPPE MINGIONE
The first deals with non-linear operators with VMO coefficients, more precisely
with equations of the type
(3.18)
div [c(x)a(Du)] = div (|F |p−2 F )
for p > 1 ,
where the vector field a : Rn → Rn satisfies (3.14) - obviously recast for the case
where there is no x-dependence, while the coefficient function c(·) satisfies (3.11).
We have
Theorem 3.6. Let u ∈ W 1,p (Ω) be a weak solution to the equation (3.18) in Ω,
where Ω is a bounded domain in Rn , and such that assumptions (3.11) and (3.14)
are satisfied. Then the assertion in (3.5) and (3.17) hold for u, and the constant c
appearing in (3.17) depends also on the coefficient c(·).
For a proof one could for instance adapt the arguments from [3, 107]. In particular, in the last paper a version of Theorem 3.6 in the so called Heisenberg group
has been obtained, while in the first one CZ estimates have been obtained for a
class of non-uniformly elliptic operators. The previous result can be also extended
to the boundary when considering the Dirichlet problem - as (3.23) below, under
very mild assumptions on the regularity of the boundary ∂Ω; we will not deal very
much with boundary regularity, and for such issues we for instance refer to [28] and
related references.
The second generalization goes in another direction; we have seen that the possibility of getting a priori regularity estimates for homogeneous equations as (3.16)
is crucial for proving related CZ estimates. As matter of fact the assumptions
guaranteeing that solutions to
(3.19)
div a(Dv) = 0 ,
are Lipschitz can be considerably relaxed with respect to those in (3.14). More
precisely we may consider a : Rn → Rn to be a continuous vector field satisfying

 ν(s2 + |z1 |2 + |z2 |2 ) p−2
2 |z − z |2 ≤ ha(z ) − a(z ), z − z i
2
1
2
1
2
1
(3.20)
p−1

|a(z)| ≤ L(s2 + |z|2 ) 2 ,
whenever z1 , z2 ∈ Rn , where p > 1 and s ∈ [0, 1]. Note that here the vector field
a(·) is not even assumed to be differentiable. We then have
Theorem 3.7. Let u ∈ W 1,p (Ω) be a weak solution to the equation
div a(Du) = div (|F |p−2 F ) ,
where Ω is a bounded domain in Rn , and such that assumptions (3.20) are satisfied.
Then (3.5) and (3.17) hold for u.
The proof can be obtained using the method in [3], but using the a priori regularity estimates for solutions to (3.19) developed for instance [85, 57] - note that
the proofs there extend to the general non-variational case of general operators
in divergence form. Finally we remark that in Theorem 3.7 we can allow VMO
coefficients as in (3.18).
3.5. The case of systems. Theorem 3.3 tells us that CZ estimates extend to the
case of systems when considering the specific p-Laplacean system. We now wonder
up which extend the results of the previous section extend to general systems. The
reason for Theorem 3.3 to hold is that, as first shown by Uhlenbeck, solutions to the
homogeneous p-Laplacean system (3.12) are actually of class C 1,α for some α > 0.
This makes the local comparison argument work, finally leading to Theorem 3.3. In
fact a major effort in [49] is to show suitable form of a priori estimates for solutions
to (3.12). We also recall that, as pointed out in the previous section, the regularity
TOWARDS A NON-LINEAR CALDERÓN-ZYGMUND THEORY
13
of solutions to associated homogeneous problems is crucial to obtain the desired
CZ estimates.
In the case of general systems as (3.13), and satisfying (3.14), we cannot expect
a theorem like 3.3 to hold, and for a very simple reason. It is known that solutions
to general homogeneous systems as
(3.21)
div a(Dv) = 0 ,
are not everywhere regular; they are C 1,α -regular only when considered outside a
closed negligible subset of Ω, in fact called the singular set of the solution. Moreover,
even for p = 2, and in the case of a smooth vector field a(·), Šverák & Yan [113]
have shown that solutions to (3.21) may even be unbounded in the interior of Ω;
for such issues see for instance the recent survey paper [103]. This rules out the
validity of Theorem 3.3 for general systems in that, should it hold, when applied to
the case (3.21) it would imply the everywhere Hölder continuity of v in Ω, clearly
contradicting the existence of unbounded solutions proved in [113].
On the other hand an intermediate version of Theorem 3.3 which is valid for
general systems holds in the following form:
Theorem 3.8 ([86]). Let u ∈ W 1,p (Ω, RN ) be a weak solution to the system
div a(x, Du) = div (|F |p−2 F ) ,
for N ≥ 1, where Ω is a bounded domain in Rn and the continuous vector field
a : Ω × RN n → RN n satisfies (3.14) when suitably recast for the vectorial case.
Then there exists δ ≡ δ(n, N, p, L/ν) > 0 such that
F ∈ Lγloc (Ω, RN n ) =⇒ Du ∈ Lγloc (Ω, RN n ) ,
whenever
2p
+δ
when
n>2,
n−2
while no upper bound is prescribed on γ in the two-dimensional case n = 2. Moreover, the local estimate (3.17) holds.
(3.22)
p≤γ <p+
Note that the previous theorem does not contradict the counterexample in [113],
since this does not apply when n = 2. The previous result comes along with a
global one. For this we shall consider the Dirichlet problem
div a(x, Du) = 0 in Ω
(3.23)
u=v
on ∂Ω
for some boundary datum v ∈ W 1,p (Ω, RN ); here we assume for simplicity that
∂Ω ∈ C 1,α , but such an assumption can be relaxed. The main result for (3.23) is
Theorem 3.9 ([86]). Let u ∈ W 1,p (Ω, RN ) be the solution to the Dirichlet problem
(3.23) for N ≥ 1, where Ω is a bounded domain in Rn and the continuous vector
field a : Ω × RN n → RN n satisfies (3.14) when suitably recast for the vectorial case.
Then there exists δ ≡ δ(n, N, p, L/ν) > 0 such that
Z
Z
|Du|γ dx ≤ c (|Dv|γ + sγ ) dx ,
Ω
Ω
holds whenever (3.22) is satisfied for n > 2, while no upper bound is imposed on γ in
the two-dimensional case n = 2; the constant c depends only on n, N, p, ν, L, γ, ∂Ω.
The previous theorem reveals to be crucial when deriving certain improved
bounds for the Hausdorff dimension of the singular set of minima of integral functions - see [86] - and when proving the existence of regular boundary points for solutions to Dirichlet problems involving non-linear elliptic systems - see [54]. Moreover
the peculiar upper bound on γ appearing in (3.22) perfectly fits with the parameters
14
GIUSEPPE MINGIONE
values in order to allow the convergence of certain technical iterations occurring in
[86, 54].
The proof of Theorems 3.8-3.9 is based on an argument different from those in
[70], but rather relying on some more recent methods used by Caffarelli & Peral [34]
in order to prove higher integrability of solutions to some homogenization problems.
Although quite different from the previous ones, such method still relies on the use
of maximal operators.
3.6. Parabolic problems. The extension to the parabolic case of the results of
the previous sections is quite non-trivial, and in fact the validity of Theorem 3.2
for the parabolic p-Laplacean system
(3.24)
ut − div (|Du|p−2 Du) = div (|F |p−2 F )
remained an open problem for a while in the case p 6= 2, even in the case of one
scalar equation N = 1; it was settled only recently in [4]. All the parabolic problems
in this section, starting by (3.24), will be considered in the cylindrical domain
(3.25)
ΩT := Ω × (0, T ) ,
where, as usual, Ω is a bounded domain in Rn , and T > 0.
Let us now explain where are the additional difficulties coming from. As we
repeatedly pointed out in the previous sections, the proof of the higher integrability
results strongly relies on the use of maximal operators. This approach is completely
rules out in the case of (3.24). This is deeply linked to the fact that the homogeneous
system
(3.26)
ut − div (|Du|p−2 Du) = 0
locally follows an intrinsic geometry dictated by the solution itself. This is
essentially DiBenedetto’s approach to the regularity of parabolic problems [46] we
are going to briefly streamline - see also Remark 3.1. The right cylinders on which
the problem (3.26) enjoys good a priori estimates when p ≥ 2 are of the type
(3.27)
Qz0 (λ2−p R2 , R) ≡ BR (x0 ) × (t0 − λ2−p R2 , t0 + λ2−p R2 ) ,
where z0 ≡ (x0 , t0 ) ∈ Rn+1 and the main point is that λ must be such that
Z
(3.28)
|Du|p ≈ λp .
Qz0 (λ2−p R2 ,R)
The last line says that Qz0 (λ2−p R2 , R) is defined in an intrinsic way. It is actually
the main core of DiBenedetto’s ideas to show that such cylinders can be constructed
and used. Now the point is very simple: since the cylinders in (3.27) depend on the
size of the solution itself, then it is not possible to associate to them, and therefore
to the problem (3.26), a universal family of cylinders - that is independent of the
solution considered. In turn this rules out the possibility of using parabolic type
maximal operators.
In the paper [4] we overcame this point by introducing a completely new technique bypassing the use of maximal operators, and giving the first Harmonic Analysis free, purely pde proof, of non-linear CZ estimates. The result is split in the
case p ≥ 2 and p < 2.
Theorem 3.10 ([4]). Let u ∈ C(0, T, L2 (Ω, RN ))∩Lp (0, T, W 1,p (Ω, RN )) be a weak
solution to the parabolic system (3.24), where Ω is a bounded domain in Rn , and
p ≥ 2. Then
(3.29)
F ∈ Lγloc (ΩT , RN n ) =⇒ Du ∈ Lγloc (ΩT , RN n )
for every γ ≥ p .
TOWARDS A NON-LINEAR CALDERÓN-ZYGMUND THEORY
15
Moreover, there exists a constant c ≡ c(n, N, p, ν, L, γ) such that for every parabolic
cylinder QR ≡ BR (x0 ) × (t0 − R2 , t0 + R2 ) b ΩT it holds that
! γ1
Z
−
|Du|γ dx dt
QR/2
(3.30)
"Z
p1 Z
p
+ −
≤ c − (|Du| + 1) dx dt
γ1 # p2
.
|F |γ dx dt
QR
QR
We note the peculiar form of the a priori estimate (3.30), which fails to be a
reverse Hölder type inequality as (3.17) due to the presence of the exponent p/2,
which is the the scaling deficit of the system (3.24). The presence of such exponent is
natural, and can be explained as follows: in fact, let us consider the case F ≡ 0, that
is (3.26). We note that if u is a solution, then, with c ∈ R being a fixed constant,
the function cu fails to be a solution of a similar system, unless p = 2. Therefore, we
cannot expect homogeneous a priori estimate of the type (3.17) to hold for solutions
to (3.24), unless p = 2, when (3.30) becomes in fact homogeneous. Instead, the
appearance of the scaling deficit exponent p/2 in (3.30) precisely reflects the lack
of homogeneity. Another sign of the lack of scaling is the presence of the additive
constant in the second integral, this is a purely parabolic fact, linked to the presence
of a diffusive term - that is ut - in the system.
Remark 3.1 (Intrinsic geometry and self-rebalancing). We will explain here some
basic principles of DiBenedetto’s intrinsic geometry [46], confining ourselves to the
case p > 2; the case p < 2 can be treated by similar means. The reason for considering cylinders as in (3.27) appears natural if we use the following heuristic argument:
the relation (3.28) roughly tells us that |Du| ≈ λ in the cylinder Qz0 (λ2−p R2 , R).
Therefore in the same cylinder we may think to system (3.26) as actually
ut − div (λp−2 Du) = 0 .
(3.31)
Now, switching from the intrinsic cylinder Qz0 (λ2−p R2 , R) to Q0 (1, 1), that is making the change of variables
v(x, t) := u(x0 + Rx, t0 + λ2−p R2 t)
(x, t) ∈ B1 × (−1, 1) ≡ Q1 ,
we note that (3.31) gives that
vt − 4v = 0 ,
(3.32)
holds in the cylinder Q1 . Therefore this argument tells us that on an intrinsic
cylinder of the type in (3.27) the solution u approximately behaves as a solution to
the standard heat system, and therefore enjoys good estimates.
Note that for p = 2 the cylinders considered in (3.27) are actually the standard
parabolic ones - that is those equivalent to the balls generated by the parabolic
metric in Rn+1
1
dpar ((x, t), (y, s)) := |x − y| + |t − s| 2 ,
x, y ∈ Rn , s, t ∈ R .
These are in turn independent of the solution, and therefore if ones wants to derive
CZ estimates for solutions to (3.24) in the case p = 2, then the standard elliptic
proof works, provided using the parabolic maximal operator, that is the one defined
by considering as defining family the one parabolic cylinders
Z
[Mpar f ](x, t) := sup
|f (y, s)| dy ds , Qr ≡ Br (x0 ) × (t0 − r2 , t0 + r2 ) .
(x,t)∈Qr
Qr
Instead, in the case p > 2 one is lead to consider intrinsic cylinders as in (3.27)
which depend on the solution themselves, and therefore do not define a universal
16
GIUSEPPE MINGIONE
family. In a sense, we are considering the locally deformed parabolic metric given
by
dpar,λ ((x, t), (y, s)) := max |x − y| + λ
(3.33)
p−2
2
1
|t − s| 2 ,
where, again the number λ depends on the solution via (3.28).
Remark 3.2 (Interpolation nature of Theorem 3.10). A closer look at the proofs
in [4, 56] reveals a more explicit structure of estimate (3.30), which actually looks
like
! γ1
Z
−
|Du|γ dx dt
QR/2
Z
p1
+ c2 (γ) −
− (|Du| + 1) dx dt
"Z
≤ c1
p
γ1 # p2
,
|F | dx dt
γ
QR
QR
where the constant c1 depends on n, N, p, ν, L, but is independent of q. Therefore,
considering the case F ≡ 0 and eventually letting γ → ∞ the previous estimate
yields
(3.34)
Z
12
p
sup |Du| ≤ c1 − (|Du| + 1) dx dt
,
QR/2
QR
which is the original L∞ -gradient estimate obtained by DiBenedetto-Friedmann [47,
48] for solutions to the homogeneous p-Laplacean system (3.12). This phenomenon
reflects the interpolation nature of Theorem 3.10, which in some sense provides
an estimate which interpolates the trivial Lp estimate - that is (3.30) with γ = p,
after absorbing the intermediate integral via standard methods - which comes from
testing the system with the solution, and the L∞ one (3.34). A similar remark
applies to the case p < 2 treated a few lines below.
We turn now to the case p < 2. This is the so called singular case since when
|Du| approaches zero, the quantity |Du|p−2 , which roughly speaking represents
the lowest eigenvalue of the operator div (|Du|p−2 Du), tends to infinity. Anyway,
this interpretation is somewhat misleadins here: we are interested in determining
the integrability rate of Du, therefore we are interested in the large values of the
gradient. Here a new phenomenon appears: we cannot consider values of p which
are arbitrarily close to 1, as described in [46]. The right condition turns out to be
(3.35)
p>
2n
,
n+2
otherwise, as shown by counterexamples, solutions to (3.26) maybe even unbounded.
This can be explained by looking at (3.26) when |Du| is very large: if p < 2, and it
is far from 2, then the regularizing effect of the elliptic part - the diffusion - is too
weak as |Du|p−2 is very small, and the evolutionary part develops singularities like
in odes, where no diffusion is involved.
For the case p < 2 the result is now
Theorem 3.11 ([4]). Let u ∈ C(0, T, L2 (Ω, RN ))∩Lp (0, T, W 1,p (Ω, RN )) be a weak
solution to the parabolic system (3.24), where Ω is a bounded domain in Rn , and
p < 2 satisfies (3.35). Then
F ∈ Lγloc (ΩT , RN n ) =⇒ Du ∈ Lγloc (ΩT , RN n )
for every γ ≥ p .
TOWARDS A NON-LINEAR CALDERÓN-ZYGMUND THEORY
17
Moreover, there exists a constant c ≡ c(n, N, p, ν, L, q) such that for every parabolic
cylinder QR ≡ BR (x0 ) × (t0 − R2 , t0 + R2 ) b ΩT it holds that
! γ1
Z
(3.36)
−
|Du|γ dx dt
QR/2
"Z
p1 Z
p
≤ c − (|Du| + 1) dx dt
+ −
QR
2p
γ1 # p(n+2)−2n
.
|F |γ dx dt
QR
Note how in the previous estimate the scaling deficit exponent p/2 in (3.30) is
replaced by 2p/(p(n + 2) − 2n), a quantity that stays finite as long as (3.35) is satisfied. Therefore estimate (3.36) exhibits in quantitative way the role of assumption
(3.35).
Theorems 3.10-3.11 admit of course several possible generalizations; a first one
concerns general parabolic equations of the type
ut − div a(Du) = div (|F |p−2 F ) ,
where the vector field a(·) satisfies (3.14). In this case Theorems 3.10-3.11 hold in
the form described above, for a constant c depending also on ν, L.
We shall now outline a couple of non-trivial extensions, recently obtained in [56].
The first concerns the evolutionary p-Laplacean system with coefficients
(3.37)
ut − div (c(x)b(t)|Du|p−2 Du) = div |F |p−2 F .
The point here is that while the function depending on the space variable c(x)
is assumed to be VMO regular, that is to satisfy assumptions (3.11), this time
dependent measurable function b(t) is assumed to satisfy only
0 < ν ≤ b(t) ≤ L < ∞ ,
while no pointwise regularity is assumed other than the obvious measurability.
Under such assumptions for solutions to (3.37) Theorem 3.10 holds exactly in the
form presented above, but for the fact that the constant c also depends on the
function c(·). This result, and the related one obtained in [56], is a far reaching
extension of recent analogous results due to Krylov [89] and his students, who
consider a similar situation in the case of linear parabolic equations (when, in
particular, p = 2, and no intrinsic geometry needs to be considered).
The second result from [56] we are presenting is the parabolic analog of Theorem
3.8, we are therefore treating general parabolic systems of the type
ut − a(x, t, Du) = div (|F |p−2 F ) ,
while the assumptions on a(·) are

2
2 1
2
2 p−1

 |a(x, t, z)| + (s + |z| ) 2 |Da(x, t, z)| ≤ L(s + |z| ) 2


p−2
(3.38)
ν(s2 + |z|2 ) 2 |λ|2 ≤ hDa(x, t, z)λ, λi



p−1

|a(x, t, z) − a(x0 , t, z)| ≤ Lω (|x − x0 |) (s2 + |z|2 ) 2 ,
whenever x, x0 ∈ Ω, t ∈ (0, T ), z, λ ∈ RN n , where p ≥ 2 and s ∈ [0, 1]. Note that,
again, we are assuming no continuity of t 7→ a(x, t, z), this being just a measurable
map. The result is finally
Theorem 3.12 ([56]). Let u ∈ C(0, T, L2 (Ω, RN )) ∩ Lp (0, T, W 1,p (Ω, RN )) be a
weak solution to the general parabolic system (3.24), where Ω is a bounded domain
in Rn , and p ≥ 2, N ≥ 1, and where the vector field satisfies (3.38). Then there
exists a positive number δ ≡ δ(n, N, p, ν, L) > 0 such that
F ∈ Lγloc (ΩT , RN n ) =⇒ Du ∈ Lγloc (ΩT , RN n ) ,
18
GIUSEPPE MINGIONE
whenever
4
+δ .
n
Moreover, there exists a constant c ≡ c(n, N, p, ν, L, γ) such that for every parabolic
cylinder QR ≡ BR (x0 ) × (t0 − R2 , t0 + R2 ) b ΩT the local estimate (3.30) holds.
p≤γ <p+
(3.39)
3.7. Open problems. For the sake of brevity we shall restrict here to the model
equation (3.4). Let us start from one simple observation. The minimum degree of
integrability required to Du and F in order to give meaning to the weak formulation
of (3.4), that is
Z
Z
|Du|p−2 DuDϕ dx =
|F |p−2 F Dϕ dx
for every ϕ ∈ C ∞
Ω
Ω
is clearly given by
u ∈ W 1,p−1 (Ω, RN )
and
F ∈ Lp−1 (Ω, RN n ) .
This leads to consider those distributional solutions to (3.4) which do not belong
to the natural space W 1,p , and therefore are not energy solutions; these are very
weak solutions. We shall encounter such solutions also later on, when dealing
with measure data problems, and we shall see that they can exist beside the usual
energy solutions.
Now a comparison between the result of Theorem 3.2 and the linear one in (2.12),
which regards solutions to (2.11), naturally leads to the following open problem,
which is actually a conjecture of Iwaniec:
Open problem 1. Prove that the results of Theorems 3.1 and 3.2 hold in the full
range p − 1 < γ.
The only result known up to now in this direction is due independently to Iwaniec
& Sbordone [72, 69], and Lewis [90], who were able to prove that the statement
of Theorem 3.2 holds in the range p − ε < γ < ∞ for some > 0, depending
on the exponent p and the dimension n, but independent of all the other entities
considered, in particular of the solution. The methods of proof in [72] uses Iwaniec’s
non-linear Hodge decomposition, a powerful and deep tool of its own interest. The
method in [90] relies instead on the truncation of maximal operators, the so-called
Lipschitz truncation method.
The conjecture above extends to solutions to the parabolic system (3.24). Again,
in this direction Kinnunen & Lewis [82, 83] proved the validity of Theorems 3.103.11 in the range p − ε < γ < p + ε, thereby finding the right extension of so
called Gehring’s lemma [63, 70] in the case of parabolic problems. Bögelein, in a
remarkably deep paper [26], recently extended the results of Kinnunen & Lewis on
very weak solutions [83] to the case of parabolic systems depending on higher order
spatial derivatives; see also [25], which features the higher order extension of [82].
We close this section noting that the weaker integrability assumption F ∈ Lγ
with γ < p, puts the right hand side of (3.4) outside the natural dual space
0
div (|F |p−2 F ) 6∈ W −1,p (Ω) ,
and therefore leads us to consider those problems, as measure data ones, for which
we face the problem of proving gradient estimates below the duality exponent.
3.8. Obstacle problems. We conclude with further integrability results, recently
obtained in [27], and concerning gradient estimates for obstacle problems. A point
of interest here is that, differently from the usual results available in the literature,
the obstacles considered here are just Sobolev functions, and therefore discontinuous, in general.
TOWARDS A NON-LINEAR CALDERÓN-ZYGMUND THEORY
19
We shall start with the simplest elliptic case, involving the minimization problem
Z
(3.40)
Min
|Dv|p dx
v∈K
Ω
where
ψ ∈ W01,p (Ω) .
K := {v ∈ W01,p (Ω) : v ≥ ψ a.e.}
The integrability result available is
Theorem 3.13. Let u ∈ W 1,p (Ω) be the unique solution to the obstacle problem
(3.40), where Ω is a bounded domain in Rn . Then
1,γ
1,γ
ψ ∈ Wloc
(Ω) =⇒ u ∈ Wloc
(Ω)
for every γ ≥ p .
Moreover, for every BR b Ω it holds that
! γ1
Z
Z
p1
Z
γ
p
−
|Du| dx
≤ c − |Du| dx
+c −
BR/2
BR
γ
γ1
|Dψ| dx
,
BR
where c ≡ c(n, p, ν, L, γ).
The previous theorem is obviously optimal, as it follows by considering the gradient integrability of u on the contact set {u ≡ ψ}, where Du and Dψ coincide
almost everywhere.
The next result concerns the evolutionary case; for the sake of simplicity we shall
confine ourselves to the case of time independent obstacles and zero right hand side,
referring to [27] for the time dependent case and more general right hand sides. We
shall consider parabolic variational inequalities in the cylinder (3.25), defined in the
class
K = {v ∈ Lp (0, T ; W01,p (Ω)) ∩ L2 (ΩT ) : v ≥ ψ a.e.}
where the exponent p is as usual assumed to satisfy (3.35), and ψ : ΩT → R is a
fixed obstacle function that for simplicity we assume to be time independent. The
variational inequality under consideration is
Z
vt (v − u) + h|Du|p−2 Du, D(v − u)i dz + (1/2)kv(·, 0)k2L2 (Ω) ≥ 0,
(3.41)
ΩT
0
0
for all v ∈ K ∩ W 1,p (0, T ; W −1,p (Ω)). Finally, for the initial values of u we shall
0
0
assume that there exists v0 ∈ K ∩ W 1,p (0, T ; W −1,p (Ω)) such that
(3.42)
u(x, 0) = v0 (x, 0)
for x ∈ Ω.
The integrability result is now
Theorem 3.14. Let u ∈ K satisfy the integral inequality (3.41) under the assump2n
. Then
tion (3.42) with p > n+2
Dψ ∈ Lγloc (ΩT ) =⇒ Du ∈ Lγloc (ΩT )
for every γ ≥ p .
Moreover, there exists a constant c = c(n, p, γ) such that for any parabolic cylinder
Q2R (z0 ) b ΩT there holds
! γ1
Z
−
|Du|γ dx dt
QR/2
p1 Z
+ −
− (|Du| + 1) dx dt
"Z
≤c
p
QR
QR
where
d≡
p
2
2p
p(n+2)−2n
if p ≥ 2
if p < 2.
γ1 #d
|Dψ| dx dt
,
γ
20
GIUSEPPE MINGIONE
Both in Theorem 3.13 and in Theorem 3.14 we have considered the model case
of the p-Laplacean operator, but more general operators can be also considered;
for this we again refer to [27]. In the same paper one can also find conditions
for treating time dependent obstacles; in this case in order to get the Lγ local
integrability of the spatial gradient it is necessary to assume at least that
γ
p−1
(ΩT ) .
∂t ψ ∈ Lloc
4. Review on measure data problems
Following a rather consolidated tradition we shall talk about measure data problems also in those cases when the datum involved is not genuinely a measure, but
also a function with low integrability properties. For the sake of simplicity we shall
concentrate on Dirichlet problems, with homogeneous boundary datum, of the type
−div a(x, Du) = µ
in Ω
(4.1)
u=0
on ∂Ω,
where Ω ⊂ Rn is a bounded open subset with n ≥ 2, while µ is a (signed) Borel
measure with finite total mass
|µ|(Ω) < ∞ .
Non-homogeneous boundary data can be dealt with by standard reductions, and
will not be treated here. Of course, it is always possible to assume that the measure
µ is defined on the whole Rn by letting µ(Rn \ Ω) = 0, therefore in the following
we shall do so. As for the structure properties of the problem, these are essentially
the standard ones prescribing growth and monotonicity properties at p-rate: in the
following a : Ω × Rn → Rn will denote a Carathèodory vector field satisfying
(
p−2
ν(s2 + |z1 |2 + |z2 |2 ) 2 |z2 − z1 |2 ≤ ha(x, z2 ) − a(x, z1 ), z2 − z1 i
(4.2)
p−1
|a(x, z)| ≤ L(s2 + |z|2 ) 2 ,
for every choice of z1 , z2 ∈ Rn , and x ∈ Ω. Unless otherwise stated, when considering (4.2) the structure constants will satisfy
2≤p≤n
(4.3)
0<ν≤1≤L
s ≥ 0.
In particular we remark that at this stage x 7→ a(x, ·) is only a measurable map.
For this reasons many of the following results readily extend to problems of the
type
−div a(x, u, Du) = µ
in Ω
(4.4)
u=0
on ∂Ω,
but again for brevity we shall confine ourselves to simpler ones as in (4.1). See
for instance [106, Section 6.4] for the treatment of such cases in the context of the
results we are going to present.
Again for brevity, we shall not deal with the sub-quadratic case 1 < p < 2, where
additional problems appear. For this we refer to [16, 51].
Now, let’s switch to the facts. How to prove the existence of a solution? And,
what kind of solutions we are talking about? This is already an issue introduced in
Section 3.7: here very weak solutions come back. We start by the following crude
distributional definition, particularizing the one in (3.3)
Definition 4.1. A solution u to the problem (4.1) under assumptions (4.2), is a
function u ∈ W01,1 (Ω) such that a(x, Du) ∈ L1 (Ω, Rn ) and
Z
Z
(4.5)
ha(x, Du), Dϕi dx =
ϕ dµ,
for every ϕ ∈ Cc∞ (Ω) .
Ω
Ω
TOWARDS A NON-LINEAR CALDERÓN-ZYGMUND THEORY
21
Energy solutions form a class in which the unique solvability of (4.1) is possible.
In fact given u, v two energy solutions to (4.1), we can test the weak formulation
Z
ha(x, Du) − a(x, Dv), Dϕi dx = 0
Ω
by ϕ = u − v, and then (4.2)1 and Poincaré inequality implies that u ≡ v.
As already mentioned in Section 3.7, solutions which are not energy solutions are
usually called very weak solutions, and they exists beside usual energy solutions,
even for simple linear homogeneous equations of the type
div (A(x)Du) = 0 ,
as shown by a classical counterexample of Serrin. In fact, in his fundamental paper
[111] Serrin showed that, for a proper choice of the strongly elliptic and bounded,
measurable matrix A(x), the previous equation admits at least two solutions: one
of them belongs to the natural energy space W 1,2 , and it is therefore an energy
solution; the other one does not belong to W 1,2 , and for this reason in a time
where the concept of very weak solution was not very familiar, was conceived as a
pathological solution. This situation immediately poses the problem of uniqueness
of solutions. In fact, the problem of finding a definition of solution allowing for
unique solvability of (4.1), similarly as what happens for energy solutions, is in
general still open, and is the most important and outstanding one in the theory of
measure data problems. For a comprensive discussion on the uniqueness problem
we refer to [43].
0
Going back to the existence problem, in the case µ ∈ (W01,p (Ω))0 ≡ W −1,p (Ω),
the dual of W01,p (Ω), then the standard monotone operator theory [98] provides us
with the existence of a solution u ∈ W01,p , which is at this point unique amongst
the energy solutions.
The first issue is therefore to establish the existence of a solution in the case µ 6∈
0
W −1,p (Ω). Let me examine a few related situations, also related to the theorems
in the following sections.
Example 1 (Measures with low density). In the case of a Borel measure µ in the
right hand side, there is a classical trace type theorem due to Adams [5] stating
that if the density condition
|µ|(BR ) . Rn−p+ε
(4.6)
0
holds for some ε > 0, then it follows that µ ∈ (W01,p (Ω))0 ≡ W −1,p (Ω). Therefore
for such measures we have the existence of a unique energy solution. We notice that
the p-capacity of a ball BR is comparable to Rn−p , therefore (4.6) implies that the
measure in question is absolutely continuous with respect to the p-capacity. Indeed
Sobolev functions are those that can be defined up to set of negligible p-capacity.
For a general uniqueness theorem concerning measures which are absolutely continuous with respect to the capacity without necessarily satisfying (4.6) we refer to
[24]; this paper concerns so called entropy solutions, a kind of solution we shall not
discuss here.
Example 2 (High integrable functions). When the measure is a function µ = f ,
0
which belongs to Lγ (Ω), then for certain values of γ we have f ∈ W −1,p (Ω). In
fact let me recall that Sobolev imbedding theorem yields, when p < n
∗
np
.
W01,p (Ω) ⊂ Lp (Ω)
p∗ :=
n−p
Therefore L(p
∗ 0
)
0
(Ω) ⊂ W −1,p (Ω). This means that if f ∈ Lγ (Ω) and
np
γ ≥ (p∗ )0 =
np − n + p
22
GIUSEPPE MINGIONE
then there is a unique energy solution to (4.1). This argument can be refined up to
Lorentz spaces - see Definition 5.4 below. In fact the improved Sobolev embedding
theorem gives
∗
W01,p (Ω) ⊂ L(p∗ , p)(Ω) $ Lp (Ω) = L(p∗ , p∗ )(Ω) .
For a proof see for instance [119]. But since (L(p∗ , p))0 = L((p∗ )0 , p0 ), we have that
if f ∈ L((p∗ )0 , q)(Ω) with q ≤ p0 , then there exists a unique energy solution to (4.1).
Example 3 (Non-linear Green’s functions). By the fundamental solution to the
p-Laplacean equation we mean the function
(
p−n
|x| p−1 if 2 ≤ p < n
(4.7)
Gp (x) ≈
log |x| if
p=n.
Compare with (2.6).
Remark 4.1. With abuse of notation we shall go on denoting by Gp any function
differing by the one in (4.7) for a multiplicative or an additive constant. This is
why we used the symbol ≈ in (4.7).
Denoting by δ the Dirac mass charging the origin, up to a re-normalization
constant depending only on n and p, the function Gp (·) properly solves the measure
data equation
(4.8)
4p u := div (|Du|p−2 Du) = δ ,
see for instance [43]. It is interesting to compute the degree of integrability of Gp (·).
Note that
1−n
|DGp | ≈ |x| p−1 ,
and therefore by (2.21) it follows that
n
(4.9)
n−p
|Gp |p−1 ∈ Mloc
(Rn )
n
and
n−1
|DGp |p−1 ∈ Mloc
(Rn ) ,
the first being meaningful of course when p < n. The previous inclusions should be
compared with (2.20).
It is important to keep in mind (4.9), as well as the integrability exponents
thereby introduced, will serve as a reference for testing the optimality of the regularity results presented later for solutions to general non-linear problems as (4.1).
Here we need a clarification; in fact in Section 4.4 below we shall see that Gp (·),
or rather, a minor modification of it via an additive constant to meet boundary
values, is the unique solution to (4.8) - with zero boundary value - in the class of
solutions obtainable via certain positive approximations - see next section.
For this reason Gp (·) is safely conjectured to be the unique distributional solution to (4.8) amongst all possible entropy or so called re-normalized solutions
[43]. Moreover, as shown later in Theorem 5.4 and Remark 5.3, Gp (·) exhibits the
worst behavior amongst the solutions with measure data problems, according to
the rough principle stating that “the more the measure concentrates, the worse
solutions behave”.
As a conclusion, in the following we shall use the regularity properties of Gp (·)
described in (4.9) to test the optimality of the regularity results obtained for approximate solutions to problems involving general measure as (4.1).
4.1. Solvability. Here we discuss the basic solvability of (4.1). The main point is
that although uniqueness is still lacking, it is always possible to solve (4.1) in the
plain sense of Definition 4.1. Since distributional solutions are not unique, at this
point there are in the literature several definitions of solution adopted towards the
settlement of the uniqueness problem - see for instance [16] for the definition of
entropy solutions, and [43] for the definition of re-normalized ones. Here we shall
TOWARDS A NON-LINEAR CALDERÓN-ZYGMUND THEORY
23
adopt the notion of solution obtained by limits of approximations (SOLA); these
are solutions obtained via an approximation scheme using solutions of regularized
problems. We adopt this notion for a number of reasons: our main emphasis on
the a priori regularity estimates; all the notions of solutions at the end turn out
to provide uniqueness in the same cases i.e. when considering measure which are
L1 -functions, and turn out to be essentially equivalent in such cases; in any case the
existence of any of such solutions is constructed via an approximation procedure.
There are anyway a few advantages in considering such SOLA: first, when deriving
regularity estimates one is dispensed from tedious calculations implied, in the cases
of different definitions of solutions, by the fact that one has to verify additional
conditions when using test functions; in the case of approximate solutions one
essentially argues on an a priori level, testing the equations as one would have
a usual energy solutions, as, in fact, the approximating solutions are. Second,
as already noted a few lines above, when dealing with SOLA solutions it is easy
to verify the uniqueness, in its Dirichlet class, of the fundamental solution (4.7),
and this allows for claiming the optimality of the regularity results obtained about
SOLA; see Section 4.4 below. Moreover, as we shall see later, in certain special
cases, approximate solutions allows to formulate additional uniqueness results; see
Section 5.5 below.
The approximation procedure has been settled by Boccardo & Gallöuet [22, 23],
see also [42]. The idea is to approximate the measure µ via a sequence of smooth
functions {fk } ⊂ L∞ (Ω), such that fk → µ weakly in the sense of measures, or
fk → µ strongly in L1 (Ω) in the case µ is a function. At this point, by standard
monotonicity methods, one finds a unique solution uk ∈ W01,p (Ω) to
−div a(x, Duk ) = fk
in Ω
(4.10)
uk = 0
on ∂Ω.
The arguments in [22] lead to establish that there exists u ∈ W01,p−1 (Ω) such that,
up to a not relabeled subsequence,
(4.11)
uk → u
and
Duk → Du
strongly in Lp−1 (Ω), and a.e.
and (4.1) is solved by u in the sense of (4.5). We have therefore found a distributional solution having the remarkable additional feature of having been selected via
an approximation argument through regular energy solutions.
It is important to notice that, as described for instance in [20, 42], in the case µ
is an L1 -function, by considering a different approximating sequence {f¯k } strongly
converging to f in L1 (Ω), we still get the same limiting solution u. As a consequence
the described approximation process allows to build a class of solutions, those in
fact obtained by approximation, in which the unique solvability of (4.1) is
possible. Related uniqueness properties follows for entropy solutions when the
measure is a function; see [24].
For the reason we have just explained, from now on, when dealing with the case
the measure µ is a actually an L1 -function, we shall talk about the solution to (4.1),
meaning by this the unique solution found by the above settled approximation
scheme.
4.2. Basic regularity results. There is a vast literature on the regularity of solutions to measure data problems. We shall confine ourselves to the first papers
dealing with general quasilinear problems of the type (4.1). The basic regularity
results obtained fall in two categories; the first deals with genuine measure data
problems and in some sense reproduce for general solutions the integrability properties of the fundamental solution (4.9). The second deals with the case the measure
24
GIUSEPPE MINGIONE
is a function enjoining extra integrability properties, and aims at reproducing in
the non-linear case the CZ linear results of the type (2.4).
Theorem 4.1 ([16, 52]). Under the assumptions (4.2) there exists a solution u ∈
W01,p−1 (Ω) such that
n
|u|p−1 ∈ M n−p (Ω)
for n < p ,
and
(4.12)
n
|Du|p−1 ∈ M n−1 (Ω) .
The result of the previous theorem has been obtained in some preliminary forms
in [22, 117]; the form above has been obtained in [16] for the case p < n, while
the case p = n, with the consequent Mn estimate, is treated in [52]. Results for
systems have been obtained in [50].
We now switch to the case when the measure is actually a function
(4.13)
µ ∈ Lγ (Ω) ,
γ ≥ 1.
For this we premise the following:
Remark 4.2 (Maximal regularity). The equations we are considering have measurable coefficients, and this means that x 7→ a(x, z) is a measurable map. The
maximal regularity in terms of gradient integrability we may expect, even for energy solutions to the homogeneous equation div a(x, Du) = 0, is at most Du ∈ Lqloc ,
for some q which is in general only slightly larger than p, and depends in a critical way on n, p, L/ν. This is basically a consequence of Gehring’s lemma [63, 70].
Therefore we are not expecting to get much more that Du ∈ Lp in general for solutions to the measure data problems considered in the following. Therefore, with
abuse of terminology, we shall consider Du ∈ Lp as the maximal regularity for the
gradient of solutions u.
The previous remark allows to restrict the range of parameters of γ, the exponent
0
appearing in (4.13). We first look for values of γ such that Lγ 6⊂ W −1,p , otherwise
p
the existence of an energy solution such Du ∈ L follows. We are in fact almost
at the maximal regularity. As a matter of fact when considering measure data
problems one is mainly interested in those solutions which are not energy ones. By
Example 2 we see that the right condition is
np
= (p∗ )0
for
p<n.
(4.14)
1<γ<
np − n + p
Theorem 4.2 ([23]). Under the assumptions (4.2) and (4.13)-(4.14), the solution
u ∈ W01,p−1 (Ω) to (4.1) is such that
nγ
|Du|p−1 ∈ L n−γ (Ω) .
Finally, a borderline case
Theorem 4.3 ([23]). Assume that (4.2) hold and that the measure µ is a function
belonging to L log L(Ω). The solution u ∈ W01,p−1 (Ω) to (4.1) is such that
n
|Du|p−1 ∈ L n−1 (Ω) .
Theorems 4.1-4.3 establish a low order CZ-theory for elliptic problems with measure data which is completely analogous to that available in the linear case and
therefore optimal in the scale of Lebesgue’s spaces - compare with (2.3) and the
other results in Section 2.
We just talked about a low-order theory since, when dealing with second order
equations, one would expect to get results on second order derivatives or the like, as
TOWARDS A NON-LINEAR CALDERÓN-ZYGMUND THEORY
25
in (2.3) for the linear case. Indeed, in the next sections we shall extend Theorems
4.1-4.3 in two different directions. We shall in fact present
1) Integrability theorems for the gradient in function spaces different from the
Lebesgue’s ones. In particular, we shall consider non rearrangement invariant function spaces (Section 5).
2) Differentiability theorems for the gradient, having in turn, when dealing with
more regular equations, weak forms of Theorems 4.1-4.3 as a corollary (Section 6).
4.3. Open problems. There are in our opinion at least two main open issues in
the theory of measure data problems. The first has been already mentioned:
Open problem 2. Find a functional class where problem (4.1) can be uniquely
solved; the solution found must be distributional.
The second open problem is concerned with a tremendous gap in the theory
of measure data problems, as it is far mote important from the regularity theory
viewpoint.
Open problem 3. Prove solvability of the Dirichlet problem (4.1) under assumptions (4.2) in the case (4.1)1 is a system, and u takes its values in RN , N > 1; in
particular prove Theorems 4.1-4.3 in the case of systems.
A weaker version of the previous one is
Open problem 4. Find classes of structure conditions on the vector field a : Ω ×
RN n → RN n , still satisfying (4.2), allowing for solvability of the Dirichlet problem
(4.1) in the case of systems.
The first attempts to prove the last two problems are in [50, 51, 52, 60]. Such
papers give a complete solution for the case of certain systems with a special structure, first introduced by Landes [91], and aimed at selecting those vector fields a(·)
whose ellipticity properties match with the gradient of certain truncated maps in
the vectorial case. Such condition is satisfied for instance by systems depending on
the gradient in a peculiar way; the so called Uhlenbeck structure, first considered
in [121] to select homogeneous systems allowing for everywhere regular solutions
(4.15)
a(Du) ≡ b(|Du|)Du ,
b(t) ≈ tp−2
is a typical example. The important case of the p-Laplacean system with measure
data
4p u := div (|Du|p−2 Du) = µ
is therefore covered. The difficulties involved in the Open problem 3, are of two
types, and they are in turn related each other. The main point is that due to
the presence of a measure in the weak formulation (4.5) we can only allow L∞ functions, and therefore in order to prove energy estimates one is led to test (4.5)
with truncations of the solutions:
w(x) := max{−k, min{u(x), k}}
k∈N.
It turns out that while in the scalar case the gradient of w is just Duχ{−k≤u(x)≤k} , in
the vectorial case it has a more delicate expression which does not fit the ellipticity
properties of a general system, but in some special cases, as, in fact, the one in
(4.15). This obstruction pops-up both at the level of getting a priori regularity
estimates, and at the level of the convergence estimates, that is those related to
(4.11); for more details see [50].
26
GIUSEPPE MINGIONE
4.4. On the uniqueness of fundamental solutions. Here we shall present a
uniqueness result for the Dirichlet problem
−4p u = δ
in B1
(4.16)
u=0
on ∂B1 ,
which is known amongst experts. More precisely we shall see that - compare with
the notation introduced in (4.7) and subsequent remark - the non-linear fundamental solution
( p−n
|x| p−1 − 1
if 2 ≤ p < n
(4.17)
Gp (x) ≈
log |x|
if
p=n,
is the unique solution to the problem (4.16) amongst those obtainable via approximation, as described in (4.10), with non-negative functions fk - compare with the
calculations made in [43, Example 2.16] where it is shown that for a suitable choice
of the constant involved in the symbol ≈ in (4.17), Gp is also a re-normalized
solution to (4.16).
This uniqueness result immediately identifies the function class where to solve,
in a unique way, problem (4.16), since the standard way to to build a sequence
of functions fk weakly converging to µ in the sense of measures, is obviously via
convolutions with standard mollifiers {φk }
fk := µ ∗ φk ,
(4.18)
which provide positive approximations data fk when µ is a non-negative measure.
As a matter of fact, when considering the approximation scheme (4.10) one always
uses the approximations settled in (4.18), that therefore should be considered as
a part of the approximation scheme in (4.10), and therefore of the definitions of
approximate solutions (SOLA).
We explicitly point out that the following argumentation singles-out in a very
clear way the difference between any distributional solution, and a solution obtained
as a limit of approximations.
We start by considering the approximating problems to (4.16), which, according
to (4.10), are now defined by
−4p uk = fk
in B1
u=0
on ∂B1 .
Moreover, up to a non-relabeled subsequence, we have that
uk → u
(4.19)
and
Duk → Du
p−1
strongly in L
and almost everywhere. This result follows from [22, 23], as already
explained in Section 4.1. As a consequence:
(4.20)
u ∈ W 1,p−1 (B1 ) .
We want to show that
(4.21)
u ≡ Gp
and now Gp is defined in (4.17).
By the very definition (4.18) it follows that fk ≥ 0, and since fk is a smooth
function we can apply the maximum principle to the energy solution uk , thereby
getting that
(4.22)
uk ≥ 0
since uk ≡ 0 on ∂B1 . Combining (4.19) and (4.22) we infer
(4.23)
u≥0,
TOWARDS A NON-LINEAR CALDERÓN-ZYGMUND THEORY
27
almost everywhere in B1 . We now get further regularity properties of u. We have
that, in a distributional sense
4p u = 0
in B1 \ {0}
and therefore standard regularity theory for solutions to the p-Laplacean equation
tells that u is continuous in B1 \ {0}; we have actually more, and this will be
important in a few lines:
1,α
u ∈ Cloc
(B1 \ {0}) ,
(4.24)
for some α ∈ (0, 1). In particular, we have that (4.23) holds everywhere. As a
corollary we also have
(4.25)
Du ∈ Lp (B1 \ Br )
for every r ∈ (0, 1) .
Using this fact together with (4.23) we may apply a classical result of Serrin [112,
Theorem 12] to infer that there exists a positive constant c such that
u
1
≤
≤c.
c
Gp
(4.26)
Now we recall the following result:
Theorem 4.4 ([75]). There exists a unique distributional solution to the problem
(4.16) u such that: u ∈ C 1 (B1 \ {0}), Du ∈ Lp−1 (B1 ), Du ∈ Lp (B1 \ Br ) for every
r ∈ (0, 1), and satisfying condition (4.26).
Since u ∈ W 1,p−1 (B1 ), by (4.20), (4.24), (4.25) and (4.26) all the assumptions of
the previous theorems are satisfied by u, and being obviously satisfied also by Gp ,
(4.21) finally follows.
5. Nonlinear Adams theorems
The results in this section follow simultaneously two different viewpoints: first,
they extend in new directions the regularity results available for measure data problems presented in Section 4, showing CZ estimates in new types of function spaces;
second, and more importantly, they show that certain classical potential theory facts
apparently linked to the linear setting, can be actually reformulated in the context
of what is called non-linear potential theory. We shall in fact present optimal nonlinear extensions of classical results of Adams [6] and Adams & Lewis [9]. Moreover,
we shall present a localization of the classical Lorentz spaces estimates obtained by
Talenti [117]. All the results presented in this and in the next section are taken
from [105, 106].
Remark 5.1. When stating our regularity results for measure data problems, we
shall state them in the form of existence and regularity results, since the uniqueness
of distributional solutions does not hold. On the other hand the regularity claimed
holds for every approximate solution to (4.1) in the sense of the approximation
scheme described in Section 4.1.
5.1. Morrey spaces. Morrey spaces provide a way of measuring the size of functions which is in some sense orthogonal to that of Lebesgue spaces. In fact, while
these read the size of the super-level sets of functions - as rearrangement invariant
spaces - Morrey spaces use instead density conditions in their formulation. Specifically, the condition is
Z
(5.1)
|w|γ dx ≤ M γ Rn−θ
and
0≤θ≤n,
BR
to be satisfied for all balls BR ⊆ Ω with radius R.
28
GIUSEPPE MINGIONE
Definition 5.1. A measurable map w : Ω → Rk , belongs to the Morrey space
Lγ,θ (Ω, Rk ) ≡ Lγ,θ (Ω) iff satisfies (5.1), and moreover one sets
Z
γ
γ
θ−n
kwkLγ,θ (Ω) := inf{M ≥ 0 : (5.1) holds} = sup R
|w|γ dx .
BR ⊆Ω
BR
Obviously Lγ,n ≡ Lγ , and Lγ,0 ≡ L∞ . The Morrey scale is orthogonal to the
one provided by Lebesgue spaces in fact
Lγ,θ ⊂ Lγ \ Lγ log L
(5.2)
for every θ > n ,
that is, no matter how close θ is to zero, therefore no matter how close Lγ,θ is to
L∞ in the Morrey scale. We recall that the space Lγ log L is that of those functions
w satisfying
Z
|w|γ log (e + |w|) dx < ∞ ,
0
so that Lγ ⊂ Lγ log L ⊂ Lγ for every γ 0 > γ ≥ 1.
In a similar way the classical Marcinkiewicz-Morrey spaces [9, 7, 109, 114] are
naturally defined.
Definition 5.2. A measurable map w : Ω → Rk belongs to the MarcinkiewiczMorrey space Mγ,θ (Ω, Rk ) ≡ Mγ,θ (Ω) iff
sup Rθ−n kwkγMγ (BR )
=
BR ⊂Ω
(5.3)
=:
sup sup λγ Rθ−n |{x ∈ BR : |w(x)| > λ}|
BR ⊂Ω λ>0
kwkγMγ,θ (Ω)
<∞.
Extending in a endpoint way previous results of Stampacchia [114], Adams
proved the following extension of Theorem 2.1:
Theorem 5.1 ([6]). Let β ∈ [0, θ); for every γ > 1 such that βγ < θ we have
(5.4)
θγ
f ∈ Lγ,θ (Rn ) =⇒ Iβ (f ) ∈ L θ−βγ ,θ (Rn ) .
When γ = 1 and β < θ we have
(5.5)
θ
f ∈ L1,θ (Rn ) =⇒ Iβ (f ) ∈ M θ−β ,θ (Rn ) ,
and
(5.6)
θ
f ∈ L1,θ (Rn ) ∩ L log L(Rn ) =⇒ Iβ (f ) ∈ L θ−β ,θ (Rn ) .
In other words, we formally obtain Theorem 5.1 by Theorem 2.1 by replacing
n with θ. In fact, as noted above, Lγ,n ≡ Lγ . As shown in [6] the embedding
spaces of Theorem 5.1 are optimal with respect to every scale. In particular Iβ (f )
in general does not belong to Lq (Ω) for any q > θγ/(θ − βγ).
Theorem 5.1 has a few immediate consequences. Via the representation formula
|u(x)| . I1 (|Du|)(x) we have the so called Sobolev-Morrey embedding theorem:
(5.7)
θγ
Du ∈ Lγ,θ =⇒ u ∈ L θ−γ ,θ
provided 1 < γ < θ .
Along the same line there are applications to the regularity of solutions to the
Poisson equation (2.1):
(5.8)
θγ
f ∈ Lγ,θ =⇒ Du ∈ L θ−γ ,θ
provided 1 < γ < θ .
TOWARDS A NON-LINEAR CALDERÓN-ZYGMUND THEORY
29
5.2. Nonlinear versions. Here we shall introduce the non-linear potential theory
versions of Theorem 5.1 and of (5.8) in the context of measure data problems:
this means that images of Riesz potentials are replaced by solutions to non-linear
equations with p-growth, for example p-harmonic functions. Specifically, we are
dealing with solutions to problems of the type (4.1); due to the fact that x →
a(x, z) is just measurable, the maximal regularity expected in terms of gradient
integrability is essentially Du ∈ Lp ; see also Remark 4.2. By the discussion on
the existence/uniqueness to solutions to (4.1) made in Section 4.1, in the following,
when the measure µ will be a function we shall talk about “the solution” meaning
the one defined by approximation according to the scheme described in Section
4.1, since in this case approximate solutions are unique. In the case µ is a genuine
measure regularity results stated will refer to the found solution - compare with
Remark 5.1.
Since after Theorem 5.1 we expect the Morrey space parameter θ to play the
role of the dimension n, in formal accordance with (4.14), we start assuming that
θp
and
p<θ≤n,
θp − θ + p
a condition whose actual role will be discussed in Remark 5.2 below. We have the
following non-linear potential theory version of (5.4):
1<γ≤
(5.9)
Theorem 5.2 ([106]). Assume (4.2), and that the measure µ is a function belonging
to Lγ,θ (Ω) with (5.9). Then the solution u ∈ W01,p−1 (Ω) to the problem (4.1) is
such that
θγ
,θ
θ−γ
|Du|p−1 ∈ Lloc
(Ω) .
(5.10)
Moreover, the local estimate
k|Du|p−1 k
θγ
L θ−γ
,θ
≤ cR
θ−γ
γ
−n
(BR/2 )
k(|Du| + s)p−1 kL1 (BR ) + ckf kLγ,θ (BR )
holds for every ball BR ⊆ Ω, with a constant c only depending on n, p, L/ν, γ.
Observe that, on one hand for p = 2 inclusion (5.10) locally gives back (5.8),
while on the other (5.10) is also the natural Morrey space extension of the nonlinear result of Theorem 4.2, to which it locally reduces for n = θ. In the relevant
p
borderline case γ = θp/(θp − θ + p) the maximal regularity Du ∈ Lp,θ
loc (Ω) ⊂ Lloc (Ω)
holds.
Remark 5.2 (Sharpness of condition (5.9)). The parameters choice in (5.9) is
optimal for the gradient integrability in the sense that the upper bound for γ is the
minimal one allowing for the maximal regularity Du ∈ Lp . In fact we have
θγ(p − 1)
θp
iff
≤ p.
θp − θ + p
θ−γ
Related to this fact is Theorem 5.3 below. Moreover, by Example 1 in Section 4 a
Borel measure µ satisfying (4.6) for some ε > 0 and for every ball BR ⊂ Rn , belongs
0
to the dual space W −1,p , and therefore (4.1) is uniquely solvable in W01,p (Ω). This
is the reason for assuming p < θ in Theorem 5.2 and Theorem 5.4 below. The case
p = θ forces γ = 1 and therefore falls in the realm of measure data problems: it will
be treated in Theorem 5.5 below. Note that Hölder’s inequality and (5.1) imply
that |µ|(BR ) = [|f | dx](BR ) . M Rn−θ/γ , and therefore, again by the mentioned
Adams’ result, in order to avoid trivialities we should also impose that pγ ≤ θ. But
keep in mind (5.9) and note that
(5.11)
(5.12)
γ≤
θp
θ
≤
θp − θ + p
p
iff
1 ≤ p ≤ θ.
30
GIUSEPPE MINGIONE
Therefore, assuming the first inequality in (5.11) together with p ≤ θ implies pγ ≤ θ.
Theorem 5.3 ([106]). Assume (4.2), and that the measure µ is a function belonging
to Lγ,θ (Ω) with γ > θp/(θp−θ+p) and p ≤ θ ≤ n. Then the solution u ∈ W01,p−1 (Ω)
to the problem (4.1) is such that
Du ∈ Lh,θ
loc (Ω),
for some h ≡ h(n, p, L/ν, γ, θ) > p .
Moreover, for every ball BR ⊆ Ω with R ≤ 1 the local estimate
θ
1
n
kDukLh,θ (BR/2 ) ≤ cR h − p−1 k(|Du| + s)kLp−1 (BR ) + ckf kLp−1
γ,θ (B )
R
holds for a constant c only depending on n, p, L/ν.
In the case γ = 1 we cannot obviously expect Theorem 5.2 to hold; instead,
imposing an L log L type integrability condition on f allows to deal with the case
γ = 1 too, obtaining the natural analog of (5.6).
Theorem 5.4 ([106]). Assume that (4.2) holds, and that the measure µ is a
function belonging to L1,θ (Ω) ∩ L log L(Ω) with p ≤ θ ≤ n. Then the solution
u ∈ W01,p−1 (Ω) to the problem (4.1) satisfies
θ
θ−1
|Du|p−1 ∈ Lloc
(Ω) .
(5.13)
Moreover, the reverse-Hölder type inequality
! θ−1
Z
Z
θ
(p−1)θ
θ−1
−
|Du|
dx
≤ c − (|Du| + s)p−1 dx
BR/2
BR
"Z
+ckf kL1,θ (BR ) −
f
|f | log e + R
−BR |f (y)| dy
BR
1
θ
(5.14)
# θ−1
θ
!
dx
holds for every ball BR ⊆ Ω, with a constant c only depending on n, p, L/ν.
The previous theorem is the natural extension to Morrey spaces of Theorem 4.3.
We note that the appearance of the L log L-type functional in the right hand side
of the last estimate is exactly what we expect in a reverse Hölder type inequality
as (5.14), since the last quantity defines a norm in L log L, as already explained in
Section 2.2.
In order to conclude the non-linear extension of Theorem 5.1, giving the analog
of (5.5), we introduce Morrey spaces of measures.
Definition 5.3. We say that a Borel measure, defined on Ω, belongs to the Morrey
space L1,θ (Ω) iff
kµkL1,θ (Ω) := sup Rθ−n |µ|(BR ) < ∞ .
(5.15)
BR ⊆Ω
The non-linear analog of (5.5) is now
Theorem 5.5 ([105]). Under the assumptions (4.2), and µ ∈ L1,θ (Ω) with p ≤ θ ≤
n, there exists a solution u ∈ W01,p−1 (Ω) to the problem (4.1) such that
θ
,θ
θ−1
|Du|p−1 ∈ Mloc
(Ω) .
(5.16)
Moreover, the local estimate
k|Du|p−1 k
θ
M θ−1
,θ
(BR/2 )
≤ cRθ−1−n k(|Du| + s)p−1 kL1 (BR ) + ckµkL1,θ (BR )
holds for every ball BR b Ω, with a constant c only depending on n, p, L/ν.
TOWARDS A NON-LINEAR CALDERÓN-ZYGMUND THEORY
31
Just a few comments: in order to avoid trivialities we assumed p ≤ θ ≤ n,
otherwise, in the case θ < p, the measure µ satisfies (4.6) and therefore belongs to
0
the dual space W −1,p (Ω) by a result of Adams [5] - compare with Example 2 - and
there exists a unique solution u ∈ W01,p (Ω), found via usual monotonicity methods
[98] - note that, obviously
θ
p
≤
.
θ−1
p−1
Theorem 5.5 incorporates in a local way - being nevertheless extendable up to
the boundary - all the main integrability results previously obtained for equations
with measure data in the literature. When θ = n, and in particular no density
information on the measure is assumed, we find back the result in (4.12). When
p = n, a case forcing θ = p = n, we find back for the case of n-Laplacean equation
the Mn -regularity results obtained in [52], and in particular the explicit Mn local
estimates subsequently obtained in [79]. In the borderline case θ = p we have in
particular Du ∈ Mp , locally, and therefore this is in perfect accordance with what
happens when θ < p: the Lp -regularity of the gradient is here just missed by a
natural Marcinkiewicz factor. Finally let us mention that original Adams’ theorem
[6] of course applies when f in (5.5) is a measure rather than a function - of course
Riesz potentials naturally act on measures too.
Remark 5.3 (Dimensional remark). The qualitative information yielded by Theorem 5.5 is somehow interesting: let us recall that a measure µ which belongs
to L1,θ , and that therefore satisfies |µ|(BR ) . Rn−θ , cannot concentrate on sets
with Hausdorff dimension larger than n − θ. Therefore Theorem 5.5 tells that the
less the measure µ concentrates, the better solutions behave, confirming that the
Dirac measure case is in some sense the worst one. when analyzing the qualitative
properties of solutions to measure data problems.
Remark 5.4 (Boundary regularity). At the end of this first series of results let us
point out that in all the foregoing result we dealt with local regularity for brevity,
but up-to-the-boundary results are still obtainable using our techniques provided
suitable assumptions are made on ∂Ω. The Lipschitz continuity of ∂Ω - or even
the p-capacitary thickness of the complement of Ω - will for instance suffice in
most of the cases. Moreover the strong p-monotonicity assumption (4.2)1 can be
significantly relaxed. The same remark applies to most of the results we are going
to deal with in the next sections.
5.3. Lorentz spaces, and finer regularity. Lorentz spaces are a two-parameter
scale of spaces which refine Lebesgue spaces in a sense that will be clear in a few
lines. Lorentz spaces can be actually released as interpolation spaces using the
K-functional interpolation theory of Gagliardo, Lions and Peetre, or using trace
theory; we shall not pursue this abstract approach in the following rather pointing
at a very straight presentation.
Definition 5.4. The Lorentz space
L(t, q)(Ω) ,
with 1 ≤ t < ∞ and
0 < q ≤ ∞,
is defined prescribing that a measurable map w belongs to L(t, q)(Ω) iff
Z ∞
q dλ
q
<∞,
(5.17)
kwkL(t,q)(Ω) := q
λt |{x ∈ Ω : |w(x)| > λ}| t
λ
0
when q < ∞; for q = ∞ we set L(t, ∞)(Ω) := Mt (Ω), and this means finding
Marcinkiewicz spaces back.
32
GIUSEPPE MINGIONE
The quantity in (5.17) is only a quasinorm, that is satisfies the triangle inequality
only up to a multiplicative factor larger than one, and we remark that in the
following, when writing L(t, q) without further specifications, we shall mean that t
and q vary in the range specified in (5.4). We nevertheless remark that there is a
canonical way to equip Lorentz spaces with a norm when t > 1, equivalent to the
quantity introduced in (5.17). Good references for Lorentz spaces are for instance
[17, 64].
Recalling that in this paper Ω has always finite measure, we remark that the
spaces L(t, q)(Ω) “decrease” in the first parameter t, while increasing in q; moreover,
they “interpolate” Lebesgue spaces as the second parameter q “tunes” t in the
following sense: for 0 < q < t < r ≤ ∞ and we have, with continuous embeddings,
that
Lr ≡ L(r, r) ⊂ L(t, q) ⊂ L(t, t) ⊂ L(t, r) ⊂ L(q, q) ≡ Lq .
Remark 5.5 (Lorentz spaces are not bizzarre). In fact Lorentz spaces serve to
describe finer scales of singularities, not achievable neither via the use of Lorentz
spaces nor of Marcikiewicz ones. We have seen that Marcinkiewicz spaces describe
in a sharp way potentials. For instance, with the ambient spaces being Rn , we have
1
γ
γ
n ∈ M (B1 ) \ L (B1 ) .
|x| γ
The perturbation of a potential via a logarithmic singularity is then described via
Lorentz spaces
1
∈ L(γ, q)(B1 )
|x| logβ |x|
n
γ
iff
q>
1
.
β
The last strict inequality tells us that Lorentz spaces are even less fine that one
would wish! Note how the inverse relation between β and q demonstrates the fact
that Lorentz spaces increase in the second index.
The “morreyzation” of the Lorentz norm leads to consider the so called LorentzMorrey spaces [9, 7]; this means coupling definition (5.17) with a density condition.
Definition 5.5. A measurable map w belongs to Lθ (t, q)(Ω), for 1 ≤ t < ∞,
0 < q < ∞ and θ ∈ [0, n], iff
kwkLθ (t,q)(Ω)
:=
sup R
θ−n
t
BR ⊆Ω
=
Z
sup q
BR ⊆Ω
0
kwkL(t,q)(BR )
∞
t
λR
θ−n
|{x ∈ BR
q dλ
: |w(x)| > λ}| t
λ
q1
< ∞.
Moreover one sets Lθ (t, ∞)(Ω) := Mt,θ (Ω) and again we find back MarcinkiewiczMorrey spaces defined in (5.3).
Remark 5.6. As already mentioned above, it follows from the definition in (5.17)
that L(t, t) ≡ Lt , in fact by Fubini’s theorem we have
Z ∞
dλ
t
kgkLt (A) = t
λt |{x ∈ A : |g(x)| > λ}|
,
λ
0
so that kgkLt (A) = kgkL(t,t)(A) . As a consequence we also have that Lt,θ ≡ Lθ (t, t),
and kgkLt,θ (A) = kgkLθ (t,t)(A) hold. Of course, Ln (t, q) ≡ L(t, q).
We finally proceed with the “morreyzation” of L log L. In view of (2.26) the
following definition holds:
TOWARDS A NON-LINEAR CALDERÓN-ZYGMUND THEORY
33
Definition 5.6. The Morrey-Orlicz space L log Lθ (Ω) for θ ∈ [0, n] is defined by
saying that a map w belongs to L log Lθ (Ω) iff
kwkL log Lθ (Ω)
:=
sup Rθ kwkL log L(BR )
BR ⊆Ω
≈
sup R
θ−n
BR ⊆Ω
Z
w
|w| log e + R
−
|w(y)|
dy
BR
BR
!
dx < ∞ .
We are now ready for the results in such finer scales of spaces. Here is a classical
result on Riesz potentials, which is actually a rather easy consequence of the offdiagonal version of Marcinkiewicz interpolation theorem and of Theorem 2.1.
Theorem 5.6. Let β ∈ [0, n]; let γ > 1 be such that βγ < n, and let q > 0. We
have
nγ
f ∈ L(γ, q)(Rn ) =⇒ Iβ (f ) ∈ L
, q (Rn ) .
n − βγ
See [106] for additional details. A result of Adams & Lewis is instead
Theorem 5.7 ([9]). Let β ∈ [0, θ); let γ > 1 be such that βγ < θ, and let q > 0.
We have
θq
θγ
θ
n
θ
,
(Rn ) .
f ∈ L (γ, q)(R ) =⇒ Iβ (f ) ∈ L
θ − βγ θ − βγ
We note two important points. In the case γ = q, when Lθ (γ, q) ≡ Lγ,θ , Theorem
5.7 reduces to Theorem 5.1, part (5.4), as obviously expected. On the other hand,
when θ = n, and therefore no Morrey space condition comes into the play, we have
Lθ (γ, q) ≡ L(γ, q) but nevertheless Theorem 5.7 does not reduce to Theorem
5.6. This is not a gap in the theory, but a genuine discontinuity phenomenon
discussed at length, and by mean of counterexamples, in [9, 7]. No surprise that a
similar discontinuity phenomenon will pop-up in the non-linear case too.
The non-linear analog of Theorem 5.7 is now
Theorem 5.8 ([106]). Assume (4.2), and that the measure µ is a function belonging to Lθ (γ, q)(Ω) with γ, θ as in (5.9) and 0 < q ≤ ∞. Then the solution
u ∈ W01,p−1 (Ω) to (4.1) satisfies
θγ
θq
|Du|p−1 ∈ Lθ
,
locally in Ω .
θ−γ θ−γ
Moreover, the local estimate
k|Du|p−1 kLθ (
θγ
θ−γ
≤ cR
θq
, θ−γ
)(BR/2 )
θ−γ
γ
−n
k(|Du| + s)p−1 kL1 (BR ) + ckf kLθ (γ,q)(BR ) ,
holds for every ball BR ⊆ Ω, where c depends only on n, p, L/ν, γ, q.
We just notice that applying the previous result with the choice γ = q, therefore
dropping the Lorentz space scale, we obtain Theorem 5.2 as a particular case.
Taking instead θ = n, therefore dropping the Morrey scale, Theorem 5.8 does
not yield the sharp result for the case of Lorentz spaces, due to the discontinuity
phenomenon described after Theorem 5.7; for this we refer again to [7, 9]. The
sharp version is anyway in Theorem 5.10 below.
Remark 5.7. It is a classical fact that Morrey spaces are not interpolation spaces
- in the usual sense; see the work [18] and related references. The discontinuity
phenomenon described before Theorem 5.8 and emphasized in [7, 9] is probably
linked to the non-interpolation nature os such spaces - intuitively clear when looking
at (5.2). We wonder whether this discontinuity phenomenon could be therefore used
to give another proof of the fact that Morrey spaces are not of interpolation type.
34
GIUSEPPE MINGIONE
Along with the higher integrability of Du comes a result about u.
Theorem 5.9 ([106]). Assume (4.2), and that
(5.18)
1 < γ < θ/p
and
p < θ ≤ n.
Assume that the measure µ is a function belonging to Lθ (γ, q)(Ω) with 0 < q ≤ ∞.
Then the solution u ∈ W01,p−1 (Ω) to the problem (4.1) is such that
θγ
θq
p−1
θ
|u|
∈L
,
locally in Ω .
θ − γp θ − γp
Moreover, the local estimate
k|u|p−1 kLθ (
θq
θγ
θ−γp , θ−γp )(BR/2 )
≤ cR
θ−γp
−n
γ
k(|u| + sR)p−1 kL1 (BR ) + ckf kLθ (γ,q)(BR )
holds for every ball BR ⊆ Ω, with a constant c only depending on n, p, L/ν, γ, θ, q.
We finally conclude with the natural completion of Theorem 5.4. The point
here is that in Theorem 5.4 the information µ ∈ L log L is added to reach the full
integrability (5.13), rather that the weak one (5.16), and acts only at this level.
To have the proper analog of (5.10), that is |Du|p−1 ∈ Lθ/(θ−1),θ , necessitates
to transfer the Morrey density information, which is available only at L1 -scale in
Theorem 5.4, at a full L log L-level, that is we have to assume µ ∈ L log Lθ .
Theorem 5.10 ([106]). Assume (4.2) and that the measure µ is a function belonging to L log Lθ (Ω) with p ≤ θ ≤ n. Then the solution u ∈ W01,p−1 (Ω) to (4.1) is
such that
θ
θ−1 ,θ
|Du|p−1 ∈ Lloc
(Ω) .
Moreover, the local estimate
k|Du|p−1 k
θ
L θ−1
,θ
(BR/2 )
≤ cRθ−1−n k|Du|p−1 kL1 (BR ) + ckf kL log Lθ (BR )
holds for every ball BR ⊆ Ω, with a constant c only depending on n, p, L/ν.
5.4. Pure Lorentz spaces regularity. In this section we abandon the case of
Morrey regularity, turning our attention to regularity results in classical Lorentz
spaces; in other words we are considering here the case θ = n. There is a rather
large literature on the topic, basically going back to the original works of Talenti
[117, 118]. Here we shall present a few results, again from [106], which feature
an alternative approach to the known estimates, and in particular to Talenti’s
one based on symmetrization. The differences are in two respects: the theorems
presented here involve explicit local estimates which previously employed methods symmetrization, truncation - do not immediately yield; second: here the borderline
cases of gradient estimates are covered. Of course, being our approach aimed at
obtaining local estimates, the problem of deriving the best constants in the a priori
estimates, which is solved in [117, 118], becomes here immaterial. This problem
can be anyway faced by symmetrization methods.
The next results is the non-linear version of Theorem 5.6. As described after
Theorem 5.7, we note a difference with respect to Theorem 5.8, in that we have a
higher gain in the second Lorentz exponent here.
Theorem 5.11 ([106]). Assume that (4.2) holds with p < n, and that the measure µ
is a function belonging to L(γ, q)(Ω), with 1 < γ ≤ np/(np − n + p) and 0 < q ≤ ∞;
then the solution u ∈ W01,p−1 (Ω) to the problem (4.1) is such that
nγ
p−1
(5.19)
|Du|
∈L
,q
locally in Ω .
n−γ
TOWARDS A NON-LINEAR CALDERÓN-ZYGMUND THEORY
35
Moreover, the local estimate
k|Du|p−1 kL(
nγ
n−γ
,q )(BR/2 )
≤ cR
n−γ
γ
−n
k(|Du| + s)p−1 kL1 (BR ) + ckf kL(γ,q)(BR )
holds for every ball BR ⊆ Ω, with a constant c only depending on n, p, L/ν, γ, q.
Inclusion (5.19) was already known in the literature [44, 12, 77], except for the
borderline case γ = np/(np − n + p) = (p∗ )0 which was left uncovered; in this case
we are “around the maximal regularity”: Du ∈ L(p, q(p − 1)), locally, whenever
q ∈ (0, ∞]. The problem for the borderline case was raised for the first time in [77]
for q = ∞, and eventually in [12] where inclusion (5.19) is conjectured for q < ∞.
A partial answer in the case q = ∞ has been given by Zhong [123, Theorem 2.30]
and an approach in the case p = 2 has been given by Boccardo [21]. Inclusion (5.19)
is actually straightforward in the case (γ, q) = ((p∗ )0 , p0 ), compare with Example 2.
As far as the integrability of u is concerned we have the optimal local version of
classical Talenti’s estimates [117].
Theorem 5.12 ([106, 117]). Assume (4.2), and that the measure µ is a function
belonging to L(γ, q)(Ω) with 1 < γ < n/p, 0 < q ≤ ∞, the solution u ∈ W01,p−1 (Ω)
to the problem (4.1) is such that
nγ
,q
locally in Ω .
|u|p−1 ∈ L
n − γp
Moreover, the local estimate
nγ
k|u|p−1 kL( n−γp
,q)(BR/2 ) ≤ cR
n−γp
−n
γ
k(|u| + sR)p−1 kL1 (BR ) + ckf kL(γ,q)(BR )
holds for every ball BR ⊆ Ω, with a constant c only depending on n, p, L/ν, γ, q.
5.5. A uniqueness result. We present here a result, whose full proof will appear
in a forthcoming paper, which is related to Theorem 5.5 when considered in the
particular case θ = p, that is when the measure in (4.1) satisfies the decay condition
(5.20)
|µ|(BR ) . Rn−p .
From now on, we shall without loss of generality consider the measure µ as defined
on the whole Rn - for instance letting µ ≡ 0 outside Ω - and we shall assume that
condition (5.20) holds for every possible ball.
In the case (5.20) holds, we saw that, in particular, Du ∈ Mploc (Ω). This is an
interesting borderline result since it happens that, given two distributional solutions
u, v ∈ W01,p−1 (Ω) such that Du, Dv ∈ Mp (Ω), then u ≡ v; this is consequence of
the Lipschitz truncation method [1, 2] applied to measure data problems as in
[51, Theorem 1.2]. Therefore we identify the class of W 1,p−1 functions having
distributional gradient in Mp , that is
(5.21)
X := {u ∈ W01,p−1 (Ω) : Du ∈ Mp (Ω, Rn )} .
The next result, which involves a boundary version of Theorem 5.5, provides a
partial solution to the uniqueness problem.
Theorem 5.13. Assume that Ω is a bounded Lipschitz domain; under the assumptions (4.2) and (5.20) with p ≤ n, there exists a unique solution to the problem
(4.1) in the class X defined in (5.21).
When in the scalar case, this theorem covers the uniqueness one given in [51]
for the case p = n; the assumption of Lipschitz continuity of Ω can be also relaxed
using p-capacitary conditions, or exterior cone conditions, or the like.
An interesting point in the previous theorem is that it provides the first uniqueness result beyond those involving a measure which is absolutely continuous with
respect to the p-capacity [24, 80], in that condition (5.20) does not ensure that µ
36
GIUSEPPE MINGIONE
does not charge p-capacity null sets. On the other hand the previous theorem can
be viewed as a borderline case of such uniqueness theorems. Note indeed that, for
p < n we have
Capp (BR ) ≈ Rn−p
and therefore
|µ|(BR ) . Rn−p+ε
for some ε > 0
0
implies that µ does not charge p-capacity null sets, and actually belongs to W −1,p
by the results in [5].
5.6. Large exponents estimates. The Morrey and Lorentz spaces regularity results presented in this sections have been stated for very weak solutions, that is, for
those problems involving data with low integrability properties. On the other hand,
combining the techniques in [106] with those for instance in [3], one can obtain Morrey and Lorentz spaces estimates also for energy solutions and large integrability
exponents. Here, for the sake of brevity we shall present a couple of model results
concerning the p-Laplacean system, referring to future work for more detailed and
general statements.
The first result concerns Lorentz spaces estimates for the non-homogeneous pLaplacean system with VMO coefficients
(5.22)
div (c(x)|Du|p−2 Du) = div (|F |p−2 F )
0 < ν ≤ c(x) ≤ L .
Theorem 5.14. Let u ∈ W 1,p (Ω, RN ) be a weak solution to the system (5.22) in
Ω with F ∈ Lp (Ω, RN n ), where Ω is a bounded domain in Rn , and such that c(·) ∈
VMO(Ω). There exists ε ≡ ε(n, N, p, ν, L) > 0 such that
F ∈ Lloc (γ, q)(Ω) =⇒ Du ∈ Lloc (γ, q)(Ω)
whenever γ ≥ p − ε , q > 0 .
Finally we consider for simplicity the plain non-homogeneous p-Laplacean system
(5.23)
div (|Du|p−2 Du) = div (|F |p−2 F ) ,
and we deal with Morrey space regularity:
Theorem 5.15. Let u ∈ W 1,p (Ω, RN ) be a weak solution to the system (5.23)
in Ω with F ∈ Lp (Ω, RN n ), where Ω is a bounded domain in Rn . There exists
ε ≡ ε(n, N, p, ν, L) > 0 such that
γ,θ
F ∈ Lγ,θ
loc (Ω) =⇒ Du ∈ Lloc (Ω)
whenever γ ≥ p − ε , θ ∈ (0, n] .
6. Beyond gradient integrability
The quasilinear elliptic problems we are dealing with are second order ones,
and it is therefore natural to look for the higher differentiability of the gradient
of solutions, in the style of linear results as (2.2)-(2.3). In fact the standard CZ
theory prescribes in (2.2) the existence of second derivatives of solution. It is
quite surprising that for non-linear measure data problems the first results in this
direction have been obtained only recently in [105]. In this section we are briefly
describing the results obtained there, and later completed in [106].
6.1. Measure data. To understand what kind of differentiability results we may
expect let us go back again to the basic equation (2.1). It is clear that when γ = 1
we cannot expect that D2 u ∈ L1 , which is in general false. The viewpoint pursued
in [105] is that although we cannot have second derivatives of solutions, we “almost”
have them. At this point, to make precise this fact, we need the following:
TOWARDS A NON-LINEAR CALDERÓN-ZYGMUND THEORY
37
Definition 6.1. For a bounded open set A ⊂ Rn and k ∈ N, parameters α ∈ (0, 1)
and q ∈ [1, ∞), the fractional Sobolev space W α,q (A, Rk ) is defined requiring that
w ∈ W α,q (A, Rk ) iff the following Gagliardo-type norm is finite:
q1
Z
q1 Z Z
|w(x) − w(y)|q
q
dx dy
kwkW α,q (A) :=
|w(x)| dx
+
n+αq
A
A A |x − y|
=: kwkLq (A) + [w]α,q;A .
Fractional Sobolev spaces are a particular case of Besov spaces, and they are
interpolations spaces, and they can also be realized as trace spaces; for basic aspects
of the theory we refer for instance to [120]. We shall only need to think of W α,q functions as those function having “derivatives of order α”, in turn integrable with
exponent q. Roughly writing, this means
Z
Z Z
|w(x) − w(y)|q
dx
dy
≈
|Dα w|q dz
0 < α < 1.
n+αq
A
A A |x − y|
The idea is now rather clear: the viewpoint developed in [105] says that if is not
possible to have second order derivatives for solutions to measure data problems,
we can still look for fractional differentiability of solutions. For this we need of
course assumptions stronger that those in (4.2), since in order to prove higher
differentiability of solutions, we have in some sense to “differentiate” the equation;
we therefore prescribe that the vector field a(·, ·) is itself differentiable. We assume
that

p−2

ν(s2 + |z1 |2 + |z2 |2 ) 2 |z2 − z1 |2 ≤ ha(x, z2 ) − a(x, z1 ), z2 − z1 i




p−2


|a(x, z2 ) − a(x, z1 )| ≤ L(s2 + |z1 |2 + |z2 |2 ) 2 |z2 − z1 |
(6.1)
p−1


|a(x, z) − a(x0 , z)| ≤ L|x − x0 |(s2 + |z|2 ) 2





|a(x, 0)| ≤ Lsp−1 ,
hold for every z1 , z2 ∈ Rn , x, x0 ∈ Ω, while the structure constants are as in (4.3).
To present the higher differentiability results we start dealing with the case p = 2.
Theorem 6.1 ([105]). Under the assumptions (6.1) with p = 2, there exists a
solution u ∈ W01,1 (Ω) to the problem (4.1) such that
1−ε,1
Du ∈ Wloc
(Ω, Rn )
(6.2)
for every ε > 0 .
Moreover, for every ε > 0 there exists a constant c ≡ c(n, L/ν, ε) such that the
following Caccioppoli type inequality holds for every ball BR b Ω:
Z
Z
Z
|Du(x) − Du(y)|
c
dx dy ≤ 1−ε
(|Du| + s) dx + cRε |µ|(BR ) .
|x − y|n+1−ε
R
BR BR
BR
2
2
The previous one is the natural-scaling energy estimate associated to the regularity result in (6.2).
Remark 6.1. Again we remark that in the linear case (2.1) the statement of
Theorem 6.1 is a consequence of the linear representation formula (2.5) and of
some lengthy computation. As usual, the point in Theorem 6.1 is that the problem
in question is non-linear a no representation formula is available.
Remark 6.2. As in Section 5, when stating our regularity results for measure
data problems, we shall simultaneously state existence and regularity results, since
the uniqueness of distributional solutions does not hold. On the other hand the
regularity claimed holds for every approximate solution to problem (4.1), in
the sense of the approximation scheme described in Section 4.1.
38
GIUSEPPE MINGIONE
We now switch to the case p 6= 2, which needs a preliminary discussion on
the higher differentiability for solutions to the homogeneous p-Laplacean equation
(3.12). For this we recall that the existence of second order derivatives of solutions
is not known, due to the degeneracy of the problem with respect to the gradient
variable. On the other hand fractional derivatives of solutions naturally appear
[102]
2
,p
p
(Ω, Rn ) .
Dv ∈ Wloc
(6.3)
Moreover let us briefly recall a sort of non-linear version of the classical uniformization of singularities principles in complex analysis and in the theory of quasiconformal mappings. This, in its classical formulation, asserts that certain functions,
which are not analytic, become analytic when raised to a suitably high power.
Something similar happens to solutions to (3.12), which are obviously related to
the of quasiconformal mappings [71], and has been first pointed out in the work of
Uhlenbeck [121] who considered the function
V0 (Dv) := |Dv|
p−2
2
Dv ,
proving that full differentiability holds
1,2
V0 (Dv) ∈ Wloc
(Ω, Rn ) .
(6.4)
We shall now see that both (6.3)-(6.4) have e precise analog in the context of
measure data problems. We start by the following measure data counterpart of
(6.3).
Theorem 6.2 ([105]). Under the assumptions (6.1) with p ∈ [2, n], there exists a
solution u ∈ W01,p−1 (Ω) to the problem (4.1) such that
1−ε
p−1
Du ∈ Wloc
(6.5)
,p−1
(Ω, Rn )
for every ε > 0 .
Moreover, for every ε > 0 there exists a constant c ≡ c(n, p, L/ν, ε) such that the
following Caccioppoli type inequality holds for every ball BR b Ω:
Z
Z
Z
|Du(x) − Du(y)|p−1
c
(|Du| + s)p−1 dx + cRε |µ|(BR ) .
dx
dy
≤
n+1−ε
1−ε
|x
−
y|
R
BR BR
BR
2
2
Remark 6.3. Let us immediately observe that when adopting the scale of fractional
Sobolev spaces the result in (6.5) is optimal: for every p ∈ [2, n], we cannot take
ε = 0 in (6.5). In fact let us recall that Sobolev embedding theorem in the fractional
case asserts
nq
W α,q ⊂ L n−αq
provided αq < n ,
and in particular
(6.6)
1
W p−1 ,p−1 ⊂ L
n(p−1)
n−1
.
Therefore, assuming by contradiction that
(6.7)
1
Du ∈ W p−1 ,p−1
we would get
n(p−1)
Du ∈ L n−1 ,
which is in general false as shown by the non-linear Green’s function in (4.7). We
recall that the uniqueness of the non-linear Green’s function Gp (·), as explained
in Section 4.4, allows to use Gp (·) to test the optimality of the regularity results
presented. Another demonstration of the fact that the limiting differentiability
exponent
1
(6.8)
p−1
TOWARDS A NON-LINEAR CALDERÓN-ZYGMUND THEORY
39
is the natural one when considering (6.5) will appear, in more extended context, in
Section 6.4 below.
The measure data counterpart of (6.4) is formulated via the map
Vs (Du) := (s2 + |Du|2 )
(6.9)
p−2
4
Du .
Theorem 6.3 ([105]). Under the assumptions (6.1) with p ∈ [2, n], there exists a
solution u ∈ W01,p−1 (Ω) to the problem (4.1) such that
p(1−ε) 2(p−1)
, p
2(p−1)
Vs (Du) ∈ Wloc
(Ω, Rn )
for every ε > 0 .
Moreover, a local estimate of the type (6.6) holds for Vs (Du).
The same argument of Remark 6.3 applies to get the optimality of Theorem 6.3.
Indeed, as
p
|Vs (Du)| ≈ |Du| 2 + 1
(6.10)
assuming
p
,
2(p−1)
p
2(p−1)
Vs (Du) ∈ Wloc
would give, via Sobolev embedding theorem, that
np(p−1)
Vs (Du) ∈ Llocn−1 ,
and taking (6.10) into account we would get (6.7) again, which is impossible as
shown by the fundamental solution Gp (·). We notice that both Theorem 6.2 and
6.3 reduce to Theorem 6.1 in the case p = 2, and also Vs (Du) ≡ Du holds for p = 2.
Remark 6.4. As in the previous section, the results presented here are local.
Their extension up to the boundary can be nevertheless achieved assuming suitable
regularity on the boundary, as for instance C 1,α regularity of ∂Ω.
We now switch to higher differentiability results for solutions to problems (4.1)
when the measure is a function with additional integrability properties.
Theorem 6.4 ([106]). Under the assumptions (6.1), assume that the measure
µ is a function belonging to Lγ (Ω) with γ as in (4.14). Then the solution u ∈
1,γ(p−1)
W0
(Ω) to (4.1) is such that
1−ε
p−1
Du ∈ Wloc
,γ(p−1)
(Ω, Rn ),
for every ε > 0 .
Moreover, for every ε > 0 there exists a constant c ≡ c(n, p, L/ν, γ, ε) such for
every ball BR ⊆ Ω it holds that
Z
Z
Z
|Du(x) − Du(y)|γ(p−1)
c
(|Du| + s)γ(p−1) dx
dx dy ≤
n+γ(1−ε)
γ(1−ε)
|x
−
y|
R
BR/2 BR/2
BR
Z
γε
+cR
|f |γ dx .
BR
Of course for γ = 1 the previous theorem formally agrees with Theorem 6.2. The
previous result raises an open problem, see Section 6.7 below.
We finally conclude this section with a pointwise result. Not so much is in fact
known concerning the pointwise behavior of the gradient of solutions to measure
data problems. On the other hand Theorems 6.2-6.4 allow to estimate the size of
the singular set of Du. Let use recall that the set of non-Lebesgue points of an
L1 (Ω)-function w is defined as
(
)
Z
Σw := x ∈ Ω : lim inf −
|w(y) − (w)x,ρ | dy > 0 or lim sup |(w)x,ρ | = ∞
ρ&0
B(x,ρ)
ρ&0
40
GIUSEPPE MINGIONE
for some q ≥ 1. A classical result in potential theory - see [8] or [102] for a fast
proof - asserts that, denoting by dim(A) the Hausdorff dimension of set A, we have
w ∈ W α,q =⇒ dim(Σw ) ≤ n − αq ,
(6.11)
provided αq < n .
Applying (6.11) together with Theorems 6.1-6.4 we conclude with
Corollary 6.1. Under the assumptions (6.1), assume that the measure µ is a
function belonging to Lγ (Ω) with γ as in (4.14), or that the measure µ is a just a
Borel measure with finite mass, and in this case let γ = 1. Let Σu denote the set
of non-Lebesgue points of the solution to (4.1) found in the previous theorems, in
the sense of
(
Z
Σu := x ∈ Ω : lim inf −
|Du(y) − (Du)x,ρ |γ(p−1) dy > 0
ρ&0
B(x,ρ)
)
or
lim sup |(Du)x,ρ | = ∞
.
ρ&0
Then its Hausdorff dimension dim(Σu ) satisfies
dim(Σu ) ≤ n − γ .
6.2. Sobolev-Morrey estimates. To conclude the results on the higher differentiability of the gradient we reproduce, in the non-linear case, the standard linear
results concerning equations with a right hand side in Morrey spaces. Such linear
results are classical [37, 31, 62], but for a general up-dated presentation we refer
to the recent work of Lieberman [94]. To review the linear result we shall confine
ourselves to the case of the Poisson equation (2.1); as a matter of fact CalderónZygmund theory extends up to cover Morrey spaces properties for highest order
derivatives in the sense that
f ∈ Lγ,θ =⇒ D2 u ∈ Lγ,θ
provided γ > 1 and θ ∈ [0, n] .
This means
Z
(6.12)
|D2 u|γ dx . Rn−θ .
BR
In order to provide the right reformulation of (6.12) in the context of non-linear
measure data problems, where the existence of second order derivatives of solutions
is missing, we have to recall the definition of so called Sobolev-Morrey spaces [35,
36, 9, 101].
Definition 6.2. For a bounded open set A ⊂ Rn and k ∈ N, parameters α ∈ (0, 1)
and q ∈ [1, ∞), and θ ∈ [0, n], a measurable map w : Ω → Rk belongs to the
Sobolev-Morrey space W α,q,θ (A, Rk ) iff w ∈ W α,q (A, Rk ) and moreover
Z Z
|w(x) − w(y)|q
θ−n
dx dy =: [w]qW α,q,θ (A) < ∞ .
sup R
n+αq
BR ⊆A
BR BR |x − y|
At this point, taking into account the result of Theorem 6.1 - and this means we
are again temporarily restricting to the case p = 2 - the natural replacement for
non-linear measure data problems of the standard result in (6.12) would read as
Z Z
|Du(x) − Du(y)|
dx dy . Rn−θ
for every ε > 0 ,
|x − y|n+1−ε
BR BR
that is the Morrey density information is fully transferred to the highest order
derivatives exactly as in (6.12). Roughly again, the previous inequality stands for
Z
|D2−ε u| dx . Rn−θ
for every ε > 0 ,
BR
TOWARDS A NON-LINEAR CALDERÓN-ZYGMUND THEORY
41
which looks like the right extension to (6.12) for the limit case γ = 1. The general
result for measure data is indeed
Theorem 6.5 ([105]). Under the assumptions (6.1), assume that µ ∈ L1,θ (Ω) in
the sense of (5.15), with θ ∈ [p, n]. Then there exists a solution u ∈ W01,p−1 (Ω) to
(4.1) which is such that
1−ε
p−1
Du ∈ Wloc
(6.13)
,p−1,θ
(Ω, Rn ),
for every ε > 0 .
Moreover, for every ε > 0 there exists a constant c ≡ c(n, p, L/ν, θ, ε) such for every
ball BR b Ω it holds that
[Du]
W
1−ε ,p−1,θ
p−1
(B
R/2 )
≤ cR
θ−(n+1)+ε
p−1
k(|Du| + s)kLp−1 (BR )
ε
1
+cR p−1 kµkLp−1
1,θ (B ) .
R
For the case of right hand side being measures which are also functions we have
Theorem 6.6 ([106]). Under the assumptions (6.1), assume that the measure µ
is a function belonging to Lγ,θ (Ω) with γ, θ as in (5.9). Then the solution u ∈
W01,p−1 (Ω) to (4.1) is such that
1−ε
p−1
Du ∈ Wloc
(6.14)
,γ(p−1),θ
(Ω, Rn ),
for every ε > 0 .
Moreover, for every ε > 0 there exists a constant c ≡ c(n, p, L/ν, γ, θ, ε) such that
for every ball BR ⊆ Ω it holds that
[Du]
W
1−ε ,γ(p−1),θ
p−1
(B
R/2 )
≤ cR
θ−γ(n+1)+γε
γ(p−1)
k(|Du| + s)kLp−1 (BR )
ε
1
+cR p−1 kf kLp−1
γ,θ (B ) .
R
Remark 6.5 (Morrey spaces properties vs fractional differentiability). The presence of ε > 0 in (6.13)-(6.14) makes such inclusions not optimal when looking at
scaling properties of local estimates - compare with the possible limit cases discussed in the last three sections. As a consequence inclusions (6.13)-(6.14) may be
slightly improved in
σ,γ(p−1),m
Du ∈ Wloc
(Ω, Rn )
for
whenever
σ<
m := θ − γ[1 − σ(p − 1)] < θ ,
1
.
p−1
For this we refer to [106, Section 8].
6.3. Other ways to measure fractional differentiability. Fractional Sobolev
spaces as briefly described in Definition 6.1 provide one possible way to express the
fact that a map is differentiable only up to a limited extent. There are of course
other ways, and, accordingly, the results in [105, 106] can be reformulated using
new scales of function spaces. It is at this point a matter of taste to formulate the
fractional differentiability properties using one space rather than another. Here we
present two possible other such scales.
Definition 6.3. For a bounded open set A ⊂ Rn and k ∈ N, parameters α ∈
(0, 1) and q ∈ [1, ∞), the Nikolskii space N α,q (A, Rk ) is defined requiring that
w ∈ N α,q (A, Rk ) iff
Z
q1
Z
q1
|w(x + h) − w(x)|q
q
+ sup
dx
kwkN α,q (A) :=
|w(x)| dx
|h|αq
x,x+h∈A
A
A
(6.15)
=: kwkLq (A) + [w]N α,q ;A < ∞ ,
42
GIUSEPPE MINGIONE
where of course |h| =
6 0.
Nikoslkij condition (6.15) actually prescribes that a function is Hölder continuous
with respect to the Lq -norm, and in other words we have that
kw(· + h) − w(·)kLq ≤ [w]N α,q |h|α .
The following inclusion holds:
W α,q & N α,q $ W α,q−ε
∀ ε > 0.
We note that taking q = ∞ corresponds to take C 0,α functions. Both fractional
Sobolev spaces and Nikolskii spaces are particular instance of a more general scale
of spaces called Besov spaces and denoted by
Btq,α ,
1≤q≤∞
0<t≤∞
α ∈ [0, 1] ,
where the second integrability index t refines the first one q exactly as for Lorentz
spaces L(q, t). In particular, with some difference with the standard notation available in the literature, we denote
Bqq,α ≡ W q,∞
Bqq,α ≡ W q,∞
q,α
B∞
≡ N q,∞
Btq,0 ≡ L(q, t) .
We shall not deal with Besov spaces here, referring the reader to one of the many
textbooks available on them; see for instance [120].
There is yet another way to describe fractional differentiability properties of
functions, that cannot be covered by Besov spaces, and that presents the nice
feature of involving pointwise estimates for differences. This way is due DeVore &
Sharpley [45], and is described in the following:
Definition 6.4 (DeVore & Sharpley [45]). For a bounded open set A ⊂ Rn and
k ∈ N, parameters α ∈ (0, 1) and q ∈ [1, ∞), the space Cqα (A, Rk ) is defined
requiring that w ∈ Cqα (A, Rk ) iff there exists a non-negative function
g ∈ Lq (A)
such that
(6.16)
|w(x) − w(y)| ≤ (g(x) + g(y)) |x − y|α
holds whenever x, y ∈ A. Accordingly
kwkCqα (A) := kwkLq (A,Rk ) + inf kgkLq (A) .
g
The previous definition, as explained by DeVore & Sharpley, is motivated by the
following fact; when considering a map w ∈ W 1,q (Rn ) it follows immediately that
|w(x) − w(y)| ≤ (g(x) + g(y)) |x − y|
with
g(x) = cM (|Du|)(x)
for a suitable absolute constant c depending only on the space dimension n; see
also [1, 2]. Moreover, when starting from the usual fractional Sobolev space W α,q
one can reproduce the same argument using, instead of the maximal function of
Du, the so called fractional sharp maximal function
Z
]
−α
Mα (w)(x) := sup R − |w − (w)BR | dx .
x∈BR
BR
Although the spaces defined by DeVore & Sharpley cannot be described using
the Besov scale, as already mentioned above, they provide an essentially equivalent
characterization of fractional differentiability as established by the following strict
inclusions:
W α,q $ Cqα $ N α,q .
TOWARDS A NON-LINEAR CALDERÓN-ZYGMUND THEORY
43
Therefore (6.16) gives the possibility of still measuring the differences of the map
w is a Hölder scale, but taking into account the possibility that at some point the
“constants blow-up” like a Lq -function. Clearly we have that
α
C∞
≡ C 0,α .
With the above definitions at our disposal we may reformulate Theorem 6.2 as
follows:
Theorem 6.7. Under the assumptions (6.1) with p ∈ [2, n], there exists a solution
u ∈ W01,p−1 (Ω) to the problem (4.1) such that
1−ε
p−1
Du ∈ Nloc
and
,p−1
(Ω, Rn )
1−ε
p−1
n
Du ∈ Cp−1
loc (Ω, R )
In particular, for p = 2 we have
(6.17)
for every ε > 0 ,
1−ε,1
Du ∈ Nloc
(Ω, Rn )
and
for every ε > 0 .
n
Du ∈ C11−ε
loc (Ω, R ) .
Similar reformulations can be obviously given for Theorems 6.3 and 6.4. Suitable
local a priori estimates can be also given following again [105, 106]. The use of
definitions as in (6.16) turn out to be very useful to describe certain non-linear
variational problems exhibiting rough coefficients [87, 88].
6.4. Weak Schauder theory - separation of scales. The previous section opens
the way to draw what we may call a “weak Schauder theory”. Let us consider the
linear elliptic equation
(6.18)
div (A(x)Du) = f ,
or even the non-linear one
(6.19)
div a(x, Du) = f ,
assuming
(
(6.20)
ν|λ|2 ≤ hDz a(x, z)λ, λi ,
|Dz a(x, z)| ≤ L
|a(x, z) − a(x0 , z)| ≤ L|x − x0 |α (1 + |z|) ,
whenever z, λ ∈ Rn , where α ∈ (0, 1), 0 < ν ≤ L, and
f ∈ L∞
loc .
In both cases we have that Du ∈ C 0,α locally, that is
(6.21)
|Du(x + h) − Du(x)| ≤ c|h|α ,
whenever x, y ∈ Ω0 b Ω, and c depends also on dist(Ω0 , ∂Ω). When referred to the
equation (6.18), and coupled with related a priori estimates, the previous result
essentially gives back the classical Schauder regularity theory for linear elliptic
equations. We note that with the notation of the previous section (6.21) can be
written as
α,∞
Du ∈ Nloc
.
For such a result see for instance [61] and related techniques.
We now want to emphasize a certain “separation of scales principle” asserting
that when decreasing the integrability properties of the right hand side in (6.18)(6.19), the rate of differentiability of the gradient is preserved, provided being red
at the proper integrability scale, that is the one established by the right hand side
datum f . As a first example we recall a result from [102], which can be easily
extended up to stating that
(6.22)
α,2
f ∈ L2loc =⇒ Du ∈ Nloc
.
44
GIUSEPPE MINGIONE
On the other hand, modifying the proofs of [105] taking into account [41], leads to
establish that, for the approximate solution of (4.1) found via the approximation
method defined via (4.10), the following implication holds:
α−ε,1
f ∈ L1loc =⇒ Du ∈ Nloc
(6.23)
for every ε > 0 .
The appearance of ε depends on the fact that we are in the borderline L1 -case.
When α = 1 we go back to Theorem 6.7 (there we have to set p = 2, of course). In
order to get rid of ε in a possible limiting case we refer to Section 6.5.
Another example of such separation of scales principle concerns the optimality
of the exponent (6.8). Let consider the radial solution to
4p u = 1
(6.24)
such u ≡ 1 on ∂B1 . We have
p
u(x) ≈ |x| p−1 ,
and therefore
1
p−1
Du ∈ Nloc
,∞
,
and in fact the right hand side of the equation in (6.24) belongs to L∞ . In other
words, the rate of differentiability for the solutions found for in Theorems 6.2 and 6.7
is the same of the one of the solution to (6.24), what it changes is the integrability
degree with which it is measured.
The viewpoint outlined in this section is fruitful when studying dimension reductions properties of singular sets of elliptic and variational vectorial problems
[104, 102]. As a matter of fact, the result in (6.22) holds for general elliptic systems and allows, as originally shown in [102] to assert that the Hausdorff dimension
of the singular set of solutions does not exceed n − 2α. At the end it comes up
that when considering vectorial problems, the gradient of solutions are not Hölder
continuous anymore, but on the other hand they are still Hölder continuous with
respect to the L2 norm, and this is linked to the dimension of the singular set.
6.5. Further issues on gradient differentiability - measure data. As pointed
out above the results of Theorems 6.1-6.2 are sharp when considered in the scale
of fractional Sobolev spaces, in the sense that we cannot allow ε = 0 in (6.2)-(6.5),
otherwise Sobolev embedding theorem would give too much integrability (6.6). On
the other hand, since “behind every ε there is a limiting space”, one may speculate
about the existence of possible limit cases beyond the results (6.2)-(6.5). Of course
this, as we shall see in a few lines, leads us to consider different function spaces, and
some readers may argue that this is already too far in the spirit of generalizations
for their own’s sake. On the other hand there is perhaps still something interesting
to observe here. More precisely, let us concentrate for the moment on the case
p = 2. It is clear in general, that, even for solutions to the basic linear equations
4u = f ∈ L1
4u = µ
we cannot assert
u ∈ W 2,1 ,
and not even locally. We see that the problem lies exactly in the initial lack of
integrability of the solutions, and not in an additional lack of full differentiability.
Indeed we notice that the lack of integrability starts at a very primitive level in
that, looking at the fundamental solution G2 (·), defined in (2.6), we have that (for
simplicity we restrict to the case n > 2)
n
n
n−2
n−2
(Rn ) \ Lloc
(Rn )
G2 ∈ Mloc
n
n
n−1
n−1
and DG2 ∈ Mloc
(Rn ) \ Lloc
(Rn )
TOWARDS A NON-LINEAR CALDERÓN-ZYGMUND THEORY
while, after differentiating twice we have
1
,
|D2 G2 (x)| .
|x|n
45
x 6= 0 ,
and so this lack of integrability propagates up to second order derivatives
2
2
D2 G2 ∈ M1loc (Rn , Rn ) \ L1loc (Rn , Rn ) .
(6.25)
We may therefore suspect, as stated above, that the absence of second derivatives
for solutions does not depend on a lack in the differentiability scale, but rather on
a lack on the integrability scale. We shall try to make this concept rigorous now.
There is now a problem in assertion (6.25): what kind of derivatives are those
in (6.25)? Certainly, not of distributional type. At the moment they are just the
usual, good old pointwise derivatives, which exist, in the case of the fundamental
solution, at every point but the origin. Therefore, in order to look for a way of
formulating a corresponding result for general approximate solutions to (4.1) we
shall propose two different approaches. Both of them are reflect the fact that we
are actually meeting a double borderline case: L1 -data and a limiting function
space, that is M1 , which does not imply integrability of its elements.
The first method deals with a priori regularity estimates for the approximate
solutions uk ∈ W01,2 (Ω) defined in (4.10). Let us recall that here it is p = 2
and therefore we have uk ∈ W02,2 (Ω). For such solutions we wonder whether the
following uniform estimates in k ∈ N:
kD2 uk kM1 (Ω0 ) ≤ c(Ω0 )
whenever Ω0 b Ω is an open subset. Note that the second derivatives in the previous
line are distributional, but, since the bound is uniform only in the space M1 we
cannot pass k → ∞ and use compactness in such space.
The other proposed way is more direct and concerns the solutions itself and
involves the definition of further function spaces.
Definition 6.5. For a bounded open set A ⊂ Rn and k ∈ N, parameters α ∈
(0, 1) and q ∈ [1, ∞), the Marcinkiewicz-Nikolskii space MN α,q (A, Rk ) is defined
requiring that w ∈ MN α,q (A, Rk ) iff w ∈ Lq (A, Rk ) and moreover
sup λq |{x ∈ A : |w(x + h) − w(x)| > λ}| . |h|α .
λ>0
In other words we measure the Hölder continuity of w using the Mq norm rather
than, as in the usual case of the standard Nikolskii spaces N α,q , the Lq one. The
point is one trying to prove or disprove that
1,1
Du ∈ MNloc
(Ω, Rn ) ,
as a borderline version of (6.17).
Accordingly, under the assumptions (6.20) a possible borderline version of (6.23)
could be
α,1
Du ∈ MNloc
(Ω, Rn ) .
For the case p > 2 one may of course wonder if
1
p−1
Du ∈ MNloc
,p−1
(Ω, Rn ) .
6.6. Non-linear uniformization of singularities. We now switch to a different
issue, concerning the case p > 2. We have seen that passing from the gradient Du
to the map Vs (Du), defined in (6.9), allows to gain differentiability, while obviously
loosing in integrability; compare Theorems 6.2 and 6.3. As noted there, this is a sort
of non-linear version of the uniformization principle from Complex Analysis, stating
that raising a function to a suitable power improves its smoothness properties, while
46
GIUSEPPE MINGIONE
obviously worsening its integrability ones. As a matter of fact, taking for instance
s = 0, we have that
p
|V0 (Du)| = |Du| 2 .
In fact when passing to Vs (Du) in Theorem 6.2 we gain p/2 in differentiability, and
loose 2/p in integrability.
We now look for an improved version of such principle allowing us to find other
non-linear quantities which exhibit maximal differentiability properties. Let us
start with an example; recall that when considering the equation
div F = f
for instance in the ball B1 , with the related compatibility condition
Z
f (x) dx = 0 ,
B1
we can find a solution F such that
f ∈ Lq (B1 ) =⇒ F ∈ W01,q (B1 , Rn )
when q > 1 .
This is usually known as Bogowski’s lemma; for more results and references see also
[30].
Keeping this example in mind, when looking at the equation
div (|Du|p−2 Du) = f
(6.26)
it is therefore natural to consider the whole quantity appearing under the divergence
symbol and proving first that
(6.27)
1−ε,1
1−ε,1
|Du|p−2 Du ∈ Wloc
(Ω, Rn ) ∩ Nloc
(Ω, Rn )
for every ε > 0 ,
and eventually that
1,1
|Du|p−2 Du ∈ MNloc
(Ω, Rn ) .
Note that in the case p = 2 the result in (6.27) would give Theorem 6.1 back.
Moreover note that
||Du|p−2 Du| = |Du|p−1 ,
and therefore note that, comparing Theorem 6.1 and (6.27), we gain p−1 in differentiability and loose 1/(p − 1) in integrability. Note also that considering |Du|p−2 Du
rather that V0 (Du) allows to gain more smoothness since for p ≥ 2 we have that
p − 1 ≥ p/2.
6.7. Further issues on gradient differentiability - function data. The last
issues we treat are concerned with the case when of a right hand side f in (4.1)
belongs to Lγ , with
np
.
1 < γ ≤ (p∗ )0 =
np − n + p
In this case we can take as a model the results available for the linear equation
4u = f .
In the general non-linear case it is clear to ask for the following improvement of
Theorem 6.4:
1
p−1
Du ∈ Wloc
,γ(p−1)
(Ω, Rn ) .
While we may also wonder whether, for solutions to (6.26), it may hold that
1,γ
|Du|p−2 Du ∈ Wloc
(Ω, Rn ) .
TOWARDS A NON-LINEAR CALDERÓN-ZYGMUND THEORY
47
References
[1] Acerbi E. & Fusco N.: Semicontinuity problems in the calculus of variations. Arch. Ration. Mech. Anal. 86 (1984), 125–145.
[2] Acerbi E. & Fusco N.: An approximation lemma for W 1,p functions. Material instabilities in
continuum mechanics (Edinburgh, 1985–1986), 1–5, Oxford Sci. Publ., Oxford Univ. Press,
New York, 1988.
[3] Acerbi E. & Mingione G.:
Gradient estimates for the p(x)-Laplacean system.
J. reine ang. Math. (Crelles J.) 584 (2005), 117–148.
[4] Acerbi E. & Mingione G.: Gradient estimates for a class of parabolic systems. Duke Math.
J. 136 (2007), 285–320.
[5] Adams D. R.: Traces of potentials arising from translation invariant operators. Ann. Scuola
Norm. Sup. Pisa Cl. Sci. (III) 25 (1971), 203–217.
[6] Adams D. R.: A note on Riesz potentials. Duke Math. J. 42 (1975), 765–778.
[7] Adams D. R.: Lectures on Lp -potential theory. Department of Mathematics, University of
Umea, 1981.
[8] Adams D. R. & Hedberg L. I.: Function spaces and potential theory. Grundlehren der Mathematischen Wissenschaften 314. Springer-Verlag, Berlin, 1996.
[9] Adams D. R. & Lewis J. L.: On Morrey-Besov inequalities. Studia Math. 74 (1982), 169–182.
[10] Adams R.A.: Sobolev Spaces. Academic Press, New York, 1975.
[11] Alberico A. & Cianchi A.: Optimal summability of solutions to nonlinear elliptic problems.
Nonlinear Analysis TMA 74 (2007), 1775–1790.
[12] Alvino A. & Ferone V. & Trombetti G.: Estimates for the gradient of solutions of nonlinear
elliptic equations with L1 -data. Ann. Mat. Pura Appl. (IV) 178 (2000), 129-142.
[13] Auscher P. & Martell J.M.: Weighted norm inequalities, off-diagonal estimates and elliptic
operators. Part I: General operator theory and weights. Adv. Math. 212 (2007), 225–276.
[14] Auscher P. & Martell J.M.: Weighted norm inequalities, off-diagonal estimates and elliptic
operators. Part II: Off-diagonal estimates on spaces of homogeneous type. J. Evolution Equat.
7 (2008), 265–316.
[15] Auscher P. & Martell J.M.: Weighted norm inequalities, off-diagonal estimates and elliptic
operators. Part III: Harmonic analysis of elliptic operators. J. Functional Analsys 241 (2008),
703–706.
[16] Benilan P. & Boccardo L. & Gallouët T. & Gariepy R. & Pierre M. & Vázquez J. L.: An
L1 -theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Scuola
Norm. Sup. Pisa Cl. Sci. (IV) 22 (1995), 241–273.
[17] Bennet C. & Sharpley R.: Interpolation of operators. Pure Appl. Math. series, 129. Academic
Press, 1988.
[18] Blasco O. & Ruiz A. & Vega L.: Non-interpolation in Morrey-Campanato and block spaces.
Ann Scu. Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), 31–40.
[19] Boccardo L.: The role of truncates in nonlinear Dirichlet problems in L1 .Nonlinear partial
differential equations; Pitman Res. Notes Math. Ser., 343 42–53.
[20] Boccardo L.: Problemi differenziali ellittici e parabolici con dati misure [Elliptic and parabolic
differential problems with measure data]. Boll. Un. Mat. Ital. A (7) 11 (1997), 439–461.
[21] Boccardo L.: Quelques problemes de Dirichlet avec donneées dans de grand espaces de
Sobolev [Some Dirichlet problems with data in large Sobolev spaces]. C. R. Acad. Sci. Paris
Sér. I Math. 325 (1997) 1269-1272.
[22] Boccardo L. & Gallouët T.: Nonlinear elliptic and parabolic equations involving measure
data. J. Funct. Anal. 87 (1989), 149–169.
[23] Boccardo L. & Gallouët T.: Nonlinear elliptic equations with right-hand side measures.
Comm. Partial Differential Equations 17 (1992), 641–655.
[24] Boccardo L. & Gallouët T. & Orsina L.: Existence and uniqueness of entropy solutions for
nonlinear elliptic equations with measure data. Ann. Inst. H. Poincarè Anal. Non Lináire
13 (1996), 539–551.
[25] Bögelein V.: Higher integrability for weak solutions of higher order degenerate parabolic
systems. Ann. Acad. Sci. Fenn. Ser. A I, to appear.
[26] Bögelein V.: Very weak solutions of higher order degenerate parabolic systems. Preprint
2008.
[27] Bögelein V. & Duzaar F. & Mingione G.: Degenerate problems with irregular obstacles.
Preprint 2008.
[28] Byun S.S. & Wang L.: Gradient estimates for elliptic systems in non-smoth domains.
Math. Ann., to appear.
[29] Bojarski B. & Iwaniec T.: Analytical foundations of the theory of quasiconformal mappings
in Rn . Ann. Acad. Sci. Fenn. Ser. A I Math. 8 (1983), 257–324.
48
GIUSEPPE MINGIONE
[30] Bourgain J. & Brezis H.: On the equation div Y = f and application to control of phases.
J. Am. Math. Soc., 16 (2002), 393-426.
[31] Caffarelli L.: Elliptic second order equations. Rend. Sem. Mat. Fis. Milano 58 (1988), 253–
284.
[32] Caffarelli L.: Interior a priori estimates for solutions of fully nonlinear equations. Ann. of
Math. (2) 130 (1989), no. 1, 189–213.
[33] Caffarelli L. & Cabré X.: Fully nonlinear elliptic equations. American Mathematical Society
Colloquium Publications, 43.
[34] Caffarelli L. & Peral I.: On W 1,p estimates for elliptic equations in divergence form.
Comm. Pure Appl. Math. 51 (1998), 1–21.
[35] Campanato S.: Proprietà di inclusione per spazi di Morrey. Ricerche Mat. 12 (1963), 67–86.
[36] Campanato S.: Proprietà di una famiglia di spazi funzionali. Ann. Scuola Norm. Sup. Pisa
(III) 18 (1964), 137–160.
[37] Campanato S.: Equazioni elittiche non variazionali a coefficienti continui. Ann. Mat. Pura
Appl. (IV) 86 (1970), 125–154.
[38] Coifman R.: A real variable characterization of H p . Studia Math. 51 (1974), 269–274.
[39] Chiarenza F. & Frasca M. & Longo P.: W 2,p -solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients. Trans. Amer. Math. Soc. 336 (1993),
841–853.
[40] Cirmi G. R. & Leonardi S.: Regularity results for the gradient of solutions to linear elliptic
equations with L1,λ data. Ann. Mat. Pura e Appl. (IV) 185 (2006), 537-553.
[41] Cirmi R. & Leonardi S.: Higher differentiability for linear elliptic systems with measure data.
In preparation.
[42] Dall’Aglio A.: Approximated solutions of equations with L1 data. Application to the Hconvergence of quasi-linear parabolic equations. Ann. Mat. Pura Appl. (IV) 170 (1996), 207–
240.
[43] Dal Maso G. & Murat F. & Orsina L. & Prignet A.: Renormalized solutions of elliptic
equations with general measure data. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (IV) 28 (1999),
741–808.
[44] Del Vecchio T.: Nonlinear elliptic equations with measure data. Potential Anal. 4 (1995),
185–203.
[45] DeVore R.A. & Sharpley R.C.: Maximal functions measuring smoothness. Mem. Amer. Math.
Soc. 47 (1984), no. 293.
[46] DiBenedetto E.: Degenerate parabolic equations. Universitext. Springer-Verlag, New York,
1993.
[47] DiBenedetto E. & Friedman A.: Hölder estimates for nonlinear degenerate parabolic systems.
J. reine ang. Math. (Crelles J.) 357 (1985), 1–22.
[48] DiBenedetto E. & Friedman A.: Regularity of solutions of nonlinear degenerate parabolic
systems. J. reine ang. Math. (Crelles J.) 349 (1984), 83–128.
[49] DiBenedetto E. & Manfredi J.J.: On the higher integrability of the gradient of weak solutions
of certain degenerate elliptic systems. Amer. J. Math. 115 (1993), 1107–1134.
[50] Dolzmann G. & Hungerbühler N. & Müller S.: The p-harmonic system with measure-valued
right hand side. Ann. Inst. H. Poincarè Anal. Non Linèaire 14 (1997), 353–364.
[51] Dolzmann G. & Hungerbühler N. & Müller S.: Non-linear elliptic systems with measurevalued right hand side. Math. Z. 226 (1997), 545–574.
[52] Dolzmann G. & Hungerbühler N. & Müller S.: Uniqueness and maximal regularity for
nonlinear elliptic systems of n-Laplace type with measure valued right hand side. J. reine
angew. Math. (Crelles J.) 520 (2000), 1–35.
[53] Duoandikoetxea J.: Fourier analysis. Graduate Studies in Mathematics, 29. American Mathematical Society, Providence, RI, 2001.
[54] Duzaar F. & Kristensen J. & Mingione G.: The existence of regular boundary points for
non-linear elliptic systems. J. reine ang. Math. (Crelles J.) 602 (2007), 17-58.
[55] Duzaar F. & Mingione G.: Parabolic Adams Theorems. Preprint (2008).
[56] Duzaar F. & Mingione G. & Steffen K.: Parabolic systems with polynomial growth and
regularity. Preprint (2008).
[57] Esposito L. & Leonetti F. & Mingione G.: Regularity results for minimizers of irregular
integrals with (p, q) growth. Forum Mathematicum 14 (2002), 245–272.
[58] Fefferman C. & Stein E.: H p spaces of several variables. Acta Math. 129 (1972), 137-192.
[59] Fiorenza A. & Sbordone C.: Existence and uniqueness results for solutions of nonlinear
equations with right hand side in L1 . Studia Math. 127 (1998), 223–231
[60] Fuchs M. & Reuling J.:
Non-linear elliptic systems involving measure data.
Rend. Mat. Appl. (7) 15 (1995), 311–319.
TOWARDS A NON-LINEAR CALDERÓN-ZYGMUND THEORY
49
[61] Giaquinta M. & Giusti E.: Global C 1,α -regularity for second order quasilinear elliptic equations in divergence form. J. Reine Angew. Math. (Crelles J.) 351 (1984), 55–65
[62] Giusti E.: Direct methods in the calculus of variations. World Scientific Publishing Co., Inc.,
River Edge, NJ, 2003.
[63] Gehring F. W.: The Lp -integrability of the partial derivatives of a quasiconformal mapping.
Acta Math. 130 (1973), 265–277.
[64] Grafakos L.: Classical and modern Fourier analysis. Pearson Edu. Inc., Upper Saddle River,
NJ, 2004.
[65] Greco L. & Iwaniec T. & Sbordone C.: Inverting the p-harmonic operator. manuscripta math.
92 (1997), 249–258.
[66] Hardy G.H. & Littlewood J.E.: Some properties of fractional integrals. Math. Z. 27 (1928),
565–606.
[67] Heinonen J. & Kilpeläinen T. & Martio O.: Nonlinear potential theory of degenerate elliptic
equations. Oxford Mathematical Monographs, New York, 1993.
[68] Iwaniec T.: Projections onto gradient fields and Lp -estimates for degenerated elliptic operators. Studia Math. 75 (1983), 293–312.
[69] Iwaniec T.: p-harmonic tensors and quasiregular mappings. Ann. Math. (2) 136 (1992), 589–
624.
[70] Iwaniec T.: The Gehring lemma. Quasiconformal mappings and analysis (Ann Arbor, MI,
1995), 181–204, Springer, New York, 1998.
[71] Iwaniec T. & Martin G.: Geometric function theory and non-linear analysis. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2001.
[72] Iwaniec T. & Sbordone C.: Weak minima of variational integrals. J. reine angew. math.
(Crelle J.) 454 (1994), 143–161.
[73] Iwaniec T. & Verde A.: On the operator L(f ) = f log |f |. J. Funct. Anal. 169 (1999), 391–420.
[74] John F. & Nirenberg L.: On functions of bounded mean oscillation. Comm. Pure
Appl. Math. 14 (1961), 415–426.
[75] Kichenassamy S. & Verón L.: Singular solutions to the p-Laplace equation. Math. Ann. 275
(1986), 599-615.
[76] Kilpeläinen T.: Hölder continuity of solutions to quasilinear elliptic equations involving measures. Potential Anal. 3 (1994), 265–272.
[77] Kilpeläinen T. & Li G.: Estimates for p-Poisson equations. Diff. Int. Equ. 13 (2000), 791–800.
[78] Kilpeläinen T. & Malý J.: The Wiener test and potential estimates for quasilinear elliptic
equations. Acta Math. 172 (1994), 137–161.
[79] Kilpeläinen T. & Shanmugalingam N. & Zhong X.: Maximal regularity via reverse Hölder
inequalities for elliptic systems of n-Laplace type involving measures. Ark. Math. 46 (2008),
77–93.
[80] Kilpeläinen T. & Xu X.: On the uniqueness problem for quasilinear elliptic equations involving measures. Rev. Mat. Iberoamericana 12 (1996), 461–475.
[81] Kinnunen J.: New thoughts on the Hardy-Littlewood maximal function. Lectures delivered
at the University of Florence, 2007.
[82] Kinnunen J. & Lewis J. L.: Higher integrability for parabolic systems of p-Laplacian type.
Duke Math. J. 102 (2000), 253–271.
[83] Kinnunen J. & Lewis J. L.: Very weak solutions of parabolic systems of p-Laplacian type.
Ark. Mat. 40 (2002), 105–132.
[84] Kinnunen J. & Zhou S.: A local estimate for nonlinear equations with discontinuous coefficients. Comm. Partial Differential Equations 24 (1999), 2043–2068.
[85] Kristensen J. & Mingione G.: The singular set of ω-minima. Arch. Ration. Mech. Anal. 177
(2005), 93-114.
[86] Kristensen J. & Mingione G.: The singular set of minima of integral functionals. Arch. Ration. Mech. Anal. 180 (2006), 331–398.
[87] Kristensen J. & Mingione G.: Boundary regularity in variational problems. Preprint, 2008.
[88] Kristensen J. & Mingione G.: Boundary regularity of minima. Rend. Lincei - Mat. Appl. 19
(2008), 265-277.
[89] Krylov N.: Parabolic and Elliptic Equations with VMO Coefficients. Comm. Part. Diff. Equ.,
32 (2007), 453-475.
[90] Lewis J. L.: On very weak solutions of certain elliptic systems. Comm. Partial Differential
Equations 18 (1993), 1515–1537.
[91] Landes R.: On the existence of weak solutions to perturbed systems with critical growth.
J. reine ang. Math. (Crelles J.), 393 (1989), 21-38.
[92] Latter R. & Uchiyama A.: The atomic decomposition for parabolic H p spaces.
Trans. Amer. Math. Soc. 253 (1979), 391–398.
50
GIUSEPPE MINGIONE
[93] Lieberman G. M.: Sharp forms of estimates for subsolutions and supersolutions of quasilinear
elliptic equations involving measures. Comm. PDE 18 (1993), 1191–1212.
[94] Lieberman G.M.: A mostly elementary proof of Morrey space estimates for elliptic and
parabolic equations with VMO coefficients. J. Funct. Anal. 201 (2003), 457–479.
[95] Lindqvist P.: On the definition and properties of p-superharmonic functions. J. reine angew.
Math. (Crelles J.) 365 (1986), 67–79.
[96] Lindqvist P.: Notes on p-Laplace equation. University of Jyväskylä - Lectures notes, 2006.
[97] Lindqvist P. & Manfredi J.J.: Note on a remarkable superposition for a nonlinear equation.
Proc. AMS 136 (2008), 133-140.
[98] Lions J. L.: Quelques méthodes de résolution des problèmes aux limites non linèaires. Dunod,
Gauthier-Villars, Paris 1969.
[99] Malý J. & Ziemer W.P.: Fine regularity of solutions of elliptic partial differential equations.
Mathematical Surveys and Monographs, 51. American Mathematical Society, Providence, RI,
1997.
[100] Manfredi J.J.: Regularity for minima of functionals with p-growth. J. Differential Equations
76 (1988), 203–212.
[101] Mazzucato A.L.: Besov-Morrey spaces: function space theory and applications to non-linear
PDE. Trans. Amer. Math. Soc. 355 (2003), 1297–1364.
[102] Mingione G.: The singular set of solutions to non-differentiable elliptic systems. Arch. Ration. Mech. Anal. 166 (2003), 287–301.
[103] Mingione G.: Regularity of minima: an invitation to the Dark side of the Calculus of
Variations. Applications of Mathematics 51 (2006), 355-425.
[104] Mingione G.: Singularities of minima: a walk on the wild side of the Calculus of Variations.
J. Global Optimization 40 (2008), 209–223.
[105] Mingione G.: The Calderón-Zygmund theory for elliptic problems with measure data.
Ann Scu. Norm. Sup. Pisa Cl. Sci. (V) 6 (2007), 195–261.
[106] Mingione G.: Gradient estimates below the duality exponent. Mathematische Annalen, to
appear.
[107] Mingione G. & Zatorska-Goldstein A. & Zhong X.: Gradient regularity for elliptic equations
in the Heisenberg group. Advances in Mathematics 222 (2009), 62-129.
[108] Mingione G. & Zatorska-Goldstein A. & Zhong X.: On the regularity of the p-harmonic
functions in the Heisenberg group. Boll. Un. Mat. It. (IX) 1 (2008), 243-253.
[109] Ross J.: A Morrey-Nikolskii inequality. Proc. Amer. Math. Soc. 78 (1980), 97–102.
[110] Sarason D.: Functions of vanishing mean oscillation. Trans. Amer. Math. Soc. 207 (1975),
391–405.
[111] Serrin J.:
Pathological solutions of elliptic differential equations. Ann. Scuola
Norm. Sup. Pisa (III) 18 (1964), 385–387.
[112] Serrin J.: Local behavior of solutions of quasi-linear equations. Acta Math. 111 (1964),
247–302.
[113] Šverák V. & Yan X.: Non-Lipschitz minimizers of smooth uniformly convex variational
integrals. Proc. Natl. Acad. Sci. USA 99/24 (2002), 15269-15276.
[114] Stampacchia G.: The spaces L(p,λ) , N (p,λ) and interpolation. Ann. Scuola Norm. Sup. Pisa
(III) 19 (1965), 443–462.
[115] Stein E. M.: Note on the class L log L. Studia Math. 32 (1969), 305–310.
[116] Stein E. M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton Math. Series, 43. Princeton University Press, Princeton, NJ, 1993.
[117] Talenti G.: Elliptic equations and rearrangements. Ann Scu. Norm. Sup. Pisa Cl. Sci. (IV)
3 (1976), 697–717.
[118] Talenti G.: Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces.
Ann. Mat. Pura Appl. (IV) 120 (1979), 160–184.
[119] Tartar L.: An introduction to Sobolev spaces and interpolation spaces.. UMI Lecture notes
3, Springer 2007.
[120] Triebel H.: Theory of function spaces. Monographs in Math., Birkhäuser, 1983.
[121] Uhlenbeck K.: Regularity for a class of non-linear elliptic systems. Acta Math. 138 (1977),
219–240.
[122] Ural’tseva N.N.: Degenerate quasilinear elliptic systems. Zap. Na. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7 (1968), 184–222.
[123] Zhong
X.:
On
nonhomogeneous
quasilinear
elliptic
equations.
Ann. Acad. Sci. Fenn. Math. Diss. 117 (1998), 46 pp.
Dipartimento di Matematica, Università di Parma, Viale G. P. Usberti 53/a, Campus,
43100 Parma, Italy; e-mail: [email protected].