Dark Side manuscript No.
(will be inserted by the editor)
Tuomo Kuusi · Giuseppe Mingione
A nonlinear Stein theorem
Received: date / Accepted: date /
c Springer-Verlag 2013
Published online: date – Abstract. For vector valued solutions u to the p-Laplacian system −4p u = F in
a domain of Rn , p > 1, n ≥ 2, we prove that if F belongs to the limiting Lorentz
space L(n, 1), then Du is continuous.
To Bernard Dacorogna on his 60th birthday
Mathematics Subject Classification (2010): 35J70, 35D10; Keywords: degenerate
systems, regularity, Lorentz spaces
1. The result
A by now classical result of Stein [22] asserts that if v ∈ W 1,1 is a Sobolev
function defined in Rn with n ≥ 2, then
Dv ∈ L(n, 1) =⇒ v is continuous .
(1)
The Lorentz space L(n, 1) appearing in the above display consists of those
measurable maps g satisfying the condition
Z ∞
|{x : |g(x)| > t}|1/n dt < ∞
0
and (1) can be regarded as the limiting case of Sobolev-Morrey embedding
theorem that asserts
Dv ∈ Ln+ε =⇒ v ∈ C 0,ε/(n+ε)
(2)
Tuomo Kuusi
Aalto University Institute of Mathematics
P.O. Box 111000, FI-00076 Aalto, Finland e-mail: [email protected]
G. Mingione: Dipartimento di Matematica e Informatica, Università di Parma
Viale delle Scienze 53/a, Campus, I-43124 Parma, Italy
e-mail: [email protected]
The authors are supported by the ERC grant 207573 “Vectorial Problems” and by
the Academy of Finland project “Regularity theory for nonlinear parabolic partial
differential equations”. The authors also thank Paolo Baroni and Neil Trudinger
for remarks on a preliminary version of the paper.
2
Tuomo Kuusi, Giuseppe Mingione
whenever ε > 0. Note indeed that Ln+ε ⊂ L(n, 1) ⊂ Ln for every ε > 0,
with all the inclusions being actually strict. Another version of Stein’s theorem concerns the regularity of solutions to the non-homogeneous Laplacian
system. Indeed, using (1) together with the standard Calderón-Zygmund
theory allows to conclude with
4u ∈ L(n, 1) =⇒ Du is continuous .
The aim of this paper is to prove the same result for solutions to the pLaplacian system (with coefficients), as indeed established in the following:
1,p
Theorem 1. Let u ∈ Wloc
(Ω, RN ) with p > 1 and N ≥ 1 be a local weak
solution to the system
−div (γ(x)|Du|p−2 Du) = F
(3)
in an open subset Ω ⊂ Rn , n ≥ 2. Assume that
– the vector field F : Ω → RN satisfies F ∈ L(n, 1) locally in Ω
– the function γ : Ω → [ν, L] is Dini-continuous, where 0 < ν ≤ L < ∞ .
Then Du is continuous in Ω.
The optimal character of the space L(n, 1) in the p-Laplacian setting also
stems from the well-known regularity result
F ∈ Ln+ε =⇒ Du is locally Hölder continuous ,
which is the p-Laplacian counterpart of (2). Anyway, counterexamples working already in the linear case, show that Du can unbounded when 4u 6∈
L(n, 1); see [2]. Moreover, we notice the relevant fact that the condition
F ∈ L(n, 1) is independent of p and this is reflected in the approach we
will actually take in the proof of Theorem 1. Indeed, the basic viewpoint
adopted here is to look at (3) as a linear system in the nonlinear vector field
γ(x)|Du|p−2 Du rather than a nonlinear system in the gradient Du. This
ultimately leads to write (3) as a decoupled system
(
−div H = F
H = γ(x)|Du|p−2 Du
so that the continuity of H eventually implies the one of Du. This viewpoint helps explaining why conditions for the continuity of Du in terms of
optimal function spaces do not depend on p. The implementation of this
heuristic argument is anyway not easy, and involves a considerable number
of technicalities.
The space L(n, 1) already appears in the study of the p-Laplacian equations and systems [3, 6, 18] in connection to gradient L∞ -bounds, that in fact
can be derived when F ∈ L(n, 1). In particular, in [18] we proved a scalar
version of Theorem 1, but this result applies only to a single equation, and
A nonlinear Stein theorem
3
not to systems, and the gradient continuity has remained an open problem
for systems. Moreover, only the case p ≥ 2 has been considered in [18]. The
novelty of Theorem 1 therefore lies not only in the fact that it covers the
vectorial case N > 1, but also in that we are treating the full range p > 1.
The approach we are developing here does not distinguish between the socalled singular case 1 < p < 2 and the degenerate one p ≥ 2. We also remark
that, already in the case of the gradient L∞ -regularity, the two dimensional
case n = 2 had remained an open issue in [3, 6], essentially for technical
reasons. This point is fixed in this paper by mean of a completely different
approach. Indeed, in contrast to [18], where the analysis was based on the
use of potential estimates (see [14, 24, 25]), here we pursue a different path
using directly certain characterisations of Lorentz spaces (see for instance
Section 2.3 below) and a careful linearisation approach.
Another feature of Theorem 1 is the presence of the coefficients - i.e.
the function γ(·) - something that did not seem to be achievable with the
known techniques. In this respect, the assumption of Dini-continuity of γ(·)
is sharp. In fact, already in the case of linear and homogeneous elliptic
equations
div (A(x)Du) = 0
(4)
the gradient of solutions is in general unbounded for continuos but not Dinicontinuous matrices A(·), as shown in [13]. We just recall that the function
γ(·) is Dini-continuous when there exists a concave, non-decreasing function
ω : [0, ∞) → [0, 1] with ω(0) = 0, satisfying
|γ(x) − γ(y)| ≤ Lω(|x − y|)
(5)
for every x, y ∈ Ω and such that
Z
d%
< ∞.
ω(%)
%
0
(6)
There has been recently a renewed interest in Dini-continuous coefficients
and in related regularity issues, see for instance [20].
Although we preferred to concentrate on the model case in (3), the result
of Theorem 1 continues to hold for systems with more general structures. For
instance, with essentially minor modifications, we can treat general quasidiagonal structures (sometime called “Uhlenbeck structure” as in [1]) of the
type
−div (g(|Du|)Du) = F ,
g(|Du|) ≈ |Du|p−2 .
More general dependences on the coefficients can be considered as well. For
instance we can treat models as
−div (hA(x)Du, Duip−2 A(x)Du) = F ,
where A(x) is a bounded and strictly elliptic matrix with Dini-continuous
entries (see for instance [15]). Yet minor modifications allow to deal with
non-degenerate structures too, as for instance
−div (γ(x)(µ2 + |Du|2 )(p−2)/2 Du) = F ,
µ > 0.
4
Tuomo Kuusi, Giuseppe Mingione
We remark that a preliminary gradient continuity result for solutions to
(3) has been obtained in [7] for the case p ≥ 2 and under the suboptimal
assumptions
Z
1
2/p d%
and
< ∞.
(7)
F ∈ L n,
[ω(%)]
p−1
%
0
In particular, the case of Dini continuity of coefficients was not covered.
Both conditions in (7) worsen when p increases. Here sharp results are
finally reached by a very careful linearisation argument that allows a better
control of the degeneracy.
2. Preparatory material
2.1. General notation
In what follows we denote by c a general positive constant, possibly varying
from line to line; special occurrences will be denoted by c1 , c2 , c̄1 , c̄2 or the
like. All such constants will always be larger or equal than one; moreover
relevant dependencies on parameters will be emphasized using parentheses,
i.e., c1 ≡ c1 (n, N, p, ν, L) means that c1 depends only on n, N, p, ν, L. We
denote by
B(x0 , r) ≡ Br (x0 ) := {x ∈ Rn : |x − x0 | < r}
the open ball with center x0 and radius r > 0; when not important, or
clear from the context, we shall omit denoting the center as follows: Br ≡
B(x0 , r); moreover, with B being a generic ball with radius r we will denote
by σB the ball concentric to B having radius σr, σ > 0. Unless otherwise
stated, different balls in the same context will have the same center. We shall
also denote B ≡ B1 = B(0, 1) if not differently specified. With O ⊂ Rn
being a measurable subset with positive measure, and with g : O → Rk ,
k ≥ 1, being a measurable map, typically a gradient in the following, we
shall denote by
Z
Z
1
g(x) dx
g(x) dx :=
(g)O ≡
|O| O
O
its integral average; here |O| denotes the Lebesgue measure of O. In the rest
of the paper we shall use several times the following elementary property of
integral averages:
Z
|g − (g)O |γ dx
1/γ
1/γ
|g − A|γ dx
,
Z
≤2
O
(8)
O
whenever A ∈ Rk and γ ≥ 1. The oscillation of g on O is instead defined as
osc g := sup |g(x) − g(x̃)| .
O
x,x̃∈O
A nonlinear Stein theorem
5
Finally, for s ≥ 1, we shall denote its Ls -excess functional as
1/s
|g − (g)O |s dx
.
Z
E(g, O) ≡ Es (g, O) :=
O
Such a quantity is bound to provide an integral measure of the oscillations
of g in O.
2.2. General setting for the proofs
In this section we build the basic set-up for the proof of Theorem 1. With
B(x0 , 2r) ⊂ Ω being a fixed ball we then set for j ≥ 0
Bj ≡ B(x0 , rj ) ,
rj := σ j r ,
σ ∈ (0, 1/4) .
(9)
We remark that the parameter σ is at the moment assumed to be just a
number belonging to (0, 1/4) and all the considerations through Section 2
will stay valid for any choice of it. Specific values of σ will be then used
in Sections 3 and 4 below. Next, for j ≥ 0, we define the maps u = wj +
W01,p (Bj , RN ) as the unique solutions to the problems
(
div (γ(x)|Dwj |p−2 Dwj ) = 0 in Bj
(10)
wj = u on ∂Bj ,
and, eventually, vj ∈ wj + W01,p ( 21 Bj , RN ) as the unique solutions to
(
div (γ(x0 )|Dvj |p−2 Dvj ) = 0
in 12 Bj
(11)
vj = wj on ∂ 12 Bj .
Before going on, let us remark that since all the results in this papers are
local in nature in the following we shall assume without loss of generality
that F : Rn → RN (this can be done for instance letting F ≡ 0 outside Ω)
and that
u ∈ W 1,p (Ω, RN )
and
F ∈ L(n, 1)(Rn , RN ) .
(12)
2.3. A relevant series and Lorentz spaces
Let us recall a few basic facts about Lorentz spaces (for which we for instance
refer to [23]). We recall that a map g : Ω → Rk belongs to the Lorentz space
L(s, γ) ≡ L(s, γ)(Ω, Rk ) for s ≥ 1 and γ > 0 iff
Z ∞
dt
γ
< ∞.
(13)
kgkL(s,γ) :=
(ts |{x ∈ Ω : |g(x)| > t}|)γ/s
t
0
The local variant of L(s, γ) is then defined, as usual, by saying that v ∈
L(s, γ) locally, iff v ∈ L(s, γ)(Ω 0 , Rk ) for every open subset Ω 0 b Ω. A
useful characterization of Lorentz spaces can be given via rearrangements;
6
Tuomo Kuusi, Giuseppe Mingione
indeed, if |g|∗ : [0, |Ω|] → [0, ∞] denotes the non-increasing rearrangement
of |g| (see [10]) and if we consider the following maximal type operator
Z
1 t ∗
∗∗
|g| (%) d% ,
t ∈ (0, |Ω|] ,
|g| (t) :=
t 0
then we have
Z
g ∈ L(s, γ) ⇐⇒
|Ω|
[%(|g|∗∗ (%))s ]γ/s
0
d%
<∞
%
if s > 1 .
Moreover, it holds that
Z |Ω|
d%
[%(|g|∗∗ (%))s ]γ/s
≤ c(s, γ)kgkγL(s,γ)
%
0
(14)
(15)
again for s > 1 (see [9, 23]).
With F being the vector field defined in Theorem 1 (but also recall
(12)), and with the balls Bj being defined in (9), we consider the following
quantity:
!1/q
Z
∞
X
q
Sq (x0 , r, σ) :=
|F | dx
rj
,
q ∈ (1, n) .
(16)
j=0
Bj
The quantity Sq (x0 , r, σ) defined in (16) plays a crucial role in the analysis
of the fine pointwise properties of the gradient of solutions to systems as in
(3). We are here interested in deriving an estimate for the quantity in the
previous display in terms of the L(n, 1)-norm of F , or, better saying, the
L(n/q, 1/q) norm of |F |q . More precisely, we have the following:
Lemma 1. If F ∈ L(n, 1), q ∈ (1, n) and σ ∈ (0, 1/4), then
Sq (x0 , r, σ) ≤ c(n, q, σ)kF kL(n,1)
holds for every r > 0 and x0 ∈ Rn and, more precisely, estimate
Z 2r
d%
Sq (x0 , r, σ) ≤ σ 1−n/q
[%q (|F |q )∗∗ (ωn %n )]1/q
≤ ckF kL(n,1)
%
0
(17)
(18)
is true for a constant c depending only on n, q, σ.
Proof. By the classical Hardy-Littlewood inequality on rearrangements [10],
and letting r−1 = 2r, it follows that for j ≥ 0 if rj ≤ % ≤ rj−1
Z
Z ωn rjn
rjq
rjq
|F |q dx ≤
(|F |q )∗ (ξ) dξ ≤ σ q−n %q (|F |q )∗∗ (ωn %n )
ωn rjn 0
Bj
holds whenever j ≥ 0 and therefore we have
!1/q
Z
rjq
|F |q dx
Bj
A nonlinear Stein theorem
7
−1 Z rj−1
d%
1
[%q (|F |q )∗∗ (ωn %n )]1/q
≤σ
log
σ
%
rj
Z rj−1
d%
≤ σ 1−n/q
[%q (|F |q )∗∗ (ωn %n )]1/q
.
%
rj
1−n/q
Summing up the previous inequalities gives (18). Here we just recall that
ωn standardly denotes the measure of the unit ball in Rn . Next, notice that
the right hand side of (18) is finite when F ∈ L(n, 1). Indeed, by the very
definition of Lorentz spaces in (13) we have that |F |q ∈ L(n/q, 1/q) and
k|F |q kL(n/q,1/q) = q q kF kqL(n,1)
then a change of variable and a repeated use of (15) give that
Z ∞
Z ∞
d%
1
d%
[%q (|F |q )∗∗ (ωn %n )]1/q
=
[%((|F |q )∗∗ (%))n/q ]1/n
1/n
%
%
0
0
nωn
1/q
≤ c(n, q)k|F |q kL(n/q,1/q)
= c(n, q)kF kL(n,1) .
Observe that we have used (14) with g = |F |q , s = n/q and γ = 1/q, since
n/q > 1. The inequality in the last display together with (18) gives (17).
2.4. Facts of algebraic nature
We shall largely use the auxiliary vector field V : RN n → RN n defined by
V (z) := |z|(p−2)/2 z ,
(19)
and which is a locally Lipschitz bjiection from RN n into itself. This will be
often employed in connection to the following inequality (see [9]):
|z1 − z2 |
|V (z1 ) − V (z2 )|
≤
≤ c1 |z1 − z2 |
c1
(|z1 | + |z2 |)(p−2)/2
(20)
which is valid for all matrixes z1 , z2 ∈ RN n that are not simultaneously
null and for every p > 1, where c1 ≡ c1 (n, N, p). The previous inequality is
relevant in manipulations involving the classical monotonicity estimate
(|z1 | + |z2 |)p−2 |z1 − z2 |2 ≤ c(n, N, p)h|z1 |p−2 z1 − |z2 |p−2 z2 , z1 − z2 i , (21)
which again holds for all matrixes z1 , z2 ∈ RN n and p > 1. We now give a
lemma for the case 1 < p ≤ 2.
Lemma 2. Let p ∈ (1, 2]. There exists a constant c, depending only on
n, N, p such that the following inequality holds whenever z1 , z2 ∈ RN n :
|z1 − z2 | ≤ c|V (z1 ) − V (z2 )|2/p + c|V (z1 ) − V (z2 )||z2 |(2−p)/2 .
(22)
8
Tuomo Kuusi, Giuseppe Mingione
Proof. We assume p 6= 2, because otherwise the assertion is trivial since
V (z) = z when p = 2. We have
|z1 − z2 | = (|z1 | + |z2 |)(p−2)/2 |z1 − z2 |(|z1 | + |z2 |)(2−p)/2
(20)
≤ c1 |V (z1 ) − V (z2 )| · (|z1 | + |z2 |)(2−p)/2
≤ c|V (z1 ) − V (z2 )| · |z1 − z2 |(2−p)/2 + |z2 |(2−p)/2 .
Then, using Young’s inequality with conjugate exponents
2
2
,
p 2−p
we gain
|z1 − z2 | ≤ (1/2)|z1 − z2 | + c|V (z1 ) − V (z2 )|2/p + c|V (z1 ) − V (z2 )||z2 |(2−p)/2
and (22) follows.
2.5. Homogeneous systems with Dini-continuous coefficients
Our recent paper [17] deals with the spatial gradient continuity properties
of solutions to nonlinear parabolic systems of p-Laplacian type
ut − div (γ(x, t)|Du|p−2 Du) = 0
and we have proved that if the coefficient function γ(·) is Dini-continuous
with respect to the space variable x then Du is continuous. Related a priori
estimates have been provided. The results obtained in [17] obviously apply
to the elliptic case div (γ(x)|Du|p−2 Du) = 0 and this section is devoted to
restate the a priori estimates derived in [17] in a way tailored to the our
needs here. The outcome is indeed the following:
Theorem 2. Let wj be as in (10) with j ≥ 0, then Dwj is continuous.
Moreover
– There exists a constant c2 ≡ c2 (n, N, p, ν, L) ≥ 1 and positive radius
R1 ≡ R1 (n, N, p, ν, L, ω(·)) such that if r ≤ R1 , then the following holds:
Z
sup |Dwj | ≤ c2
|Dwj | dx .
(23)
1
2 Bj
Bj
– Assume that the inequality
sup |Dwj | ≤ Aλ
(24)
1
2 Bj
holds for some A ≥ 1 and λ > 0. Then, for any δ ∈ (0, 1) there exists a
positive constant σ1 ≡ σ1 (n, N, p, ν, L, ω(·), A, δ) ∈ (0, 1/4) such that
osc Dwj ≤ δλ .
σ 1 Bj
(25)
A nonlinear Stein theorem
9
Proof. The proof is actually an adaptation of the relevant arguments from
[17]. We start from the a priori estimate (23). Using [17, Theorem 1.1]
together with a standard covering argument it follows that there exists a
radius R1 ≡ R1 (n, N, p, ν, L, ω(·)) > 0, such that
Z
where c ≡ c n, N, p, ν, L,
|Dwj |p dx
sup |Dwj | ≤ c
1
2B
B
holds whenever B ⊂ Bj is a ball (not necessarily concentric to Bj ) with
radius smaller of equal than R1 . By a standard rescaling procedure, that is
considering the new solution to div (γ(x)|Dw̃|p−2 Dw̃) = 0 defined by
Z
Z
−1/p
1/p
p
p
wj =⇒
= 1,
w̃ :=
|Dwj | dx
|Dw̃| dx
B
B
we have that
sup |Dw̃| ≤ c ≡ c (n, N, p, ν, L) .
1
2B
Scaling back to wj yields
Z
p
sup |Dwj | ≤ c
1
2B
1/p
|Dwj | dx
B
with c ≡ c(n, N, p, ν, L), whenever B is a ball contained in Bj . In turn, a
by now standard interpolation/covering argument leads to the fact that the
exponent in the previous inequality can be lowered:
Z
1/t
t
|Dwj | dx
holds for every t > 0 ,
sup |Dwj | ≤ c(t)
1
2B
B
from which (23) follows by taking B ≡ Bj and t = 1. The adaptation of
(25) from the parabolic arguments of [17] needs more care. In particular
the so-called “intrinsic geometry” needed in [17] and taken from the work
of DiBenedetto [4], becomes immaterial in this context. In this respect, we
recall that in [17] we worked with intrinsic cylinders with vertex (x0 , t0 ) of
the type Qλ% (x0 , t0 ) = B(x, %)×(t0 −λ2−p %2 , t0 ), for a number λ > 0 which is
suitably related to the size of the gradient on the same cylinder Qλ% (x0 , t0 ).
When dealing with elliptic problems such peculiar cylinders disappear and
they are simply replaced by standard Euclidean balls B% (x0 ). We shall now
prove (25) by showing that for a suitable choice of σ1 as described in the
statement, we have
|Dwj (x̃) − Dwj (ỹ)| ≤ δλ
∀ x̃, ỹ ∈ σ1 Bj .
(26)
Therefore following the arguments in [17, (5.15)] we find that for every ε ∈
(0, 1) there exists a positive radius Rε ≡ Rε (n, N, p, ν, L, ω(·), ε) ∈ (0, 1/16)
such that
|(V (Dwj ))Bτ (x̃) − (V (Dwj ))B% (x̃) | ≤ (Aλ)p/2 ε
(27)
10
Tuomo Kuusi, Giuseppe Mingione
and
!1/2
Z
2
≤ (Aλ)p/2 ε
|V (Dwj ) − (V (Dwj ))B% (x̃) | dx
(28)
B% (x̃)
hold whenever x̃ ∈ 41 Bj and 0 < τ ≤ % ≤ Rε rj ; notice that Rε is in
particular independent of λ, A and the considered cylinder Bj . The vector
field V (·) has been introduced in (19). Since Dwj is continuous we let τ → 0
in (27) thereby obtaining
|V (Dwj (x̃)) − (V (Dwj ))B% (x̃) | ≤ (Aλ)p/2 ε
∀ % ∈ (0, Rε rj ] .
(29)
Now, observe that if ỹ, x̃ ∈ σ1 Bj and σ1 ≤ Rε /32 it obviously follows that
BRε rj /8 (ỹ) ⊂ BRε rj (x̃) ⊂
1
Bj .
4
(30)
Then, using also Jensen’s inequality we have
|(V (Dwj ))BRε rj /8 (ỹ) − (V (Dwj ))BRε rj /8 (x̃) |
Z
≤
|V (Dwj ) − (V (Dwj ))BRε rj /8 (x̃) | dx
BRε rj /8 (ỹ)
(8)
Z
≤ 2
BRε rj /8 (ỹ)
(30)
≤ 16n
|V (Dwj ) − (V (Dwj ))BRε rj (x̃) | dx
Z
BRε rj (x̃)
n
|V (Dwj ) − (V (Dwj ))BRε rj (x̃) | dx
!1/2
Z
2
≤ 16
BRε rj (x̃)
|V (Dwj ) − (V (Dwj ))BRε rj (x̃) | dx
(28)
≤ 16n (Aλ)p/2 ε .
By using the previous estimate and (29) twice (centered in ỹ and x̃) together
with triangle inequality we easily gain
|V (Dwj (x̃)) − V (Dwj (ỹ))| ≤ 48n (Aλ)p/2 ε .
(31)
In order to show the validity of (26) we now distinguish between two cases.
The first is when p ≥ 2. Let us observe that we can assume that
max{|Dwj (x̃)|, |Dwj (ỹ)|} ≥ δλ/2
(32)
holds otherwise we are done. We then have
(20)
|Dwj (x̃) − Dwj (ỹ)| ≤ c1
(32)
|V (Dwj (x̃)) − V (Dwj (ỹ))|
(|Dwj (x̃)| + |Dwj (ỹ)|)(p−2)/2
≤ c1 2p/2−1 δ 1−p/2 λ1−p/2 |V (Dwj (x̃)) − V (Dwj (ỹ))|
A nonlinear Stein theorem
11
(31)
≤ c1 48n 2p/2−1 δ 1−p/2 Ap/2 λε .
In the case 1 < p ≤ 2 we instead have
(20)
|Dwj (x̃) − Dwj (ỹ)| ≤ c1
|V (Dwj (x̃)) − V (Dwj (ỹ))|
(|Dwj (x̃)| + |Dwj (ỹ)|)(p−2)/2
(24)
≤ c1 21−p/2 A1−p/2 λ1−p/2 |V (Dwj (x̃)) − V (Dwj (ỹ))|
(31)
≤ c1 48n 21−p/2 Aλε .
Then, in the case p ≥ 2 we choose
ε :=
δ p/2
.
c1 48n 2p/2−1 Ap/2
This determines the radius Rε ≡ Rε (n, N, p, ν, L, ω(·), A, δ) > 0 for which
(27) and (28) work and therefore we conclude with (26) with the choice
σ1 := Rε /32. In the case 1 < p < 2 we instead choose
ε :=
δ
c1 48n 21−p/2 A
and conclude similarly.
2.6. Regularity for the p-Laplacian system
We here recall some basic properties of solutions to the p-Laplacian system;
our basic reference here are [5, 8, 9]. We again restate all the results for the
maps vj defined in (11).
Theorem 3. Let vj be as in (11) with j ≥ 0, then Dvj is continuous.
Moreover
– There exists a constant c3 ≥ 1 depending only on n, N, p such that
Z
sup |Dvj | ≤ c3
|Dvj | dx
(33)
1
2 Bj
1
4 Bj
– For every A ≥ 1 there exist constants
c4 ≡ c4 (n, N, p, A) ≡ c̃4 (n, N, p)A
and α ≡ α(n, N, p) ∈ (0, 1) such that
sup |Dvj | ≤ Aλ
1
4 Bj
=⇒
osc Dvj ≤ c4 τ α λ
τ Bj
∀ τ ∈ (0, 1/4) .
(34)
12
Tuomo Kuusi, Giuseppe Mingione
Proof. Estimate (33) is a slightly stronger version of a classical estimate
going back to work of Uhlenbeck [26]. In the form presented here it can
be retrieved using (23) as vj solves a systems with constants - and hence
Dini-continuous - coefficients. As for (34) we just point out that this an
immediate consequence of a classical estimate that again goes back to the
work of Uhlenbeck [26]. Indeed, in [26] the following inequality is proved to
hold for solutions to the homogenous p-Laplacian system:
1/p α
Z
|x̃ − ỹ|
p
,
|Dvj | dx
|Dvj (x̃) − Dvj (ỹ)| ≤ c(n, N, p)
R
BR
whenever BR ⊂ Bj and x̃, ỹ ∈ BR/2 . See also [5, 8, 9, 19] for more genertal
cases covering the full range p > 1.
The next lemma presents estimates involving excess functionals defined for
exponents smaller that the natural one i.e. p; estimates of such kind are
usually called estimates below the natural growth exponent.
Lemma 3. Let vj be as in (11) for j ≥ 0. For every choice of ε̄ > 0 and
A ≥ 1 there exists a constant σ2 ∈ (0, 1/4), depending only on n, N, p, ε̄, A,
such that if σ ∈ (0, σ2 ] and
λ
≤ sup |Dvj | ≤ sup |Dvj | ≤ Aλ
1
A σBj
Bj
(35)
4
holds, then
!1/t
Z
|Dvj − (Dvj )σBj |t dx
σBj
!1/t
Z
≤ ε̄
1
2 Bj
|Dvj − (Dvj ) 12 Bj |t dx
(36)
also holds, whenever t ∈ [1, 2].
Proof. The proof goes in two steps. In the first one we recall some basic facts
from classical regularity theory for degenerate elliptic systems, asserting a
decay estimate of the type in (36) for an excess functional of, let’s say,
traditional type, that is involving the vector field V (·) introduced in (19).
In the second step we show how to use the bounds in (35) to commute the
result of the first step in the decay estimate we want, that is (36).
Step 1: A preliminary decay estimate. The vector field V (·) is useful
in reformulating the regularity properties of solutions to the p-Laplacian
system, indeed a standard result (see for instance [8, 9]) is the following
decay estimate:
!1/2
Z
|V (Dvj ) − (V (Dvj ))B% |2 dx
B%
≤c
1/2
|V (Dvj ) − (V (Dvj ))BR |2 dx
% β Z
R
BR
(37)
A nonlinear Stein theorem
13
that holds for a constant c ≡ c(n, N, p), whenever B% ⊂ BR are concentric
balls contained in 21 Bj (but not necessarily concentric to Bj ). See for instance [5, 8, 9]. The first remark to do is that the previous inequality can be
actually reformulated in terms of lower exponents, that is
!1/t
Z
|V (Dvj ) − (V (Dvj ))B% |t dx
B%
≤c
1/t
|V (Dvj ) − (V (Dvj ))BR |t dx
% β Z
R
(38)
BR
for every t ∈ [1, 2] and again for c ≡ c(n, N, p). For the sake of completeness,
let us briefly recall the proof of (38) since it is a consequence of a few
facts from the regularity theory of general elliptic systems that are not
commonly used in the literature. We start by the following reverse Hölder
type inequality:
!1/(2χ)
Z
Z
1/2
|V (Dvj ) − z0 |2χ dx
|V (Dvj ) − z0 |2 dx
≤c
BR/2
,
BR
(39)
which holds whenever BR ⊂ 12 Bj and z0 ∈ RN n , for a constant c ≡
c(n, N, p), where χ = n/(n − 1); see for instance [8, 16]. The well-known
self improving property of reverse Hölder type inequalities also reflects in
the fact that the range of exponents for which the reverse property holds
can be actually extended in terms of the exponents involved. Indeed, proceeding as in [21, Lemmas 3.1 and 3.2] we have that (39) implies that the
following inequality holds uniformly in t ∈ [1, 2] for every ball BR ⊂ 21 Bj :
!1/2
Z
2
|V (Dvj ) − z0 | dx
1/t
|V (Dvj ) − z0 |t dx
Z
≤c
BR/2
(40)
BR
with c ≡ (n, N, p). Using estimates (37) and (40) with the specific choice
z0 = (V (Dvj ))BR , and yet Hölder’s inequality we indeed deduce, for 0 <
% ≤ R/2
!1/t
Z
t
|V (Dvj ) − (V (Dvj ))B% | dx
B%
!1/2
Z
2
|V (Dvj ) − (V (Dvj ))B% | dx
≤
B%
≤c
≤c
% β
!1/2
Z
R
|V (Dvj ) − (V (Dvj ))BR |2 dx
BR/2
% β Z
R
|V (Dvj ) − (V (Dvj ))BR |t dx
BR
1/t
.
14
Tuomo Kuusi, Giuseppe Mingione
We have also used (8) in the second-last estimate and note that the constant
c depends, as usual, only on n, N, p. On the other hand, (38) follows by
trivial means when R/2 ≤ % ≤ R so that (38) is completely proved.
Step 2: Proof of (36). The value of σ2 will be determined in due
course of the proof, while we shall argue under the assumption (35) for
some σ ≤ σ2 . Towards the determination of the number σ2 we start looking
at (34) and define the radius r > 0 as
r ≡ r(n, N, p, A) :=
1
16c̃4 A2
1/α
=⇒
λ
.
4A
osc Dvj ≤
rBj
(41)
With σ ≤ σ2 ≤ r, and σ2 yet to be determined, if x ∈ rBj , then we have
|Dvj (x)| = sup |Dvj | + |Dvj (x)| − sup |Dvj |
σBj
(35)
σBj
(41)
λ
λ
− osc Dvj ≥
A rBj
4A
≥
so that, again taking (35) into account we conclude with
λ
≤ inf |Dvj | ≤ sup |Dvj | ≤ Aλ .
1
4A rBj
Bj
(42)
4
Let us now first consider the case p ≥ 2. As V (·) is a bjiection of RN n we
can define the matrix z(σ) ∈ RN n such that
z(σ) := V −1 ((V (Dvj ))σBj ) ⇐⇒ V (z(σ)) = (V (Dvj ))σBj .
(43)
As a straightforward consequence of (42) and of the definition of V (·), we
have that there exists a constant c ≡ c(n, N, p) ≥ 4 such that
λ
≤ |z(σ)| ≤ cAλ .
cA
(44)
Keeping in mind the content of the last two displays we now have that
!1/t
Z
t
|Dvj − (Dvj )σBj | dx
σBj
(8)
!1/t
Z
t
≤ 2
|Dvj − z(σ)| dx
σBj
(20)
!1/t
Z
≤ c
(|Dvj | + |z(σ)|)
(2−p)t/2
t
|V (Dvj ) − V (z(σ))| dx
σBj
(43),(44)
≤
(p−2)/2
A
c
λ
!1/t
Z
t
|V (Dvj ) − V ((Dvj ))σBj | dx
σBj
A nonlinear Stein theorem
15
(p−2)/2 σ β
A
≤ c
λ
r
Z
(p−2)/2 A
σ β
≤ c
λ
r
Z
(38)
|V (Dvj ) − (V (Dvj ))rBj | dx
rBj
(8)
(20)
≤ c
!1/t
t
!1/t
rBj
|V (Dvj ) − V ((Dvj ) 14 Bj )|t dx
(p−2)/2 σ β
A
λ
r
!1/t
Z
(p−2)t/2
t
·
|Dvj − (Dvj ) 41 Bj | dx
(|Dvj | + |(Dvj ) 14 Bj |)
rBj
(42)
≤ cAp−2
(8)
≤ cA
(41)
r
p−2 −n/t
r
!1/t
Z
σ β
rBj
σ β
|Dvj − (Dvj ) 41 Bj |t dx
r
t
1
2 Bj
p−2+2(n+β)/α β
≤ cA
!1/t
Z
|Dvj − (Dvj ) 21 Bj | dx
!1/t
Z
t
σ
1
2 Bj
≤
cAp−2+2(n+β)/α σ2β
|Dvj − (Dvj ) 12 Bj | dx
!1/t
Z
t
1
2 Bj
|Dvj − (Dvj ) 12 Bj | dx
for a constant c ≡ c(n, N, p). Therefore is it now sufficient to take σ2 such
that
(
)
1/β 1 1/α
ε̄
σ2 :=
≤r
,
16c̃4 A2
cAp−2+2(n+β)/α
and (36) follows in the case p ≥ 2. The proof in the case 1 < p ≤ 2 is
completely similar, and in some sense dual, as it just requires to use the
inequalities in (42) and (44) in an order which is reversed with respect to
the one used in the case p ≥ 2. The final outcome, again for a constant
c ≡ c(n, N, p), is the inequality
!1/t
Z
|Dvj − (Dvj )σBj |t dx
σBj
≤
cA2−p+2(n+β)/α σ2β
!1/t
Z
t
1
2 Bj
|Dvj − (Dvj ) 12 Bj | dx
and we again conclude with (36) by taking σ2 as
(
)
1/β 1 1/α
ε̄
σ2 :=
,
≤ r.
16c̃4 A2
cA2−p+2(n+β)/α
The proof is complete.
16
Tuomo Kuusi, Giuseppe Mingione
2.7. Preliminary estimates
With p > 1 given, in this paper we denote by p∗ the usual Sobolev conjugate
exponent in the sense of

np

if n > p
∗
n−p
(45)
p :=
 any number larger than p if p ≥ n .
Moreover, the exponent p0 = p/(p − 1) is the usual conjugate exponent of
p. We start with a first comparison estimate.
Lemma 4. Let u be as in Theorem 1 with p ≥ 2 and wj , vj as in (10)-(11),
respectively, with j ≥ 0. There exists a constant c5 ≡ c5 (n, N, p, ν, q), such
that
!1/p
!1/[q(p−1)]
Z
Z
q
p
q
≤ c5 rj
(46)
|Du − Dwj | dx
|F | dx
Bj
Bj
holds for every q ≥ (p∗ )0 when p < n, and for every q > 1 when p ≥ n.
Moreover, when j ≥ 1 and for another constant c6 ≡ c6 (n, N, p, ν, q, σ), it
holds that
!1/p
!1/[q(p−1)]
Z
Z
q
p
q
|Dwj−1 − Dwj | dx
≤ c6 rj−1
|F | dx
. (47)
Bj
Bj−1
Proof. Using the weak formulations of (3) and (10) we have
Z
Z
p−2
p−2
γ(x)h|Du| Du − |Dwj | Dwj , Dϕi dx =
hF, ϕi dx ,
Bj
Bj
which is valid for every ϕ ∈ W01,p (Bj , RN ); there we choose ϕ = u − wj .
Using (21) we find that the inequality
Z
Z
p−2
2
(|Dwj | + |Du|) |Du − Dwj | dx ≤ c
|F ||u − wj | dx
(48)
Bj
Bj
holds for p > 1 and c ≡ c(n, N, p, ν). Since p ≥ 2, we have
Z
Z
|Du − Dwj |p dx ≤
|F ||u − wj | dx
Bj
Bj
!1/p∗
Z
|u − wj |p
≤
∗
Bj
!1/(p∗ )0
Z
|F |(p
∗ 0
)
dx
Bj
!1/p
Z
p
!1/q
Z
q
|Du − Dwj |
≤ crj
Bj
|F | dx
Bj
so that (57) follows. Notice that here we are implicitly using the fact that,
with q > 1 and p ≥ n, by (45) we can choose p∗ such that 1 < (p∗ )0 ≤ q;
A nonlinear Stein theorem
17
the dependence of the constant c5 on q stems from this fact. As for (47),
this follows applying (46) on Bj−1 and Bj
Z
|Dwj−1 − Dwj |p dx
Bj
c
≤ n
σ
≤ cσ
Z
Z
p
|Du − Dwj |p dx
|Du − Dwj−1 | dx + c
Bj−1
Bj
p(n−q)
−n− q(p−1)
q
rj−1
!p/[q(p−1)]
Z
q
|F | dx
.
Bj−1
The proof is complete.
Lemma 5. Let wj , vj as in (10)-(11), respectively, with j ≥ 0. There exists
a constant c ≡ c(n, N, p, ν, L) such that
Z
Z
2
2
|V (Dvj ) − V (Dwj )| dx ≤ c[ω(rj )]
|Dwj |p dx
(49)
1
2 Bj
1
2 Bj
holds. Moreover, when p ≥ 2 there exists a constant c7 ≡ c7 (n, N, p, ν, L)
such that the following holds too:
!1/p
Z
p
|Dvj − Dwj | dx
1
2 Bj
2/p
!1/p
Z
p
≤ c7 [ω(rj )]
|Dwj | dx
. (50)
1
2 Bj
Proof. The weak formulations of (10) and (11) give
Z
hγ(x0 )|Dvj |p−2 Dvj − γ(x)|Dwj |p−2 Dwj , Dϕi dx = 0 ,
1
2 Bj
which holds whenever ϕ ∈ W01,p ( 12 Bj , RN ) and can be re-written as
Z
γ(x0 )h|Dvj |p−2 Dvj − |Dwj |p−2 Dwj , Dϕi dx
I1 :=
1
Z2
Bj
(γ(x) − γ(x0 ))h|Dwj |p−2 Dwj , Dϕi dx =: I2 ,
=
1
2 Bj
where we have chosen ϕ = vj − wj . By using (20)-(21) we have
Z
|V (Dvj ) − V (Dwj )|2 dx ≤ cI1 ≤ c|I2 |
(51)
1
2 Bj
with c ≡ c(n, N, p, ν). We estimate I2 as follows, using also the concavity of
ω(·) and the fact that ω(·) ≤ 1:
Z
(5)
|I2 | ≤ cω(rj )
|Dwj |p−1 |Dvj − Dwj | dx
1
2 Bj
18
Tuomo Kuusi, Giuseppe Mingione
Z
(|Dwj | + |Dvj |)p−1 |Dvj − Dwj | dx
≤ cω(rj )
1
2 Bj
Z
(|Dwj | + |Dvj |)p/2 (|Dwj | + |Dvj |)(p−2)/2
= cω(rj )
1
2 Bj
·|Dvj − Dwj | dx
(20)
Z
(|Dwj | + |Dvj |)p/2 |V (Dvj ) − V (Dwj )| dx
≤ cω(rj )
1
2 Bj
Z
|V (Dvj ) − V (Dwj )|2 dx
≤ ε
1
2 Bj
+
c[ω(rj )]2
ε
Z
(|Dwj | + |Dvj |)p dx
1
2 Bj
c[ω(rj )]2
≤ ε
|V (Dvj ) − V (Dwj )| dx +
1
ε
2 Bj
Z
2
Z
|Dwj |p dx ,
1
2 Bj
where c ≡ c(n, N, p, L). Notice that in the last estimate we have used the
inequality
Z
Z
p
|Dvj | dx ≤
|Dwj |p dx ,
(52)
1
2 Bj
1
2 Bj
which is a consequence of the fact that vj minimizes the p-energy
Z
z 7→
|Dz|p dx
1
2 Bj
in its Dirichlet class wj + W01,p ( 12 Bj , RN ). Connecting the above estimate
found for I2 to (51) yields (49), by standardly choosing ε ≡ ε(n, N, p, ν, L)
suitably small and reabsorbing terms. When p ≥ 2, again applying (20) we
obtain
!1/p
!1/p
Z
Z
|Dvj − Dwj |p dx
|V (Dvj ) − V (Dwj )|2 dx
≤
1
2 Bj
1
2 Bj
so that (50) readily follows from (49).
2.8. Basic comparison estimate for p ≥ 2
For the case p ≥ 2 we now fix the number q as
 np

if n > 2 or p > 2
q := n + p

3/2 if n = p = 2 .
We notice that with this definition we have that

np

= (p∗ )0 ≤ q < n if p < n
np − n + p

1<q<n
if p ≥ n
(53)
(54)
A nonlinear Stein theorem
and
19

0
 np if n > 2 or p > 2
q 0 := n − p0

3 if n = p = 2 .
(55)
The value of the exponent q will remain fixed as above for the rest of the
paper as long as we shall be considering the case p ≥ 2. The value of q will
be defined in a different way when 1 < p < 2 (see (68) below).
Lemma 6. Let u be as in Theorem 1 with p ≥ 2 and wj , vj as in (10)-(11),
respectively, for j ≥ 1, and finally let the number q be as in (53). Assume
that for λ > 0 it holds that
!1/q
Z
|F |q dx
rj−1
≤ λp−1
(56)
Bj−1
and that for a constant A ≥ 1 the inequalities
sup |Dwj | ≤ Aλ
λ
≤ |Dwj−1 | ≤ Aλ
A
and
1
2 Bj
in Bj
(57)
also hold. Then there exists a constant c8 ≡ c8 (n, N, p, ν, L, A, σ) such that
!1/p0
Z
0
|Du − Dvj |p dx
1
2 Bj
≤ c8 ω(rj )λ + c8 λ2−p rj−1
!1/q
Z
|F |q dx
.
(58)
Bj−1
Proof. We will first prove that there exists a constant c ≡ c(n, p, ν, A, σ)
such that
!1/q
!1/p0
Z
Z
p0
q
2−p
|Du − Dwj | dx
|F | dx
.
(59)
≤ cλ rj−1
Bj
Bj−1
Then we shall prove that there exists c ≡ c(n, N, p, ν, L, A, σ) such that
!1/p0
Z
0
|Dvj − Dwj |p dx
1
2 Bj
≤ cω(rj )λ + cλ
2−p
!1/q
Z
q
|F | dx
rj−1
.
(60)
Bj−1
At this point (58) follows combining (59) and (60). The rest of the proof
accordingly splits into two steps.
Step 1: Proof of (59). We will use the following rescaled maps:
w̄j−1 :=
wj−1
λ
and
w̄j :=
wj
.
λ
(61)
20
Tuomo Kuusi, Giuseppe Mingione
Then we estimate, by mean of (57)
!1/p0
Z
0
|Du − Dwj |p dx
Bj
!1/p0
Z
≤ Ap−2
0
0
|Dw̄j−1 |p (p−2) |Du − Dwj |p dx
Bj
!1/p0
Z
p0 (p−2)
|Dw̄j − Dw̄j−1 |
≤c
p0
|Du − Dwj | dx
Bj
!1/p0
Z
0
0
|Dw̄j |p (p−2) |Du − Dwj |p dx
+c
=: cI3 + cI4 (62)
Bj
where the constant c depends on p, A. Then we find
I3
(61)
λ2−p
=
!1/p0
Z
0
0
|Dwj − Dwj−1 |p (p−2) |Du − Dwj |p dx
Bj
≤
!(p−2)/p
Z
2−p
p
|Dwj − Dwj−1 | dx
λ
Bj
!1/p
Z
p
·
|Du − Dwj | dx
(63)
Bj
(46),(47)
≤
cλ
2−p
q
rj−1
!1/q
Z
q
|F | dx
,
Bj−1
where c depends on n, N, p, ν, σ. We proceed with the estimation of I4 ; we
write
!1/p0
Z
(p−2)/(p−1)
|Dw̄j |
I4 =
p0
p−2
|Du − Dwj | |Dw̄j |
dx
Bj
and apply Hölder’s inequality with respect to the measure µ = |Dw̄j |p−2 dx
with conjugate exponents 2/p0 and 2/(2 − p0 ) (when p > 2), to get
!1/2
Z
I4 ≤
|Dw̄j |
p−2
2
!(p−2)/(2p)
Z
p
|Du − Dwj | dx
Bj
|Dw̄j | dx
.
Bj
(64)
Now we observe that
Z
(61)
|Dw̄j |p dx ≤
Bj
c
λp
Z
p
Z
|Dw̄j−1 |p dx
|Dwj − Dwj−1 | dx + c
Bj
Bj
A nonlinear Stein theorem
(47),(57)
≤
21
c
λp
q
rj−1
p
! q(p−1)
Z
|F |q dx
(56)
+c ≤ c
Bj−1
and the constant c depends on n, N, p, ν, A, σ. Merging the last estimate
with (64) and recalling (61) yields
!1/2
Z
I4 ≤ cλ(2−p)/2
|Dwj |p−2 |Du − Dwj |2 dx
Bj
≤ cλ
(2−p)/2
!1/2
Z
p−2
(|Dwj | + |Du|)
2
|Du − Dwj | dx
Bj
(48)
≤ cλ
(2−p)/2
!1/2
Z
|F ||u − wj | dx
(65)
Bj
again for c ≡ c(n, N, p, ν, A, σ). We estimate the last integral appearing in
the above display as follows:
!1/q0 Z
!1/q
Z
Z
0
|u − wj |q dx
|F ||u − wj | dx ≤
Bj
|F |q dx
Bj
(55)
Bj
!1/p0
Z
p0
≤ c
rjq
|Du − Dwj | dx
!1/q
Z
q
|F | dx
Bj
.
Bj
Of course in the last inequality we have used Sobolev embedding theorem.
Connecting the last estimate with (65) and in turn with (62), (63) yields
!1/q
!1/p0
Z
Z
0
|Du − Dwj |p dx
≤ cλ2−p
q
rj−1
Bj
|F |q dx
Bj−1
!1/p0

Z
0
|Du − Dwj |p dx
+c 
λ2−p
q
rj−1
Bj
1
≤
2
!1/q 1/2
Z
|F |q dx

Bj−1
!1/p0
Z
0
|Du − Dwj |p dx
+ cλ2−p
Bj
q
rj−1
!1/q
Z
|F |q dx
Bj−1
for a constant c depending only on n, N, p, ν, A, σ and (59) follows.
Step 2: Proof of (60). Using the first inequality in (57) in (49) and
(50) yields
Z
|V (Dvj ) − V (Dwj )|2 dx ≤ c[ω(rj )]2 (Aλ)p
(66)
1
2 Bj
and
!1/p
Z
p
|Dvj − Dwj | dx
1
2 Bj
≤ c[ω(rj )]2/p Aλ ,
(67)
22
Tuomo Kuusi, Giuseppe Mingione
respectively. Recalling the definitions in (61) we write
!1/p0
Z
0
|Dvj − Dwj |p dx
1
2 Bj
(57)
!1/p0
Z
≤ Ap−2
0
0
|Dw̄j−1 |p (p−2) |Dvj − Dwj |p dx
1
2 Bj
!1/p0
Z
p0 (p−2)
≤c
|Dw̄j − Dw̄j−1 |
p0
|Dvj − Dwj | dx
1
2 Bj
!1/p0
Z
p0 (p−2)
|Dw̄j |
+c
p0
|Dvj − Dwj | dx
=: cI5 + cI6 .
1
2 Bj
To estimate I5 we argue as follows:
!(p−2)/p
Z
I5 ≤
p
|Dw̄j − Dw̄j−1 | dx
|Dvj − Dwj | dx
1
2 Bj
(67)
!1/p
Z
p
1
2 Bj
!(p−2)/p
Z
p
[ω(rj )]2/p λ
|Dw̄j − Dw̄j−1 | dx
≤
1
2 Bj
Z
|Dw̄j − Dw̄j−1 |p dx
≤ cω(rj )λ + cλ
1
2 Bj
Z
(61)
= cω(rj )λ + cλ1−p
|Dwj − Dwj−1 |p dx
1
2 Bj
(47)
≤ cω(rj )λ + cλ
q
rj−1
1−p
!p/[q(p−1)]
Z
q
|F | dx
Bj−1
(56)
≤ cω(rj )λ + cλ
2−p
!1/q
Z
q
|F | dx
rj−1
.
Bj−1
As for I6 , observe that, since |Dw̄j | ≤ A in 12 Bj , then
I6 ≤ A
!1/p0
Z
(p−2)/2
p0 (p−2)/2
|Dw̄j |
p0
|Dvj − Dwj | dx
1
2 Bj
!1/2
Z
|Dw̄j |
≤ c
p−2
2
|Dvj − Dwj | dx
1
2 Bj
(61)
= cλ
(2−p)/2
!1/2
Z
p−2
|Dwj |
2
|Dvj − Dwj | dx
1
2 Bj
≤ cλ
(2−p)/2
!1/2
Z
p−2
(|Dvj | + |Dwj |)
1
2 Bj
2
|Dvj − Dwj | dx
A nonlinear Stein theorem
(20)
≤ cλ
(2−p)/2
23
!1/2
Z
2
|V (Dvj ) − V (Dwj )| dx
1
2 Bj
(66)
≤ cω(rj )λ .
Merging the estimates found for I5 and I6 with (62) finally yields (60).
2.9. Basic comparison estimate for 1 < p ≤ 2
We fix the exponent q for the case 1 < p ≤ 2 as follows:

np

= (p∗ )0 if 1 < p < 2
q := np − n + p

3/2
if n = p = 2 .
(68)
With this definition condition (54) is still satisfied. Moreover, note that the
above definition of q coincides with the one in (53) when p = 2. In this case
we are giving the definition twice since we want to show that the different
exponents used in the cases p ≥ 2 and 1 < p ≤ 2 actually coincide for
p = 2, and that all the estimates we have for the p-Laplacian operator are
stable when p approaches 2. For the same reason, the constants c5 and c8
appearing in the next lemma have the same name of the similar constants
introduced in (46) and (58), respectively; this is due to the fact that the
constants below will play a similar role in the case 1 < p ≤ 2 and the of
those in (46) and (58) when p ≥ 2.
Lemma 7. Let u be as in Theorem 1 with 1 < p ≤ 2 and wj , vj as in
(10)-(11), respectively, for j ≥ 1, and finally let the number q be as in (68).
Assume that for λ > 0 the inequalities
!1/q
Z
|F |q dx
rj
≤ λp−1
(69)
Bj
and
!1/p
Z
p
|Du| dx
≤λ
(70)
Bj
hold. Then there exists a constant c8 ≡ c8 (n, N, p, ν, L, A, σ) such that
!1/p
Z
|Du − Dvj |p dx
1
2 Bj
≤ c8 ω(rj )λ + c8 λ
2−p
!1/q
Z
q
|F | dx
rj−1
.
(71)
.
(72)
Bj−1
Moreover, there exists a constant c5 ≡ c5 (n, N, p, ν) such that
!1/p
!1/q
Z
Z
|Du − Dwj |p dx
Bj
≤ c5 λ2−p rj
|F |q dx
Bj
24
Tuomo Kuusi, Giuseppe Mingione
Proof. We will first prove (72) and then we will prove
!1/p
Z
p
|Dvj − Dwj | dx
≤ cλω(rj ) + cλ
2−p
!1/q
Z
q
|F | dx
rj
1
2 Bj
(73)
Bj
for a constant c ≡ c(n, N, p, ν, L, A, σ), so that (71) will follow via elementary manipulations and triangle inequality. We start applying (22) in order
to have
|Du − Dwj |p ≤ c|V (Du) − V (Dwj )|2 + c|V (Du) − V (Dwj )|p |Du|(2−p)p/2
for c ≡ c(n, N, p), so that, by means of Hölder’s inequality we have
!1/p
Z
|Du − Dwj |p dx
!1/p
Z
|V (Du) − V (Dwj )|2 dx
≤c
Bj
Bj
!1/2
Z
!(2−p)/2p
Z
2
p
|V (Du) − V (Dwj )| dx
+c
|Du| dx
Bj
Bj
=: cI7 + cI8 .
For I7 we instead get
(20),(48)
I7
≤
!1/p
Z
|F ||u − wj | dx
c
Bj
!1/pq0
Z
≤
q0
|F | dx
Bj
≤
q
|u − wj | dx
c
(68)
!1/pq
Z
Bj
!1/p2
Z
rjq
p
|Du − Dwj | dx
c
q
|F | dx
Bj
Bj
!1/p
Z
≤
!1/pq
Z
p
|Du − Dwj | dx
ε
rjq
+c
≤
q
|F | dx
Bj
(69)
!1/[q(p−1)]
Z
Bj
!1/p
Z
p
|Du − Dwj | dx
ε
+ cλ
2−p
!1/q
Z
q
|F | dx
rj
Bj
,
Bj
where c ≡ c(n, N, p, ν). Proceeding similarly as in the last display, for I8 we
obtain
!1/2
Z
(70)
I8
≤
λ(2−p)/2
|V (Du) − V (Dwj )|2 dx
Bj
(20),(48)
≤
cλ
(2−p)/2
!1/2
Z
|F ||u − wj | dx
Bj
A nonlinear Stein theorem
≤
cλ
25
!1/2p
Z
(2−p)/2
rjq
p
|Du − Dwj | dx
≤
Bj
!1/p
|Du − Dwj |p dx
ε
q
|F | dx
Bj
Z
!1/2q
Z
!1/q
Z
+ cλ2−p rj
Bj
|F |q dx
.
Bj
Connecting the estimates in the last three displays, choosingε ≡ ε(n, N,
p, ν) small enough and reabsorbing terms yields (72). Next we turn to the
proof of (73). From the proof of Lemma 6 we recall that the content of
display (51) still holds in the case 1 < p ≤ 2. Then we use (20) to have
|Dvj − Dwj |p ≤ c|V (Dvj ) − V (Dwj )|p (|Dvj | + |Dwj |)p(2−p)/2
so that, using Hölder’s inequality we conclude with (73) as follows and
therefore the proof of the lemma is complete:
!1/p
Z
p
|Dvj − Dwj | dx
1
2 Bj
!1/2
Z
2
|V (Dvj ) − V (Dwj )| dx
≤c
1
2 Bj
!(2−p)/2p
Z
(|Dvj | + |Dwj |)p dx
·
1
2 Bj
(49),(52)
≤
!1/p
Z
|Dwj |p dx
cω(rj )
1
2 Bj
!1/p
Z
p
|Du − Dwj | dx
≤ cω(rj )
≤
cλ
2−p
p
|Du| dx
+ cω(rj )
1
2 Bj
(72),(70)
!1/p
Z
1
2 Bj
!1/q
Z
q
|F | dx
rj
+ cω(rj )λ .
Bj
3. A pointwise gradient bound
We start with a gradient L∞ -bound which has its own interest for essentially
three reasons. First, L∞ -bounds for gradients of solutions are available for
p-Laplacian systems without coefficients (i.e. γ(·) is a constant function)
as derived in [3, 6], but the known techniques do not apparently extend to
treat the case when coefficients are involved, at least in the case they are not
differentiable, as here. Second, already in the case when no coefficients are
present the two dimensional case n = 2 remained an open issue, essentially
for technical reasons. Third, and most importantly for us, many of the
arguments developed here are at the origin of those used in the proof of the
26
Tuomo Kuusi, Giuseppe Mingione
main Theorem 1 in the final Section 4. To formulate the result we introduce
the exponents s and q as follows:
( 0
p if p ≥ 2
s :=
(74)
p if 1 < p ≤ 2
and
(
q :=
the number in (53) if
p≥2
the number in (68) if 1 < p ≤ 2 .
(75)
We again notice an overlapping in the two definitions above when p = 2;
this is not by chance, but it is done with the purpose to show that whenever
we shall distinguish between the cases p ≥ 2 and 1 < p ≤ 2, all the methods
will coincide for the case p = 2. The gradient bound is now featured in the
following:
Theorem 4. Let u be as in Theorem 1; then Du is locally bounded in Ω.
Moreover, there exists a constant c ≥ 1 and a positive radius R0 , both
depending only on n, N, p, ν, L, ω(·), such that the pointwise estimate
!1/s
Z
|Du|s dx
|Du(x0 )| ≤ c
B(x0 ,r)
1/(p−1)
+ ckF kL(n,1)
(76)
holds whenever B(x0 , 2r) ⊂ Ω, x0 is a Lebesgue point of Du and 2r ≤ R0 .
Estimate (76) holds with no restriction on r when the coefficient function
γ(·) is constant.
Remark 1. Theorem 4 is the first step towards the proof of the gradient
continuity stated in Theorem 1; on the other hand, once Theorem 1 is
gained, it is clear that estimate (76) holds for every point x0 . Moreover, a
standard covering argument based on (76) gives that the following estimate:
1/p
Z
1/(p−1)
kDukL∞ (Br/2 ) ≤ c
|Du|p dx
+ ckF kL(n,1)
Br
holds for every Br ⊂ Ω.
Proof (of Theorem 4). Let us first briefly unveil the skeleton of the proof. We
shall use the quantity S(x0 , r, σ) ≡ Sq (x0 , r, σ) defined in (16), the number
q introduced in (75), and the chain of shrinking balls Bi considered in (9).
This chain will have as starting ball the one considered in the statement
of the Theorem, that is, B(x0 , r) with r ≤ R0 . Both the parameter σ ∈
(0, 1/4) appearing in (9) and the radius R0 will be chosen in a few lines as
a functions depending only on n, N, p, ν, L, ω(·), but not on the solution u.
More precisely, we shall prove that
!1/s
Z
|Du|s dx
|Du(x0 )| ≤ λ := H1
B(x0 ,r)
+ H2 [S(x0 , r, σ)]1/(p−1) , (77)
A nonlinear Stein theorem
27
where the large constants H1 , H2 are again going to be chosen in a few
lines as functions depending only on n, N, p, ν, L, ω(·). The proof of (76)
then follows by using (17) while the local boundedness of Du follows via
a standard covering argument. In order to prove (77) we show that the
inequality |(Du)Bi | ≤ λ holds at least for a subsequence of indexes i. Since
x0 is assumed to be a Lebesgue point of Du, (77) therefore follows. We tell in
advance that in what follows certain auxiliary constants will be deliberately
chosen larger/smaller than necessary both for the sake of readability and to
recall the reader that their role is exactly to be large/small. The rest of the
proof proceeds now in three steps. Needless to say, in (77) we may assume
that λ > 0, otherwise the statement of the Theorem 4 follows trivially.
Step 1: Choice of the constants and exit time. Keeping Theorems
2 and 3 in mind, we set
A := 105n max{c2 , c3 } ,
δ := 10−5 ,
ε̄ := 4−(n+4) ,
(78)
where c2 ≡ c2 (n, N, p, ν, L) and c3 ≡ c3 (n, N, p) are as in Theorem 2 and
Theorem 3, respectively; hence, A depends only on n, N, p, ν, L. The choice
in (78) allows to determine the constant σ1 ≡ σ1 (n, N, p, ν, L, ω(·)) in Theorem 2, the constants c4 ≡ c4 (n, N, p, A) and α ≡ α(n, N, p) > 0 in Theorem
3 and σ2 ≡ σ2 (n, N, p, ν, L) in Lemma 3. We now define the quantity
σ := min {σ1 , σ2 } ∈ (0, 1/4)
(79)
that again depends only on n, N, p, ν, L, ω(·) by taking into account all the
previous dependencies. Finally, with such a choice of σ and q we determine
the constants c5 , c6 , c7 , c8 in Lemmas 4-7, as once again functions depending
on n, N, p, ν, L, ω(·). We only notice that the constants c5 and c8 have been
defined twice (Lemmas 4-6 and Lemma 7, respectively), since they play a
similar role according to which of the cases p ≥ 2 and 1 < p ≤ 2 is actually
occurring. We then define k ≡ k(n, N, p, ν, L, ω(·)) as the smallest integer
satisfying
σn
(80)
c4 σ kα ≤ 6 and k ≥ 2 .
10
Finally, to put order in all the constant whose name starts by “c”, we
introduce
8
Y
ci ,
(81)
cf =
i=2
which is of course larger than all the constants contributing to the above
product as all of them are larger than one. Again, with the choice made
above cf only depends on n, N, ν, L, ω(·). We now proceed with the choice
of H1 , H2 and R0 . We set


H1 := 105n σ −2n

h
imax{1,1/(p−1)}
(82)

 H2 := 6n 106 σ −n(k+5) cf
.
28
Tuomo Kuusi, Giuseppe Mingione
In this way also the constants H1 and H2 ultimately depend only on n, N,
p, ν, L and ω(·). As for R0 , we recall the definition of R1 ≡ R1 (n, N, p,
ν, L, ω(·)) from Theorem 2 and then we fix a radius R2 to be small enough
to verify
Z R2
d%
σ 2n
−n(k+4)
min{1,2/p}
−1
ω(%)
≤ n 6
(83)
σ
[ω(R2 )]
+σ
%
6 10 cf
0
is satisfied. Notice that this is possible by (6). Finally we set
R0 := min{R1 , R2 }/4
and again we have a quantity that depends on n, N, p, ν, L, ω(·). This means
that, in the rest of the proof, when considering the ball in the statement
of Theorem 4, and when adopting the setting of Section 2.2 with the maps
wj , vj in (10)-(11), we will always have 2r ≤ R0 . In particular, for the
shrinking balls Bi defined in (9) we use the constant σ that has been determined in (79). Accordingly, we define the following relevant quantities for
i ≥ 0:

ai := |(Du)Bi |


1/s
Z
(84)
s

 Ei := Es (Du, Bi ) =
|Du − (Du)Bi | dx
Bi
1/q
q
|F | dx
,
Z
µi := ri
so that S(x0 , r, σ) =
Bi
∞
X
µi .
(85)
i=0
Finally, for integers i ≥ 1, we define
!1/s
Z
s
|Du| dx
Ci :=
1/s
|Du|s dx
+ 2σ −n Ei .
Z
+
Bi−1
Bi
Now we notice that the choice of H1 in (82) allows to conclude that
C1 ≤ 4σ
−2n
!1/s
Z
s
|Du| dx
≤
B(x0 ,r)
λ
4σ −2n λ
≤
.
H1
1000
We further observe that, thanks to the inequality in the above display, we
may proceed under the additional assumption that there exists an exit time
index ie ≥ 1 such that
Cie ≤
λ
1000
and
Ci >
λ
1000
∀ i > ie .
(86)
Indeed, were this not the case, there would exist a subsequence {ij }j of
indexes such that Cij ≤ λ/1000, that would in turn imply that
|Du(x0 )| = lim aij ≤ lim sup Cij ≤ λ ,
j
j
A nonlinear Stein theorem
29
because x0 is a Lebesgue point of Du. Hence (76) would follow. Therefore
the rest of the proof proceeds assuming (86). Before going on let us record
another couple of estimates that are going to be used in the following. The
first one is
σ −n(k+4) [ω(r0 )]min{1,2/p} +
∞
X
ω(ri ) ≤
i=0
σ 2n
6n 106 c
.
(87)
f
Estimate (87) is a consequence of (83) and of the next computation
Z R2
Z R2
∞ Z
d% X ri
d%
d%
ω(%)
=
ω(%)
+
ω(%)
%
%
%
0
r0
i=0 ri+1
X
∞
∞
X
1
≥ log
ω(ri+1 ) + log 4ω(r0 ) ≥ σ
ω(ri ) (88)
σ i=0
i=0
where we used the fact that r ≡ r0 ∈ (0, R2 /4] and that σ ≤ 1/4. The
second estimate follows directly from the definition of S(x0 , r, σ), (77) and
(82), and is the following:
!1/q
Z
≤ λp−1
|F |q dx
µj = rj
for every j ≥ 0 .
(89)
Bj
This inequality will be used to apply Lemmas 6 and 7. Finally we remark
that due to the exit time argument used above, from now on we shall restrict
our attention to the indexes j ≥ ie .
Step 2: Upper and lower bounds, and a decay estimate. For
j ≥ ie we now consider the following condition:

!1/s Z
!1/s 

 Z
|Du|s dx
,
|Du|s dx
≤ λ (90)
Ind(j) :
max

 Bj−1
Bj
and determine a few consequences of it to be proven shortly. The first two
are the following comparison estimates:
!1/s
Z
σ n(k+5) λ
Ind(j) =⇒
|Du − Dwj−1 |s dx
≤
(91)
2n 106
Bj−1
and
!1/s
Z
s
|Du − Dvj | dx
Ind(j) =⇒
1
2 Bj
≤
σ n(k+5) λ
.
2n 106
(92)
We also prove a series of upper and lower bounds, namely
Ind(j) =⇒
(93)
sup |Dwj−1 | ≤ Aλ , sup |Dwj | ≤ Aλ and sup |Dvj | ≤ Aλ
1
2 Bj−1
1
2 Bj
1
4 Bj
30
Tuomo Kuusi, Giuseppe Mingione
λ
≤ inf |Dwj−1 | and
Bj
A
Ind(j) =⇒
λ
≤ sup |Dvj | .
A Bj+1
(94)
Finally, we prove the following decay estimate for the excess functional E(·):
Ind(j) =⇒ Ej+1 ≤
2c8 λ2−p
2c8 λ
1
µj−1 .
Ej + n ω(rj ) +
4
σ
σn
(95)
We now proceed with the proofs, starting by the one of (91) in the case
p ≥ 2, when by definition of s it is s = p0 ≤ p:
!1/s
Z
s
|Du − Dwj−1 | dx
(46)
1/(p−1)
≤ c5 µj−1
Bj−1
(85)
≤ c5 [S(x0 , r, σ)]1/(p−1)
(77)
c5 λ
H2
(82) σ n(k+5) λ
.
≤
6n 106
≤
In the case p ∈ (1, 2], when instead s = p, we have
!1/s
Z
s
(72)
≤ c5 λ2−p µj−1
|Du − Dwj−1 | dx
Bj−1
(85)
≤ c5 λ2−p S(x0 , r, σ)
(77)
≤
(82)
≤
c5 λ
H2p−1
σ n(k+5) λ
.
6n 106
Hence (91) holds in the full range p > 1. For future convenience, we also
record a similar estimate, that is
!1/s
Z
s
|Du − Dwj | dx
≤
Bj
σ n(k+5) λ
.
6n 106
(96)
The proof of (92) will also involve the first two inequalities appearing in
(93), so we start proving them. Using (96) gives
!1/s
Z
|Dwj |s dx
Bj
!1/s
Z
|Du|s dx
≤
Bj
≤ 2λ .
!1/s
Z
|Du − Dwj |s dx
+
Bj
A nonlinear Stein theorem
31
Applying (23) yields
!1/s
Z
s
sup |Dwj | ≤ c2
|Dwj | dx
1
2 Bj
(78)
≤ 2c2 λ ≤ Aλ
(97)
Bj
so that the second inequality in (93) follows. Obviously, the first inequality
in (93) follows applying the same argument to wj−1 in Bj−1 . We are now
ready for the proof of (92); we start by treating the case p ≥ 2, when
s = p0 ≤ p. We have
!1/s
Z
s
|Du − Dvj | dx
1
2 Bj
!1/s
Z
s
≤
|Du − Dwj | dx
≤
≤
p
|Dvj − Dwj | dx
+
1
2 Bj
(96),(50)
!1/p
Z
1
2 Bj
σ n(k+5) λ
+ c7 [ω(rj )]2/p
6n 106
!1/p
Z
p
|Dwj | dx
1
2 Bj
σ n(k+5) λ
+ c7 [ω(rj )]2/p sup |Dwj |
1
6n 106
Bj
2
(97)
≤
σ
n(k+5)
6n 106
λ
+ 2c2 c7 [ω(rj )]2/p λ
σ n(k+5) λ σ n(k+5) λ
+
6n 106
4n 106
n(k+5)
σ
λ
.
≤
2n 106
(87)
≤
In the case 1 < p ≤ 2, we recall that s = p and get
!1/s
Z
s
|Du − Dvj | dx
(71)
≤
c8 ω(rj )λ + c8 λ2−p µj−1
1
2 Bj
(87),(85)
≤
(77)
≤
(82)
≤
≤
σ n(k+5) λ
+ c8 λ2−p S(x0 , r, σ)
6n 106
σ n(k+5) λ
c8 λ
+ p−1
n
6
6 10
H2
σ n(k+5) λ σ n(k+5) λ
+
6n 106
6n 106
n(k+5)
σ
λ
2n 106
so that (92) is finally proved in the full range p > 1.
32
Tuomo Kuusi, Giuseppe Mingione
As for (93), it remains to prove the third estimate, since the first two
have been obtained in due course of the proof of (92). We have
!1/s
Z
s
|Dvj | dx
!1/s
Z
n
s
≤2
|Du| dx
1
2 Bj
Bj
!1/s
Z
s
(92)
≤ 2n+1 λ .
|Du − Dvj | dx
+
1
2 Bj
Therefore, by using this time (33) we have
!1/s
Z
|Dvj |s dx
sup |Dvj | ≤ c3
(78)
≤ 2n+1 c3 λ ≤ Aλ ,
1
2 Bj
1
4 Bj
i.e., the third inequality in (93), which is now completely established. We
now turn to (94). By (93) we can apply Theorem 2 to wj−1 , and recalling
that Bj = σBj−1 ⊂ σ1 Bj−1 , we have
osc Dwj−1 ≤
Bj
λ
.
105
(98)
For a similar bound on Dvj we use again (93) that in turn allows to employ
(34) thereby concluding with
(80)
osc Dvj ≤ c4 σ αk λ ≤
Bj+k
σn λ
.
106
(99)
Therefore, using also (8) we have
2σ −n Ej+k
4σ −n E(Dvj , Bj+k ) + 4σ −n
≤
!1/s
Z
|Du − Dvj |s dx
Bj+k
≤
4σ
(99),(92)
≤
−n
osc Dvj + 4σ
−n−k
!1/s
Z
s
|Du − Dvj | dx
Bj+k
1
2 Bj
4λ
λ
λ
+ 6 ≤ 5.
106
10
10
Recalling that Cj+k > λ/1000 for j ≥ ie , we then have
0
X
!1/s
Z
s
|Du| dx
m=−1
≥
Bj+m+k
λ
.
2000
Triangle inequality then yields
λ
≤
2000
0
X
m=−1
!1/s
Z
s
|Du| dx
Bj+m+k
A nonlinear Stein theorem
≤
33
!1/s
Z
2
σ n(k+1)
+
s
|Du − Dwj−1 | dx
Bj−1
0
X
!1/s
Z
|Dwj−1 |s dx
Bj+m+k
m=−1
!1/s
Z
0
X
λ
s
+
≤
|Dwj−1 | dx
106 m=−1
Bj+m+k
(91)
(100)
so that, as k ≥ 2, we also get
2 sup |Dwj−1 | ≥
Bj
0
X
!1/s
Z
s
|Dwj−1 | dx
≥
Bj+m+k
m=−1
3λ
λ
λ
≥ 4.
−
2000 106
10
At this point the oscillation control in (98) gives λ/105 ≤ |Dwj−1 | in Bj ,
that in turn, recalling (78), implies the first inequality in (94). Arguing as
for (100), but using (92), this time we gain
!1/s
Z
0
X
λ
s
≤
|Du| dx
2000 m=−1
Bj+m+k
!1/s
Z
0
X
λ
|Dvj |s dx
≤ 6+
10
Bj+m+k
m=−1
and therefore
2 sup |Dvj | ≥
Bj+1
0
X
m=−1
!1/s
Z
s
|Dvj | dx
Bj+m+k
≥
λ
103
so that the second inequality in (94) follows recalling the definition in (78). It
remains to check (95). We will do this by using all the inequalities appearing
in (93) and (94). Keeping in mind the definition of σ in (79) we can apply
Lemma 3 (with the choice t = p0 when p ≥ 2 and t = p when 1 < p < 2)
that gives
1
E(Dvj , Bj+1 ) ≤ n+4 E(Dvj , (1/2)Bj ) .
4
Now, note that by (89) and since we are assuming Ind(j) we are able to
apply Lemmas 6 and 7; therefore, by also using (8) we obtain
E(Du, Bj+1 )
≤
2E(Dvj , Bj+1 ) + σ
−n
!1/s
Z
s
|Du − Dvj | dx
1
2 Bj
(58),(71)
≤
2E(Dvj , Bj+1 ) +
c8 λ
c8 λ2−p
ω(r
)
+
µj−1
j
σn
σn
34
Tuomo Kuusi, Giuseppe Mingione
and, similarly,
≤
E(Dvj , (1/2)Bj )
2
n+1
!1/s
Z
s
|Du − Dvj | dx
E(Du, Bj ) +
1
2 Bj
(58),(71)
≤
2n+1 E(Du, Bj ) + c8 ω(rj )λ + c8 λ2−p µj−1 .
Combining the estimates in the above three displays yields (95).
Step 3: Final induction. Here we prove, by induction, that the inequality
aj + Ej ≤ λ
(101)
holds whenever j ≥ ie , thereby proving the theorem since x0 is a Lebesgue
point of Du and so
lim aj = |Du(x0 )| .
j→∞
Notice that (86) implies that (101) holds for j = ie . Therefore, let us assume
that (101) holds whenever j ∈ {ie , . . . , i} for some i ≥ ie and prove that it
holds for j = i + 1 too. We notice that this implies the validity of Ind(j)
for every j ∈ {ie , . . . , i}. Indeed, notice first that if j = ie then Ind(ie ) is
a direct consequence of (86), while when j > ie then Ind(j) follows in a
straightforward manner from (101) and the definition of Ej :
!1/s
Z
|Du|s dx
≤ aj + Ej ≤ λ
Bj
and
!1/s
Z
s
|Du| dx
≤ aj−1 + Ej−1 ≤ λ .
Bj−1
With Ind(j) being in force for j ∈ {ie , . . . , i} the inequality appearing in
(95) holds for the corresponding indexes j. Summing up then yields
i+1
X
j=ie
i
i
i−1
1X
2c8 X
2c8 λ2−p X
Ej ≤
Ej + n
ω(rj ) λ +
µj
2 i=i
σ i=i
σ n i=i −1
+1
e
e
e
and therefore, recalling the definition of S(x0 , r, σ) we have
i+1
X
j=ie
Ej ≤ 2Eie +
∞
4c8 λ2−p
4c8 X
ω(r
)
λ
+
S(x0 , r, σ)
j
σ n i=0
σn
≤ σ n Cie +
σn λ
4c8 λ
σn λ
+
≤
.
105
500
σ n H2p−1
(102)
Notice that in order to perform the last estimation we have also used (86),
(87) and (82). On the other hand, notice that
ai+1 − aie =
i
X
j=ie
(aj+1 − aj )
A nonlinear Stein theorem
35
≤
i Z
X
j=ie
= σ −n
|Du − (Du)Bj | dx
Bj+1
i
X
(102)
Ej ≤
j=ie
λ
500
and therefore (86) gives
ai+1 ≤ aie +
λ
λ
λ
≤ Cie +
≤ .
500
500
2
Connecting the last inequality with (102) yields ai+1 + Ei+1 ≤ λ so that the
induction step is verified and (101) holds for every j ≥ ie . This completes
the proof of Theorem 4.
4. Proof of Theorem 1
By Theorem 4 we know that Du is locally bounded; since we are proving
a local result, up to passing to open subsets compactly contained in Ω, we
can assume w.l.o.g. that the gradient is globally bounded, thereby letting
λ := kDukL∞ (Ω) + 1 .
(103)
The strategy of the proof consists of gaining the continuity of Du by showing
that Du is the locally uniform limit of a net of continuous maps - actually
defined via averages
x0 7→ (Du)B% (x0 ) .
(104)
To do this, we consider an open subset Ω0 b Ω and prove that for every
ε > 0 there exists a radius
rε ≤ dist (Ω0 , ∂Ω)/100 =: R∗ ,
(105)
depending only on n, N, p, ν, L, ω(·), kF kL(n,1) , ε, such that
|(Du)B% (x0 ) − (Du)Bρ (x0 ) | ≤ λε
holds for every %, ρ ∈ (0, rε ]
(106)
whenever x0 ∈ Ω0 . This proves that Du is the uniform limit of continuous
maps defined in display (104) and hence it is continuous. The rest of the
proof sees now ε > 0 and Ω0 defined as above, and it is now devoted
to establish (106). The numbers s and q stay fixed as in (74) and (75),
respectively.
Step 1: A decay estimate. We start as in Theorem 4, where the choice
of the constants in (78) is now replaced by
A :=
10005n max{c2 , c3 }
,
ε
δ :=
ε
,
105
ε̄ :=
ε
,
4n+4
(107)
where, exactly as for Theorem 4, the constants c2 ≡ c2 (n, N, p, ν, L) and
c3 ≡ c3 (n, N, p) come from Theorem 2 and Theorem 3, respectively; in
36
Tuomo Kuusi, Giuseppe Mingione
this way we have A ≡ A(n, N, p, ν, L, ε). The choice in (107) allows to
determine σ1 ≡ σ1 (n, N, p, ν, L, ω(·), ε) in Theorem 2, the constants c4 ≡
c4 (n, N, p, ν, L, ε) and α ≡ α(n, N, p) > 0 in Theorem 3 and σ2 ≡ σ2 (n, N, p,
ν, L, ε) in Lemma 3. Similarly to as done in (79), we fix σ as
σ := min {σ1 , σ2 } ∈ (0, 1/4) .
(108)
In turn, this determines the constants c5 , c6 , c7 , c8 in Lemmas 4-7. All in all
we have determined constants depending only on n, N, p, ν, L, ω(·), ε. Next,
we fix some limitations on the radii considered; R1 ≡ R1 (n, N, p, ν, L, ω(·))
still denotes the radius considered in Theorem 2. Thanks to (6) and Lemma
1 we can select a new radius R3 ≡ R3 (n, N, p, ν, L, ω(·), kF kL(n,1) , ε) in such
a way that the following smallness conditions hold:
4n max{1,p−1}
Z 2R3
σ ε
d%
λp−1 (109)
≤
[%q (|F |q )∗∗ (ωn %n )]1/q
σ 1−n/q
n
%
6 106 cf
0
and
[ω(R3 )]min{1,2/p} +
Z
4R3
ω(%)
0
σ 5n ε
d%
≤ n 6
%
6 10 cf
and we this time set
R̄0 := min{R∗ , R1 , R3 }/4 ,
with R∗ that has been defined in (105). The constant cf here is defined
as in (81) with the new values of the various factors c2 , . . . , c8 determined
here; therefore an additional dependence on ε appears for cf . Notice that
the possibility of satisfying the last inequality follows by (6) while (109) is
allowed by (18). Notice also that, again as a consequence of the content of
Section 2.3, we have R̄0 ≡ R̄0 (n, N, p, ν, L, ω(·), kF kL(n,1) , ε, R∗ ).
Remark 2. The above lines contain a certain abuse of notation: when we say
that the radius R3 (and therefore R̄0 ) depends on the quantity kF kL(n,1)
we are not really precise. Indeed, the dependence should be credited as
simply on F (·) rather than on kF kL(n,1) as the choice of R3 in (109) is
made through the smallness of the left hand side in (109). On the other
hand, since the integral appearing in (109) relates to the norm kF kL(n,1)
via the discussion in Section 2.3, and in particular through Lemma 1, we
then prefer to keep this ambiguity here as also in the following.
We now fix a ball B ≡ B(x0 , r) ⊂ Ω and accordingly to the set-up of
Section 2.2 define
Bj ≡ B(x0 , rj ) ,
rj := σ j r ,
with r ∈ (σ R̄0 , R̄0 ]
(110)
for j ≥ 0, while the maps wj and vj are defined as in (10) and (11), respectively. We also keep the notation introduced in (84), in particular we
have
Z
1/s
Z
1/q
Ei :=
|Du − (Du)Bi |s dx
, µi := ri
|F |q dx
. (111)
Bi
Bi
A nonlinear Stein theorem
37
We now proceed with the proof, and notice that all the forthcoming arguments and constants are independent of the choice of the starting radius r
in (110), i.e., they stay uniformly bounded as long as r varies in the range
[σ R̄0 , R̄0 ]; this fact will play a crucial role later. Applying (109) together
with (18) we have
sup
sup S(x, %, σ) ≤
0<%≤R3 x∈Ω0
max{1,p−1}
σ 4n ε
6n 106 cf
λp−1 ≤ λp−1 .
(112)
Moreover, a computation similar to that in (88) then gives
[ω(R3 )]min{1,2/p} +
∞
X
ω(ri ) ≤
i=0
σ 4n ε
6n 106 c
.
(113)
f
With the above definitions, and using (112) and (113), it turns out that the
inequalities in (91) and (92) can be now replaced by
!1/s
Z
s
≤
|Du − Dwj−1 | dx
Bj−1
σ 4n λε
6n 106
(114)
and
!1/s
Z
|Du − Dvj |s dx
≤
1
2 Bj
σ 4n λε
,
2n 106
(115)
respectively, with exactly the same arguments as the ones for (91)-(92). We
now consider the following condition:
!1/s
Z
|Du|s dx
Ind*(j) :
Bj+1
≥
λε
50
(116)
for every j ≥ 1 and start proving that
Ind*(j) =⇒ Ej+1 ≤
ε
2c8 λ
2c8 λ2−p
Ej + n ω(rj ) +
µj−1 .
4
σ
σn
(117)
In order to prove (117) we observe that the setting adopted here is completely similar to the one of Theorem 4 the only differences lying in two
main points: the different choice of the constants in (107)-(108) with respect to (78)-(79) and the fact that the exit time information in (86) is not
available. On the other hand, to rebalance such a lack of information we
notice two further things: first, we notice that that (107)-(108) reduce to
(78)-(79) for ε = 1 and therefore many of the inequalities implied by (78)(79) can be proved in this setting since condition Ind(j) in (90) is satisfied
for every j by (103). Second, the absence of (86) will be compensated by
the new condition Ind*(j), which tells about a lower bound for |Du| similar
to the one eventually implied by (86) in the proof of Theorem 1. We now
38
Tuomo Kuusi, Giuseppe Mingione
go for the details, starting by (93); by using (114)-(115) we see that all the
inequalities appearing in (93), that is
sup |Dwj−1 | ≤ Aλ ,
1
2 Bj−1
sup |Dwj | ≤ Aλ ,
sup |Dvj | ≤ Aλ
1
2 Bj
1
4 Bj
(118)
can be obtained here with exactly the same proof, and for the value of A
fixed in (107). We now pass to the analogs of (94), thereby proceeding as
for (98)-(99). We take the new choices in (107)-(108) and into account and
use (118); therefore applying Theorem 2 to wj−1 gives
λε
.
105
osc Dwj−1 ≤
Bj
(119)
Observe now that
λε
50
Ind*(j)
!1/s
Z
s
≤
|Du| dx
Bj+1
≤
σ
!1/s
Z
−2n
s
|Du − Dwj−1 | dx
≤
s
|Dwj−1 | dx
+
Bj−1
(114)
!1/s
Z
Bj+1
λε
+ sup |Dwj−1 |
106
Bj
so that we infer the existence of a point x̃ ∈ Bj such that |Dwj−1 (x̃)| ≥
λε/200. This, together with the first inequality in (119) gives that
λ
λε
≤
≤ inf |Dwj−1 | .
Bj
A
400
Similarly, we have
λε
50
Ind*(j)
!1/s
Z
s
≤
|Du| dx
Bj+1
≤
σ
−n
!1/s
Z
s
|Du − Dvj | dx
1
2 Bj
(115)
≤
!1/s
Z
s
|Dvj | dx
+
Bj+1
λε
+ sup |Dvj |
106 Bj+1
so that, recalling the choice in (107) we have
λ
≤ sup |Dvj | .
A Bj+1
Summarizing, we have proved that all the inequalities appearing in (93)(94) hold for the new choice of A made in (107), and this in turn suffices
to reproduce the proof for the estimate in (95), with the new choice of
A nonlinear Stein theorem
39
the constant ε̄ made in (107) and keeping in mind the new definition of σ
given in (108). But this is exactly the inequality appearing in (117), that is
therefore proved.
Step 2: Smallness of the excess. We shall prove that for every ε ∈
(0, 1), there exists a radius rε ≡ rε (n, N, p, ν, L, ω(·), kF kL(n,1) , ε, R∗ ) > 0
such that
E(Du, B% ) < λε
holds whenever % ∈ (0, rε ]
(120)
and B% ⊂ Ω. Let us first observe that
holds for every j ∈ N ∩ [1, ∞)
Ej+1 < λε
(121)
with the meaning fixed in (111). This is indeed a consequence of the estimates made in the previous step. If Ind∗ (j) does not hold then (121) follows
trivially; if on the other hand Ind∗ (j) holds then we can apply (117) that,
together with (103) and (112)-(113), yields
2c8 λ2−p
2c8 λ
ε
µj−1
Ej + n ω(rj ) +
4
σ
σn
λε
λε
λε
≤
+
+
≤ λε
2
100 100
Ej+1 ≤
that is (121). Now (120) follows with the choice rε := σ 2 R̄0 . Indeed, consider
% ≤ σ 2 R̄0 ; this means there exists an integer m ≥ 2 such that σ m+1 R̄0 <
% ≤ σ m R̄0 . Therefore we have % = σ m r for some r ∈ (σ R̄0 , R̄0 ] and (120)
follows from (121) with this particular choice of r in (110).
Step 3: Proof of (106). Thanks to (120), with ε ∈ (0, 1) always being
the one fixed in (106), we can select a radius
R4 ≡ R4 (n, N, p, ν, L, ω(·), kF kL(n,1) , ε) > 0
such that the following inequality is satisfied:
sup
sup E(Du, B(x, %)) ≤
0<%≤R4 x∈B0
σ 4n λε
.
105
(122)
Observe that it is possible to make the choice in (122) thanks to the result
in (120), that also ensures that R4 can be chosen in a way that makes it
depending only on n, N, p, ν, L, ω(·), kF kL(n,1) , ε. This time we set
R̃0 := min{R∗ , R1 , R3 , R4 }/4 ≤ R̄0
and take everywhere r ≤ R̃0 ; notice that R̃0 ultimately depends again on
n, N, p, ν, L, ω(·), kF kL(n,1) , ε, R∗ only. The sequence of shrinking balls {Bj }
corresponding to the set-up defined in Section 2.2 is now defined as
Bj ≡ B(x0 , rj ) ,
rj = σ j R̃0
(123)
40
Tuomo Kuusi, Giuseppe Mingione
for j ≥ 0. Again, we keep the notation in (111). Notice that since R̃0 ≤ R̄0
all the inequalities and the arguments developed through of Steps 1 and 2
apply here for the set-up in (123). Let us now assume for a moment that
|(Du)Bh − (Du)Bk | ≤
λε
12
holds whenever 2 ≤ k ≤ h .
(124)
Then (106) follows with the choice rε := σ 2 R̃0 . Indeed, whenever 0 < ρ <
% ≤ rε there exist two integers, 2 ≤ k ≤ h, such that
σ k+1 R̃0 < % ≤ σ k R̃0
and σ h+1 R̃0 < ρ ≤ σ h R̃0 .
Applying (122) we get, via Hölder’s inequality
Z
|(Du)B% (x0 ) − (Du)Bk+1 | ≤
|Du − (Du)B% (x0 ) | dx
Bk+1
≤
|B% (x0 )|
|Bk+1 |
Z
|Du − (Du)B% (x0 ) | dx
B% (x0 )
≤ σ −n E(Du, B% (x0 ))
(122)
≤
λε
,
10
and, similarly,
|(Du)Bρ (x0 ) − (Du)Bh+1 | ≤
λε
.
10
Using the last two estimates and (124) we conclude with (106). Therefore,
in order to complete the proof we just need to establish (124). To this aim
let us consider the set


!1/s
Z


λε
L := j ∈ N :
|Du|s dx
<
,

50 
Bj
and
Cim = {j ∈ N : i ≤ j ≤ i + m, i ∈ L, j 6∈ L if j > i}
for m ∈ N ∪ {∞}
and, finally, the number je := min L. Note that it may happen that je = ∞;
this means that the inequality in (116), considered for the balls Bj defined in
(123), holds for every j ≥ 1. We proceed with the proof of (124), obviously
assuming k < h, and treating three different cases. The first case we analyze
is when k < h ≤ je ; this means that Ind*(j) from (116) holds whenever
j ∈ {k − 1, . . . , h − 2} and, as a consequence, by (117) we have that
Ej+1 ≤
ε
2c8
2c8 λ2−p
Ej + n ω(rj )λ +
µj−1
4
σ
σn
(125)
A nonlinear Stein theorem
41
holds for every j ∈ {k − 1, . . . , h − 2}. Summing up the previous inequalities
and using (112)-(113) and (122) easily yields
h−1
X
Ei ≤ Ek−1 +
i=k
∞
σ 2n λε
4c8 λ2−p
4c8 X
S(x
,
r,
σ)
≤
ω(r
)
λ
+
0
j
σ n j=0
σn
50
therefore (124) follows since
|(Du)Bh − (Du)Bk | ≤
h−1
XZ
i=k
≤ σ −n
|Du − (Du)Bi | dx
Bi+1
h−1
X
i=k
Ei ≤
λε
.
50
(126)
The second case we consider is when je ≤ k < h, where we prove (124)
through the inequalities
|(Du)Bh | ≤
λε
25
|(Du)Bk | ≤
and
λε
.
25
(127)
In (127), we prove the former, the argument for the latter being the same
when k > je , otherwise |(Du)Bk | ≤ λε/25 is trivial if k = je ∈ L. If h ∈ L,
the first inequality in (127) follows immediately from the definition of L. On
the other hand, if h 6∈ L, then, as h > je , it is possible to consider a set Cimh h
with mh > 0, such that h ∈ Cimh h ; notice that h > ih as h 6∈ L 3 ih . Then
(117) gives that (125) holds whenever j ∈ {ih , . . . , ih + mh − 1}. Summing
up the resulting versions of (125) and the usual elementary manipulations
give
ihX
+mh
i=ih
Ei ≤ 2Eih +
∞
4c8 λ2−p
σ 2n λε
4c8 X
ω(r
)
λ
+
S(x
,
r,
σ)
≤
j
0
σ n j=0
σn
50
where again we have used (112)-(113) and (122). Therefore, as in (126), we
have
ihX
+mh
h−1
X
λε
|(Du)Bh − (Du)Bih | ≤ σ −n
Ei ≤ σ −n
Ei ≤
50
i=i
i=i
h
h
and then, using that |(Du)Bih | ≤ λε/50 as ih ∈ L, we have
|(Du)Bh | ≤ |(Du)Bih | + |(Du)Bh − (Du)Bih | ≤
λε
25
that is (127). The third and last case is when k < je < h, that can be
actually treated by a combination of the first two. It suffices to prove that
the inequalities in display (127) still hold. The former follows exactly as in
the second case. As for the latter, let us remark that, since je ∈ L then
42
Tuomo Kuusi, Giuseppe Mingione
|(Du)Bje | ≤ λε/50. On the other hand, we can use the first case k < h ≤ je
with h = je and this yields
|(Du)Bje − (Du)Bk | ≤
λε
.
50
At this stage the second estimate in (127) follows via triangle inequality.
The proof of Theorem 1 is complete.
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