ON THE EXISTENCE OF MINIMA FOR NONCONVEX VARIATIONAL
PROBLEMS
P. CELADA – G. CUPINI – M. GUIDORZI
Abstract. We consider the problem of minimizing autonomous, multiple integrals like
Z
(P)
min
f (u , ∇u) dx : u ∈ u0 + W01,p (Ω)
Ω
where f : R × RN → [0 , ∞) is a continuous, possibly nonconvex function of the gradient variable
∇u. We discuss examples where the minimum is not attained, comment on the main results on this
subject and sketch the proof of a quite general existence result recently obtained by the authors.
Eventually, we discuss some open problems and possible directions of research in this field.
1. Introduction
Consider the following variational problem
(P)
min
Z
I(u) =
Ω
f (u(x) , ∇u(x)) dx : u ∈ u0 + W01,p (Ω) ,
where Ω is a bounded open subset of RN with N ≥ 2, 1 < p < +∞, f : R×RN → [0 , +∞) is a
continuous function, possibly nonconvex with respect to its last argument and the boundary datum
u0 is in W 1,p (Ω).
The lack of convexity of f (u , ∇u) with respect to the gradient variable ∇u inhibits the lower
semicontinuity of I and rules out the possibility of proving the existence of minimizers of (P) via
the Direct Method of the Calculus of Variations. In general, in this nonconvex framework, even
under very demanding smoothness and growth assumptions on f , the minimum problem (P) need
not have solutions. Nevertheless, in many relevant examples of nonconvex problems, the minimum
is actually achieved and the question of establishing which conditions on f , other than convexity,
ensure the existence of minimizers for (P) has been receiving increasing attention in recent years.
Among the many related papers, we mention [23], [38], [32], [28], [33], [2], [14], [15], [31], [30],
[42], [11], [12], [24], [16], [13], [4], [46], [45], [10], [3], [47], [44], [6], [7] and [8], though some of these
papers deal with one dimensional problems, a special case which is itself the subject of a very broad
literature for which we refer to [31] and the references therein. See also the references in [9] for
recent years.
In this framework of nonconvex minimum problems, the standard way of proving existence results
goes as follows: consider f ∗∗ (u , ∇u), the bipolar of f (u , ∇u) with respect to the gradient variable
∇u, i.e. the greatest convex function with respect to ∇u satisfying f ∗∗ (u , ∇u) ≤ f (u , ∇u). Then,
a suitable growth assumption on f yields the coercivity of the relaxed integral
∗∗
Z
I (u) =
f ∗∗ (u , ∇u) dx,
Ω
on the space of feasible functions u0 + W01,p (Ω) and hence a solution v for the minimum problem for
this auxiliary functional I ∗∗ exists by the Direct Method because its integrand f ∗∗ (u , ∇u) is now
2000 Mathematics Subject Classification. Primary: 49J10; Secondary: 49K10.
Key words and phrases. Nonconvex variational problems, existence of minimizers, convex integration.
1
2
P. CELADA – G. CUPINI – M. GUIDORZI
convex with respect to ∇u. If it happens that the equality f ∗∗ (v , ∇v) = f (v , ∇v) holds almost
everywhere on Ω, then v turns out to be a miminimizer of the original integral I as well. Otherwise,
one tries to modify v so as to find a new minimizer of I ∗∗ , say u, satisfying the required equality
f ∗∗ (u , ∇u) = f (u , ∇u) almost everywhere.
Obviously, the crucial and difficult link in this chain of reasoning is how to define u out of v
satisfying the equality f ∗∗ (u , ∇u) = f (u , ∇u) almost everywhere while retaining the minimality
of the original function v. To highlight how this can be accomplished, let us define the so called
detachment set where f and its convex envelope f ∗∗ are different, i.e.
n
o
D = (η , ξ) ∈ R×RN : f ∗∗ (η , ξ) < f (η , ξ)
and let D(η) be the set-valued mapping defined as the η-fixed section of D. Under mild smoothness
hypotheses on f and f ∗∗ , the detachment set D is open and, roughly speaking, the sketch of the proof
outlined above calls for showing that, if ∇v(x0 ) ∈ D(v(x0 )), then v can be locally modified around
the point x0 obtaining a new function u satisfying the differential inclusion ∇u(x) ∈ ∂D(u(x))
in a neighbourhood of x0 while, at the same time, keeping track of the requirement that u be a
minimizer. Thus, the proof is based upon a differential inclusion result and it is not surprising
that, so far, the most widely investigated cases concern Lagrangean functions f featuring a special
structure: either f (u , ∇u) = g(u) + h(∇u) or f (u , ∇u) = g(u)h(∇) with nonconvex functions
h. Indeed, in either cases, the detachment set D takes the most simple form D = R × D where
D = {ξ : h∗∗ (ξ) < h(ξ)} is the detachment set of the nonconvex function h, i.e. the set-valued
mapping η → D(η) turns out to be constant.
For these special sum-like or product-like integrals, a fairly complete understanding of attainment
and nonattainment phenomena is now available, see [11], [12], [24], [44], [47], [6] and [8]. Indeed,
whenever g and h satisfy mild regularity and growth assumptions, minimizers do exist for a very
broad class of functions g, provided the convex envelope h∗∗ of h is affine on each connected
component of the detachment set {h∗∗ < h}. Otherwise minimizers do not likely exist, even
if f (u , ∇u) = h(∇u) and the boundary datum is affine. This important result was established
independently by A. Cellina ([11] and [12]) and G. Friesecke ([24]). They proved that for lower
semicontinuous functions h with superlinear growth at infinity, i.e. h(ξ)/|ξ| → +∞ as |ξ| → +∞,
the nonconvex variational problem
Z
min
Ω
h (∇u) dx : u ∈ ua,b +
W01,1 (Ω)
with an affine boundary condition ua,b (x) = ha , xi+b has solution if and only if one of the following
two conditions is satisfied: either h is convex at a, i.e. h∗∗ (a) = h(a), or the gradient a of the affine
boundary datum ua,b is in the projection of the relative interior of a N -dimensional face of the
epigraph of the convex envelope h∗∗ of h.
As we aim at analyzing the issue of the existence of solutions to (P) for every boundary datum, we
are thus naturally lead to assume in the sequel that, whenever (η , ξ) is in D, the vector ξ is in the
projection of the relative interior of a N -dimensional face of the epigraph of the convex mapping
ξ → f ∗∗ (η , ξ), a requirement that turns out to be equivalent to the assumption that ξ → f ∗∗ (η , ξ)
be affine on each connected component of the sections D(η) of the detachment set.
A well known example of nonattainment, due to the failure of this property is the following.
2 where ξ 0 = (ξ , . . . , ξ
Example 1.1. Let Ω = (−1 , 1)N and let f (ξ) = (|ξ 0 |2 − 1)2 + ξN
1
N −1 ). Then,
1,4
there is no minimizer of I on W0 (Ω).
In this example, the infimum of I on W01,4 (Ω) is zero, but no Sobolev function u exists such that
f (∇u) = 0 almost everywhere. In fact, if f (∇u(x)) = 0 for a.e. x, then the partial derivative DN u
ON THE EXISTENCE OF MINIMA FOR NONCONVEX VARIATIONAL PROBLEMS
3
would be zero almost everywhere. Since u is zero at the boundary, then u would vanish in Ω and
we have a contradiction since I(0) = |Ω| = 2N . Note that the convex envelope of f is given by
(
∗∗
f (ξ) =
f (ξ)
2
ξN
if |ξ 0 | ≥ 1,
if |ξ 0 | < 1,
thus f ∗∗ is not affine on the detachment set {ξ : |ξ 0 | < 1}.
By contrast, in the one dimensional case, the assumption that f ∗∗ (u , u0 ) be affine as a function of
u0 on the connected components of the sections of the detachment set is automatically satisfied.
However, even in this case, the existence of solutions to (P) cannot be granted when an explicit
dependence on u is present as the following well-known Bolza’s type examples show.
Example 1.2. Let Ω = (−1 , 1)N and let f (η , ξ) = η 2 + (|ξ|2 − 1)2 . Then, there is no minimizer
of I on W01,4 (Ω).
Example 1.3. Let Ω = (−1 , 1)N and let f (η , ξ) = g(η) + (|ξ|2 − 1)2 , where

η sin 1
if η 6= 0 and η sin η1 ≥ 0,
η
g(η) =

0
else.
Then, there is no minimizer of I on W01,4 (Ω).
Some comments on the above two examples are in order. In either cases, the infimum of I on the
set of feasible functions is zero and hence a minimizer u of I would yield f (u , ∇u) = 0 almost
everywhere. In Example 1.2, this means u = 0 and |∇u| = 1 almost everywhere which is contradictory. In the other Example 1.3, the same equality would give g(u) = 0 almost everywhere. As
u is continuous by a classical regularity result (see [27]), it would follow that u takes values only
in a connected component of the level set {g = 0}. Then, the boundary condition implies that u
must vanish and the conclusion follows as in the previous example.
Recalling that f (η , ξ) = g(η) + h(ξ) and h(ξ) = (|ξ|2 − 1)2 , the main features of the previous
examples are the following:
(a)
(b)
(c)
(d)
h∗∗ is affine on {h∗∗ < h};
h fails to be convex at zero, i.e. h∗∗ (0) < h(0);
in Example 1.2, g has a strict (local) minimum;
in Example 1.3, g fails to be monotone on any closed interval whose endpoint is the origin.
It is obvious that, in the previous examples, the nonattainment is a consequence of the interplay
between (b) and either (c) or (d).
We have already pointed out the role of hypothesis (a). As regards the lack of convexity at zero,
the different behaviour of one dimensional, sum-like integrals that are convex at zero versus those
that are not is striking as the convexity at zero rules out Bolza’s type examples. This issue was
thorougly investigated in [25] where the authors address the studying of this class of problems and
prove that minimizers of I exist for “almost” every lower semicontinuous and positive function
g provided h is lower semicontinuous, superlinear at infinity and convex at zero. An additional
technical assumption on g was later removed in [39].
Though this result of [25] might be considered predictable to some extent, its proof is delicate and
is based upon the DuBois-Reymond differential inclusion, see [1], and up to now it is not clear if
this argument can be generalized to the multidimensional case N > 1.
As regards nonconvex variational problems for sum-like multiple integrals the most general existence
result is the following ([6]). Here, p∗ is the Sobolev conjugate exponent to p.
4
P. CELADA – G. CUPINI – M. GUIDORZI
Theorem 1.4. Let g : R → R and h : RN → R be a continuous and a lower semicontinuous
function, respectively, and let h∗∗ be the convex envelope of h. For 1 < p < +∞, suppose that the
following conditions hold, according to the value of p.
(a) If 1 < p ≤ N , h and g satisfy
(1.1)
c1 |ξ|p − c2 ≤ h(ξ) ≤ c3 |ξ|p + c4 ,
ξ ∈ RN ,
(1.2)
0 < g0 ≤ g(η) ≤ g1 (1 + |η|q ) ,
η ∈ R,
for some constants c3 ≥ c1 > 0, c2 , c4 ≥ 0 and g1 ≥ g0 > 0 with 1 ≤ q < p∗ ;
(b) If N < p < ∞, h and g satisfy
(1.3)
c1 |ξ|p − c2 ≤ h(ξ),
ξ ∈ RN ,
(1.4)
0 < g0 ≤ g(η),
η ∈ R,
for some constants c1 > 0, c2 ≥ 0 and g0 > 0;
(c) h∗∗ is affine on each connected component of {h∗∗ < h};
(d) g has no strict local minima;
(e) for every η ∈ R, there exists δ > 0 such that g is monotone on the two intervals [η − δ , η]
and [η , η + δ].
Then, the variational problem (P) with f (η , ξ) = g(η) + h(ξ) admits a solution for every boundary
datum u0 ∈ W 1,p (Ω).
The growth assumptions from below in (a) and (b) imply that every minimizing sequence for
(P ∗∗ )
min
Z
∗∗
I (u) =
f
∗∗
(u , ∇u) dx : u ∈ u0 +
Ω
W01,p (Ω)
has weakly convergent subsequences. This, and the lower semicontinuity of I ∗∗ along weakly convergent sequences of feasible functions (see Theorem 3.4, Chapter 3 in[17]), yields the existence of
solutions to the relaxed problem (P ∗∗ ). As described at the beginning, the proof that there exists
a solution u of (P ∗∗ ) which is also a solution to the original nonconvex problem (P) relies on the
construction of special families of solutions to differential inclusions with constant right hand side.
An important ingredient of this construction is the smoothness of solutions to (P ∗∗ ): continuity
and almost everywhere differentiability in the classical sense. For p > N , these properties are
shared by all Sobolev functions but, for 1 < p ≤ N , these properties follow from minimality and in
particular solutions to (P ∗∗ ) are locally Hölder continuous by the classical result of M. Giaquinta
and E. Giusti (see [27]) and also almost everywhere differentiable in the classical sense (see [6]).
Indeed, the following theorem hold (see [6]).
Theorem 1.5. Let 1 < p ≤ N and let f : Ω×R×RN → R be a Carathéodory function such that
c1 |ξ|p − c2 (|η|q + 1) ≤ f (x , η , ξ) ≤ c3 |ξ|p + c4 (|η|q + 1)
holds for every (x , η , ξ) ∈ Ω×R×RN and for some p ≤ q < p∗ , c3 ≥ c1 > 0 and c2 , c4 ≥ 0. Then,
every minimizer of
Z
J(u) =
Ω
f (x , u , ∇u) dx,
u ∈ u0 + W01,p (Ω),
is locally Hölder continuous and almost everywhere differentiable in the classical sense.
ON THE EXISTENCE OF MINIMA FOR NONCONVEX VARIATIONAL PROBLEMS
5
If we turn to the general case of integrals whose nonconvex Lagrangean has no special structure,
the available results concern mostly the one dimensional case, see [41], [40], [31], [26] and [9]. In
particular, in this last paper, it is shown that the two assumptions (d) and (e) on the function g
considered in Theorem 1.4, appropriately rewritten for f , ensure the existence of solutions to the
minimum problem (P). The same result was then extended by the authors to the case of multiple
integrals in [5], provided we assume in addition that f ∗∗ is affine on the connected components of
the detachment set D. We wish to emphasize that this attainment result of [5] is quite general and
holds under smoothness and structure hypotheses on the Lagrangean f that are essentially sharp:
in particular, it does not require neither any particular smoothness assumption besides continuity
of f and f ∗∗ nor any qualified convexity hypothesis at ∞ and outside D as in [21]. The proof of
this existence result is described in detail in the following Sections 2 to 3 in the special case of a
smooth function f ∗∗ .
Indeed, in this smooth case, we consider the function q : R×RN → R defined by
q(η , ξ) = f ∗∗ (η , ξ) − h∇ξ f ∗∗ (η , ξ) , ξi ,
(η , ξ) ∈ R×RN ,
where ∇ξ stands for the gradient with respect to the components of the vector ξ. The value of
q at the point (η , ξ) is then the value at the origin of the tangent affine function to the graph of
ξ → f ∗∗ (η , ξ) through the point (η , ξ). Note also that q reduces to f ∗∗ itself for ξ = 0.
Then, the hypotheses (d) and (e) considered before for sum-like integrals turn into the following
assumptions on the function q on the detachment set D:
(d’) if (η0 , ξ0 ) ∈ D, there exists δ = δ(η0 , ξ0 ) > 0 such that the function η → q(η , ξ0 ) is
monotone on both intervals [η0 − δ , η0 ] and [η0 , η0 + δ];
(e’) the function η → f ∗∗ (η , 0) has no strict, local minima on the section of D corresponding
to ξ = 0.
We refer to the following Section 2 for the exact statement of the attainment result for (P). Here, we
wish to point out only that, when f (η , 0) = f ∗∗ (η , 0) for every η, then the only condition we have
to require is (d’) which is a very weak requirement on the behaviour of f . Moreover, when f (η , ξ) =
g(η)+h(ξ) for instance and h∗∗ is smooth, it turns out that q is given by h∗∗ (ξ)−hξ , ∇h∗∗ (ξ)i+g(η)
and we thus recover the previous assumptions (d) and (e) and the attainment result of Theorem 1.4
of [6]. The same is true for the product-like case, see [8].
Finally, in the last Section 4, we discuss some open problems related to this kind of nonconvex
variational problems for multiple integrals and some possible directions of development of this field
of investigation.
2. Notations and statement of the main result
In this section, we recall some elementary definitions, notations and results, mostly from convex
analysis and measure theory.
We denote the norm of a vector ξ in RN by |ξ| and the scalar product of ξ and ζ by hξ , ζi. We also
denote the standard basis of RN by{e1 , . . . , eN } and the open ball of radius ρ > 0 in RN centered
at x0 by Bρ (x0 ). If A is a set in RN , we let int (A), A and ∂A be the interior, the closure and the
boundary of A respectively and we recall that the dimension of a convex set C is the dimension of
the smallest affine subspace containing it.
Throughout the paper, we shall consider points and subsets of R×RN . We shall write (η , ξ) for
such points and, whenever E is a subset of R×RN , we denote its sections with either η or ξ fixed
by
n
o
n
o
Eη = ξ ∈ RN : (η , ξ) ∈ E
and
E ξ = η ∈ R : (η , ξ) ∈ E
respectively and, for every point (η , ξ), the connected component of Eη containing ξ by Eη (ξ).
6
P. CELADA – G. CUPINI – M. GUIDORZI
As to measure theoretic notations and results, we denote the Lebesgue measure of a measurable
subset E of RN by |E|. We recall that a family K of compact sets containing a given point x ∈ RN
is said to shrink nicely at x if
(a) inf {|K| : K ∈ K} = 0;
and (b) sup
|K|
: K ⊂ B, B closed ball ≥ c,
|B|
K ∈ K;
for some constant c > 0 and that a Vitali covering of a measurable set E is a family of compact
sets K containing, for a.e. x ∈ E, a sequence that shrinks nicely at x itself. Then, Vitali’s covering
theorem (see [43]) states that every such covering contains a (at most) countable subfamily of sets
{Kn }n consisting of pairwise disjoint sets that cover E up to a negligible set, i.e. |E \ (∪n Kn )| = 0.
As regards functional theoretic notations, we let Ω be an open, bounded set in RN and we use
standard notations for the spaces of continuously differentiable functions on Ω and for Lebesgue
and Sobolev spaces of functions and their norms. In particular, we let p∗ be the Sobolev exponent
relative to 1 ≤ p ≤ N , i.e. p∗ = pN/(N − p) for 1 ≤ p < N and p∗ = +∞ for p = N .
Now, we introduce the class of integral functionals that we are going to consider in the sequel. Let
f : R×RN → [0 , +∞) be a continuous function. We consider the following integral functional
Z
f (u(x) , ∇u(x)) dx,
I(u) =
u ∈ W 1,p (Ω),
Ω
and the associated minimum problem
n
min I(u) : u ∈ u0 + W01,p (Ω)
(P)
o
where 1 < p < +∞ and u0 is in W 1,p (Ω). We denote the convex envelope of f with respect to the
second variable ξ by f ∗∗ : R × RN → [0 , +∞), i.e. ξ → f ∗∗ (η , ξ) is, for every η ∈ R, the largest
convex function which is everywhere less or equal than ξ → f (η , ξ). Then, f ∗∗ is Borel measurable
and we consider also the auxiliary functional
I ∗∗ (u) =
Z
f ∗∗ (u(x) , ∇u(x)) dx,
u ∈ W 1,p (Ω),
Ω
and the associated minimum problem
(P ∗∗ )
n
o
min I ∗∗ (u) : u ∈ u0 + W01,p (Ω) .
This auxiliary functional I ∗∗ coincides with the relaxed functional of I with respect to the weak
topology of W 1,p (Ω) (see Theorem 3.8, Chapter 10 in [20]) as soon as f satisfies suitable growth
assumptions like those in the first formula of (H2p ) below. Even if this does not hold, it is plain
that I ∗∗ ≤ I on W 1,p (Ω) so that any solution u to (P ∗∗ ) satisfying f ∗∗ (u , ∇u) = f (u , ∇u) almost
everywhere on Ω is a solution to (P) as well.
Next, we describe the assumptions that we are going to consider on the function f and its convex
envelope f ∗∗ . As regards the regularity of f and f ∗∗ , for the sake of simplicity we assume here that
(H1)
f ∈ C(R×RN ) and f ∗∗ ∈ C 1 (R×RN );
and we refer to [5] where the attainment result presented here (Theorem 2.1 below) is proved when
f ∗∗ is only continuous.
As regards the growth at infinity of f , we assume that f satisfies the following assumption (H2p )
according to the value of 1 < p < +∞: if 1 < p ≤ N , f is supposed to be bounded from above and
below by
(H2p )
c1 |ξ|p − c2 (1 + |η|q ) ≤ f (η , ξ) ≤ c3 |ξ|p + c2 (1 + |η|q ) ,
(η , ξ) ∈ R×RN ,
ON THE EXISTENCE OF MINIMA FOR NONCONVEX VARIATIONAL PROBLEMS
7
for some constants c3 ≥ c1 > 0 and c2 ≥ 0 with 1 ≤ q < p∗ whereas, if N < p < ∞, f is only
supposed to be such that
(H2p )
c1 |ξ|p − c2 (1 + |η|q ) ≤ f (η , ξ),
(η , ξ) ∈ R×RN ,
for some constants c1 > 0, c2 ≥ 0 and q ≥ 1.
Clearly f ∗∗ satisfies the very same growth properties as f .
The growth assumption from below ensures that every minimizing sequence for the problem (P ∗∗ )
has weakly convergent subsequences. This, and the weak lower semicontinuity of I ∗∗ on u0 +W01,p (Ω)
(see Theorem 3.4, Chapter 3 in [17]), ensure the existence of solutions to (P ∗∗ ). As regards the
regularity of these solutions, we have already recalled in the Introduction that, when 1 < p ≤ N ,
minimizers are locally Hölder continuous (see [27]) and almost everywhere differentiable in the
classical sense by Theorem 1.5 (see [6]). Moreover, the growth assumption from above can be
somewhat relaxed when the boundary datum u0 is (essentially) bounded, see the remark following
the statement of Theorem 2.1. Finally, for p > N , the same smoothness properties are shared by
all Sobolev functions.
As a consequence of (H1), the detachment set D defined by
n
o
D = (η , ξ) ∈ R×RN : f ∗∗ (η , ξ) < f (η , ξ)
is open and, as we agreed upon at the beginning, we denote its sections with either η or ξ fixed by
Dη and Dξ respectively and the connected component of Dη containing ξ by Dη (ξ).
Finally, recalling the discussion of the Introduction, we assume that f ∗∗ features the following
qualitative behaviour on the sections of the detachment set D:
(H3)
for every η ∈ R, the function ξ → f ∗∗ (η , ξ) is affine on each connected component of Dη .
To be explicit, this means that, because of (H1), the gradient ∇ξ f ∗∗ of f ∗∗ with respect to ξ is
constant on each connected component of Dη0 , i.e. for every point (η0 , ξ0 ) in D, there is a vector
d0 such that ∇ξ f ∗∗ (η0 , ξ) = d0 for every ξ ∈ Dη0 (ξ0 ). Thus,
(2.1)
f ∗∗ (η0 , ξ) = hd0 , ξi + q0 ,
ξ ∈ Dη0 (ξ0 ),
for some constant q0 ∈ R.
The constant q0 appearing in the previous formula plays an important role in our attainment result.
Its value can be computed by considering the function q : R×RN → R defined by
(2.2)
q(η , ξ) = f ∗∗ (η , ξ) − h∇ξ f ∗∗ (η , ξ) , ξi ,
(η , ξ) ∈ R×RN .
As it was mentioned in the Introduction, it is easy to check that the value of q at the point (η , ξ)
is the value at the origin of the tangent affine function to the graph of ξ → f ∗∗ (η , ξ) through the
point ξ. Moreover, it is clear that q0 = q(η0 , ξ) for every ξ ∈ Dη0 (ξ0 ) whenever (η0 , ξ0 ) ∈ D, i.e.
ξ → q(η0 , ξ) too is constant on each connected component of Dη0 .
Now, the preliminaries are over and we can state the attainment result for the nonconvex problem
(P).
Theorem 2.1. Let f : R×RN → [0 , +∞) satisfy (H1), (H2p ) for some 1 < p < +∞ and (H3).
Let q be defined by (2.2) and assume also that the following properties hold:
(2.3) for every (η0 , ξ0 ) ∈ D, there is δ = δ(η0 , ξ0 ) > 0 such that [η0 − δ , η0 + δ] ⊂ Dξ0 and such
that the restriction η ∈ [η0 − δ , η0 + δ] → q(η , ξ0 ) is monotone on each interval [η0 − δ , η0 ]
and [η0 , η0 + δ];
(2.4) if D0 6= ∅, the restriction η ∈ D0 → f ∗∗ (η , 0) has no strict, local minima on D0 .
Then, the nonconvex problem (P) has a solution for every boundary datum u0 ∈ W 1,p (Ω).
8
P. CELADA – G. CUPINI – M. GUIDORZI
It is clear from (2.2) that f ∗∗ (η , 0) = q(η , 0) for every η so that (2.4) can be equivalently stated in
terms of q by requiring that the restriction η ∈ D0 → q(η , 0) of q on the nonempty section D0 has
no strict, local minima. Note also for future purposes that the other hypothesis (2.3) implies that
(2.5) if Dξ 6= ∅, the restriction η ∈ Dξ → q(η , ξ) has only finitely many strict, local extrema in
every compact subinterval of Dξ .
As regards the hypotheses of this attainment result, we have already pointed out that the growth
assumption (H2p ) is related only with the coercivity of I ∗∗ and the regularity of minimizers of
(P ∗∗ ). Moreover, if 1 < p ≤ N and the boundary datum u0 is in L∞ (Ω) ∩ W 1,p (Ω), it can be
relaxed by requiring only that
(H20p )
c1 |ξ|p − c2 (1 + |η|q ) ≤ f (η , ξ) ≤ c(η)|ξ|p + c2 (1 + |η|q ) ,
(η , ξ) ∈ R×RN ,
for some constants c1 > 0, c2 ≥ 0 and for some function c ∈ C(R) such that c(η) ≥ c1 for every η ∈ R
with 1 ≤ q < p∗ as before. Then, every minimizer u of (P ∗∗ ) is again locally Hölder continuous
and almost everywhere differentiable in the classical sense on Ω (see Theorem 2.1 of [8]) and the
rest of the proof of Theorem 2.1 remains unchanged.
As to the main qualitative hypotheses (H3), (2.3) and (2.4), we have already remarked that they
cannot be dropped in general without affecting attainment for (P) as the examples mentioned in
the Introduction show. Thus, all the assumptions of our attainment result but (H1) - which can be
weakened to the much less demanding assumption of continuity (see [5]) - are needed in the sense
specified above and we wish to emphasize once again that the only truly demanding ones among
them are (H3) and (2.4) - provided D0 is nonempty - which are well known necessary assumptions
for nonconvex variational problems involving multiple integrals. By contrast, (2.3) is a very weak
assumption on the behaviour of f ∗∗ and its fulfillment just requires that q does not oscillate on
finer and finer scales as a function of η.
3. Proof of the main result
This section contains the proof of Theorem 2.1, the attainment result of [5] that we prove here in
the special case of a smooth convex envelope f ∗∗ .
We begin by considering again the properties of f ∗∗ and we introduce the set where f ∗∗ is affine
as a function of ξ. The motivation for doing this is the fact that the connected components of the
sections Dη of the detachment set need not be convex whereas the connected components of the
sets where ξ → f ∗∗ (η , ξ) is affine do have this property. This simple remark plays an important
role in the existence result for the related differential inclusions, see Proposition 3.1 below.
Therefore, let A be the subset of R×RN defined by the following property: for every η ∈ R, the
section Aη of A is the union of all maximal, compact, N -dimensional convex sets {Ki } such that f ∗∗
is affine on Ki as a function of ξ, i.e. f ∗∗ (η , ξ) = hdi , ξi + qi for every ξ ∈ Ki for some di ∈ RN and
qi ∈ R. Here, maximal means that there is no other compact, N -dimensional convex set containing
any Ki where the above formula for f ∗∗ holds. Note that the fact that each maximal convex set Ki
where ξ → f ∗∗ (η , ξ) is affine is compact follows from the superlinear growth of f ∗∗ as |ξ| → +∞,
see (H2p ). Each section Aη is the union of at most countably many sets Ki with this property
and every two such sets have pairwise disjoint interiors. Again, when (η , ξ) is a point of A, the
connected component of Aη containing ξ will be denoted by Aη (ξ).
Thus, the assumption (H3) can be stated equivalently by saying that Dη0 (ξ0 ) ⊂ Aη0 (ξ0 ) for every
point (η0 , ξ0 ) in D. Moreover, the formula (2.1) holds on the larger set Aη0 (ξ0 ), i.e.
(3.1)
f ∗∗ (η0 , ξ) = hd0 , ξi + q0 ,
ξ ∈ Aη0 (ξ0 ),
ON THE EXISTENCE OF MINIMA FOR NONCONVEX VARIATIONAL PROBLEMS
9
where d0 = ∇ξ f ∗∗ (η0 , ξ0 ) and q0 = q(η0 , ξ0 ) and it is easy to check that the set Aη0 (ξ0 ) is just the
set of points ξ where the previous formula holds.
Then, recall that the program outlined in the Introduction calls for considering a solution v to the
relaxed problem (P ∗∗ ) such that (v(x0 ) , ∇v(x0 )) ∈ D for some x0 and for defining new solutions to
(P ∗∗ ) which stay on the boundary of D, i.e. where f and f ∗∗ coincide, on a neighbourhood of x0 .
This is the main technical point of the paper and, following the works initiated by De Blasi and
Pianigiani on the Baire category method in [19] and by Müller and Šverák in [35] on the convex
integration of partial differential relations of Gromov ([29]), it will be accomplished by applying this
method to find special families of solutions to autonomous, first order partial differential equations
in implicit form like
H(w(x) , ∇w(x)) = 0
(3.2)
for a.e. x ∈ Ω,
where the Hamiltonian function H is given by H = f ∗∗ − f . As we said, this latter problem has
been receiving much attention in recent years and we refer to [35], [19], [18], [36] and [37] for an
extensive and systematic discussion of this kind of equations though all these papers mainly deal
with the case of vector-valued solutions w. Here, we just remark that, by contrast, we deal with
the much simpler case of real-valued solutions but we want to select, among all such solutions w to
(3.2), those featuring some kind of order-related property with respect to the original solution v to
(P ∗∗ ).
Proposition 3.1. Let f : R×RN → [0 , +∞) satisfy (H1), (H2p ) for some 1 < p < +∞ and (H3).
Let also v ∈ W 1,p (Ω) be a continuous, almost everywhere differentiable function on Ω such that
(a) v is differentiable at some point x0 ∈ Ω with gradient ξ0 = ∇v(x0 );
(b) f ∗∗ (v(x0 ) , ∇v(x0 )) < f (v(x0 ) , ∇v(x0 )).
Then, there exist two families of compact subsets Kx±0 = {Kx±0 ,ε }ε of Ω such that
(3.3)
each set Kx±0 ,ε is a neighbourhood of x0 and each family Kx±0 shrinks nicely at x0 ;
and two corresponding families of continuous functions Vx±0 = {vx±0 ,ε }ε in W 1,p (Ω) such that the
following properties hold for every ε:
(3.4)
vx±0 ,ε = v on Ω \ int Kx±0 ,ε ;
(3.5+) v(x) < vx+0 ,ε (x) ≤ v(x) + ε for every x ∈ int Kx+0 ,ε ;
(3.5−) v(x) − ε ≤ vx−0 ,ε (x) < v(x) for every x ∈ int Kx−0 ,ε ;
(3.6)
f ∗∗ vx±0 ,ε (x) , ∇vx±0 ,ε (x) = f vx±0 ,ε (x) , ∇vx±0 ,ε (x) for a.e. x ∈ Kx±0 ,ε ;
(3.7)
∇vx±0 ,ε (x) ∈ Avx± ,ε (x) (ξ0 ) for a.e. x ∈ Kx±0 ,ε ;
0
Z
(3.8)
Kx±0 ,ε
D h
vx±0 ,ε (x)
E
, ∇vx±0 ,ε (x) dx =
Z
Kx±0 ,ε
hh(v(x)) , ∇v(x)i dx for every h ∈ C(R , RN ).
The refer to [5] for the proof. Here, we just remark that the main tool of the proof – besides
Gromov’s method of convex integration as modified and described in [35] – is a rather elementary
but powerful construction which enables to glue piecewise affine and differentiable functions, thus
modifying the gradient of the latter. This kind of argument was introduced first by De Blasi and
Pianigiani in case of Lipschitz continuous functions and improved versions of this result have been
given also in [44], [47] and [8].
10
P. CELADA – G. CUPINI – M. GUIDORZI
Proof of Theorem 2.1. Recalling the definition of q in (2.2) and the hypothesis (2.3), we begin by
considering the set M of those points (η , ξ) ∈ D such that η is a strict, local extremum point of
the restriction of q to the section Dξ , i.e.
(3.9)
n
o
M = (η , ξ) ∈ D : η is a strict, local extremum point of η 0 ∈ Dξ → q(η 0 , ξ) .
It is easy to check that the hypotheses of Theorem 2.1 imply that M is a countable union of
connected components of η-sections of D, i.e.
M=
(3.10)
[
({mj }×Dj ) ,
j
where Dj is a connected component of Dmj . Moreover, D \ M is open. We refer to Proposition 4.1
in [5] for the details of the proof of these properties.
Now, let v be a solution to (P ∗∗ ) such that I ∗∗ (v) < +∞. Then, v is locally Hölder continuous and
almost everywhere differentiable on Ω in the classical sense, either because it is in W 1,p (Ω) and
p > N or because of Theorem 1.5 if 1 < p ≤ N .
As it was explained in the Introduction, the proof will be obtained by finding a new solution u to
(P ∗∗ ) such that
(3.11)
f (u(x) , ∇u(x)) = f ∗∗ (u(x) ∇u(x))
for a.e. x ∈ Ω
To this aim, recalling that f and f ∗∗ are different on D only, we first modify v so as to find this
new solution u to (P ∗∗ ) with the further property that the set
(3.12)
Eu = {x ∈ Ω : u is differentiable at x and (u(x) , ∇u(x)) ∈ D \ M}
is negligible. Here, M is the set of strict, local extrema of q defined above. Then, we show that
the set
Fu = {x ∈ Ω : u is differentiable at x and (u(x) , ∇u(x)) ∈ M}
is negligible too so that (3.11) follows. We divide the proof into two claims.
Claim 1.
There exists a solution u to (P ∗∗ ) such that the set Eu is negligible.
Proof of Claim 1. Let v be the solution to (P ∗∗ ) considered at the beginning and let Ev be the
set defined by (3.12) with v instead of u. Assume that Ev has positive measure otherwise we set
u = v and the claim is proved. Let also x0 ∈ Ω be a density point of Ev and set η0 = v(x0 ) and
ξ0 = ∇v(x0 ). Thus, (η0 , ξ0 ) is in D \ M by definition and, since this set is open, there is δ > 0 such
that (η , ξ0 ) is in D \ M too for every η ∈ I, I = [η0 − δ , η0 + δ]. Thus, recalling (3.1), we have
(3.13)
f ∗∗ (η , ξ) = hd(η) , ξi + q(η),
ξ ∈ Aη (ξ0 ),
for every η ∈ I where we have set d(η) = ∇ξ f ∗∗ (η , ξ0 ) and q(η) = q(η , ξ0 ) for every η ∈ I because
of the properties of ∇ξ f ∗∗ and q that follow from (H3). Moreover, q is monotone on the interval
I by the choice of δ. Now, since v is differentiable at x0 and f ∗∗ (η0 , ξ0 ) < f (η0 , ξ0 ), we apply
Proposition 3.1 and we thus find two families of compact neighbourhoods Kx±0 = {Kx±0 ,ε }ε of x0
that shrink nicely at x0 and two corresponding families of continuous functions Vx±0 = {vx±0 ,ε }ε in
W 1,p (Ω) featuring the properties stated in Proposition 3.1. Moreover, for small enough ε, we can
assume
|v(x) − η0 | ≤ δ
for every x ∈ Kx±0 ,ε and every ε.
and |vx±0 ,ε (x) − η0 | ≤ δ
ON THE EXISTENCE OF MINIMA FOR NONCONVEX VARIATIONAL PROBLEMS
11
Every modified function vx±0 ,ε is feasible for (P ∗∗ ) by (3.4) and we compare the values of I ∗∗ at v and
vx±0 ,ε . By (3.4) it is enough to compare the integrals on the sets Kx±0 ,ε only. Since ∇vx±0 ,ε ∈ Avx± ,ε (ξ0 )
0
for a.e. x ∈ Kx±0 ,ε by (3.7), it follows from (3.13) and (3.8) that
Z
Kx±0 ,ε
f ∗∗ vx±0 ,ε (x) , ∇vx±0 ,ε (x) dx =
Z
hD Kx±0 ,ε
Z
=
E
i
d vx±0 ,ε (x) , ∇vx±0 ,ε (x) + q vx±0 ,ε (x)
h
Kx±0 ,ε
i
hd(v(x)) , ∇v(x)i + q vx±0 ,ε (x)
dx =
dx.
Then, f ∗∗ (η , ξ) ≥ hd(η) , ξi + q(η) for every ξ ∈ RN and η ∈ I where d(η) = d(η , ξ0 ) for every η ∈ I
and similarly for q. Thus,
Z
Kx±0 ,ε
(3.14)
≤
Z
Kx±0 ,ε
f ∗∗ vx±0 ,ε (x) , ∇vx±0 ,ε (x) dx ≤
f ∗∗ (v(x) , ∇v(x)) dx +
Z
h Kx±0 ,ε
i
q vx±0 ,ε (x) − q(v(x)) dx.
Since q is monotone on the interval I, we can choose either the + or the − modified function from
vx±0 ,ε according to the monotonicity of q so that the last summand in (3.14) is nonpositive.
We have thus proved that for every density point x0 of Ev there is σ = σ(x0 ) chosen between +
and − such that the corresponding functions from Vxσ0 are still solutions to (P ∗∗ ) with the further
property that each function vxσ0 ,ε from Vxσ0 satisfies
f ∗∗ vxσ0 ,ε (x) , ∇vxσ0 ,ε (x) = f vxσ0 ,ε (x) , ∇vxσ0 ,ε (x)
(3.15)
for a.e. x ∈ Kxσ0 ,ε
because of (3.7). Now, recalling that the corresponding sets Kxσ0 shrink nicely at x0 , we apply
Vitali’s covering theorem thus finding countably many density points xh of Ev and numbers εh > 0
and symbols σh = σ(xh ) ∈ {+ , −} such that the sets Kh = Kxσhh,εh are pairwise disjoint and cover
Ev up to a negligible set, i.e.
Ev \
(3.16)
[
h
!
Kh = 0.
Then, let vh = vxσhh,εh be the corresponding new solutions to (P ∗∗ ) and notice that (3.15) turns into
f ∗∗ (vh (x) , ∇vh (x)) = f (vh (x) , ∇vh (x))
(3.17)
for a.e. x ∈ Kh .
Then, we set
(3.18)
u(x) = v(x) +
X
[vh (x) − v(x)] ,
x ∈ Ω,
h
and we check that it is a solution to (P ∗∗ ) for which |Eu | = 0.
First, recall that, by (3.4), all functions vh − v have pairwise disjoint supports. Hence, at every
point x, there is at most one nonvanishing summand of (3.18) so that u is pointwise defined. Then,
we show that u is in W 1,p (Ω). Indeed, the very same property of the supports of the functions
vh − v together with either (3.5+) or (3.5−) yield that
Z
Ω
|u|p dx ≤ 2p−1
Z
Ω
|v|p dx + 2p−1
XZ
h
Kh
|vh − v|p dx ≤ 2p−1
Z
Ω
|v|p dx + |Ω| .
12
P. CELADA – G. CUPINI – M. GUIDORZI
As to the gradients, notice that each partial sum of the series defining u is itself a minimizer of I ∗∗
by construction so that


I ∗∗ v +
(3.19)
(vh − v) = I ∗∗ (v)
X
1≤h≤k
holds for every k. Hence, a standard argument based on the growth assumption (H2p ) and SobolevPoincare’s inequality yields that
XZ
h
Ω
p
|∇vh − ∇v| dx =
XZ
h
Kh
|∇vh − ∇v|p dx < +∞.
This implies that u is in W 1,p (Ω) and that the series (3.18) converges strongly to it in W 1,p (Ω).
Moreover, u is feasible for (P ∗∗ ) and the sequential weak lower semicontinuity of I ∗∗ together with
(3.19) show that u is a solution to (P ∗∗ ) as well. In particular, as a minimizer of I ∗∗ , u must be
locally Hölder continuous and almost everywhere differentiable on Ω in the classical sense. Finally,
u = uh on each set Kh and u = v off the union of the sets Kh and the same holds true almost
everywhere for the gradients. Hence, (3.17) and (3.16) show that Eu is negligible and this completes
the proof of the claim.
Claim 2.
The set Fu = {x ∈ Ω : u is differentiable at x and (u(x) , ∇u(x)) ∈ M} is negligible.
Proof of Claim 2. For the sake of simplicity, we assume that M = {m} × D, where D is a
connected set. We refer to [5] for the general case (3.10).
Assume by contradiction that |Fu | > 0. As Fu is a level set of u, ∇u = 0 almost everywhere on
Fu and hence 0 ∈ D. Now, choose a density point x0 in Fu where ∇u(x0 ) = 0. By (H1), there
exists δ > 0 such that J ×Bδ (0) ⊂ D, where J is the interval [m − 2δ , m + 2δ]. In particular, J is
contained in D0 and Bδ (0) ⊂ D. Then, recall that q(η , 0) = f ∗∗ (η , 0) and that q(η , 0) = q(η , ξ)
for every η ∈ J and ξ ∈ Bδ (0) because of the properties of q. In the following, to simplify the
notations, we write q(η) instead of q(η , 0).
The inclusion of J in D0 and (2.4) force m to be a strict local maximum point of the section
η ∈ J → q(η). In addition, recalling (2.3), we can assume that q is increasing on the interval
[m − 2δ , m] and decreasing on [m , m + 2δ].
We claim that we can find two families of compact neighbourhoods {A±
x0 ,ε }ε of x0 contained in Ω
such that
(3.20)
Br1 ε (x0 ) ⊂ A±
x0 ,ε ⊂ Br2 ε (x0 ) ⊂⊂ Ω
for every ε and for suitable numbers r1 = r1 (x0 ) and r2 = r2 (x0 ) with 0 < r1 < r2 with the following
additional property: there exist also two families of corresponding continuous, almost everywhere
1,p (Ω) such that the following properties hold for every ε:
differentiable functions {u±
x0 ,ε }ε in W
(3.21)
±
u±
x0 ,ε = u on Ω \ int (Ax0 ,ε );
(3.22+)
+
u(x) < u+
x0 ,ε (x) < u(x) + 2ε for every x ∈ int (Ax0 ,ε );
(3.22−)
−
u(x) − 2ε < u−
x0 ,ε (x) < u(x) for every x ∈ int (Ax0 ,ε );
(3.23+)
ε ≥ u+
x0 ,ε (x) − m ≥ ε/2 for every x ∈ Br1 ε (x0 );
(3.23−)
−ε/2 ≥ u−
x0 ,ε (x) − m ≥ −ε for every x ∈ Br1 ε (x0 );
(3.24)
±
∇u±
x0 ,ε (x) ∈ ∂Bδ (0) for a.e. x ∈ Ax0 ,ε .
For the proof of these properties, see Lemma 3.5 in [5].
ON THE EXISTENCE OF MINIMA FOR NONCONVEX VARIATIONAL PROBLEMS
13
By the continuity of u and recalling that u(x0 ) = m, we may suppose that ε0 ≤ δ/2 is small enough
±
so as to have |u(x) − m| ≤ δ for every x ∈ A±
x0 ,ε and ε. Moreover, |ux0 ,ε (x) − u(x)| ≤ δ by either
±
(3.22+) or (3.22−). This fact, together with (3.24) implies that u±
x0 ,ε (x) , ∇ux0 ,ε (x) ∈ J × D for
a.e. x ∈ A±
x0 ,ε and for every ε.
±
±
Now, we wish to compare the values of I ∗∗ at u and u±
x0 ,ε . As u = ux0 ,ε outside Ax0 ,ε by (3.21), it
is enough we compare the integrals
Z
A±
x0 ,ε
Z
±
f ∗∗ u±
x0 ,ε (x) , ∇ux0 ,ε (x) dx
and
A±
x0 ,ε
f ∗∗ (u(x) , ∇u(x)) dx.
Arguing as in the proof of Claim 1, we get
Z
A±
x0 ,ε
≤
Z
A±
x0 ,ε
±
f ∗∗ u±
x0 ,ε (x) , ∇ux0 ,ε (x) dx ≤
f ∗∗ (u(x) , ∇u(x)) dx +
Z
A±
x0 ,ε
h i
q u±
x0 ,ε (x) − q(u(x)) dx
and we claim that we can choose + or − and ε such that the second term at the right hand side of
the previous formula is negative, thus contradicting the minimality of u. As the argument is the
very same of Step 2 of the proof of Theorem 1.1 in [6], we just sketch the main points of the proof
and we refer to this paper for the details.
Indeed, choose a sequence {εk }k in (0 , ε0 ] that goes to zero and set
ηk =
1
sup {|u(x) − m| : |x − x0 | < 2r2 εk }
k
for every k
where r2 is the positive number appearing in (3.20). Obviously, ηk → 0+ since u is differentiable
at x0 with ∇u(x0 ) = 0 by assumption and we can assume also that 0 < ηk k ≤ δ for every k.
Then, recalling that m is a strict, local maximum point of q and possibly extracting a subsequence
that we relabel as {k }k , we can assume in addition that the minimum between q(m − ηk k ) and
q(m + ηk k ) is actually achieved for every k by terms that always have the same sign inside, say
q(m + ηk k ), so that
0 < q(m) − q(m + ηk k ) = max {q(m) − q(m − ηk k ) , q(m) − q(m + ηk k )}
(3.25)
holds for every k.
According to this assumption, we choose the + functions and, to simplify the notations, we set
+
uk = u+
x0 ,k and Ak = Ax0 ,k for every k. Of course, should the minimum between q(m − ηk k ) and
q(m + ηk k ) be achieved at q(m − ηk k ), we would choose the − functions.
Finally, set Bi,k = Bri εk (x0 ) for i = 1 and 2 and every k so that (3.20) turns into
B1,k ⊂ Ak ⊂ B2,k .
(3.26)
We prove the claim by setting
Jk1
1
=
|Ak |
Z
Ak
[q(m) − q(uk (x))] dx
and
Jk2
1
=
|Ak |
Z
[q(m) − q(u(x))] dx
Ak
and showing that Jk1 − Jk2 > 0 for some k.
Indeed, note first that (3.23+) reduces to k /2 ≤ uk (x) − m ≤ k for every x ∈ B1,k . Hence,
recalling (3.26) and that q is decreasing on the interval [m , m + 2δ], we find that
Jk1
1
≥
|B2,k |
Z
B1,k
[q(m) − q(uk (x))] dx ≥
14
P. CELADA – G. CUPINI – M. GUIDORZI
1
≥
|B2,k |
Z
B1,k
[q(m) − q(m + k /2)] dx =
r1
r2
N
[q(m) − q(m + k /2)]
for every k. As to Jk2 , we have
Jk2 =
1
|Ak |
Z
[q(m) − q(u(x))] dx
Ak \Fu
for every k and |m − u(x)| ≤ ηk k for every x ∈ Ak by the very definition of ηk . Hence,
0 ≤ q(m) − q(u(x)) ≤ max {q(m) − q(m − ηk k ) , q(m) − q(m + ηk k )} =
= q(m) − q(m + ηk k )
for every x ∈ Ak and every k because of the behaviour of q around m and by (3.25) whence
0 ≤ Jk2 ≤
|Ak \ Fu |
[q(m) − q(m + ηk k )]
|Ak |
for every k.
Since ηk → 0+ , it follows that eventually q(m) − q(m + k /2) ≥ q(m) − q(m + ηk k ) > 0 since q is
increasing on [m , m + 2δ]. As x0 is a density point of Fu , (3.26) shows that the ratio |Ak \ Fu |/|Ak |
goes to zero and the conclusion follows.
4. Open problems
The question of the existence of solutions to non convex variational problems has thus been completely settled at least when the energy densities satisfy standard polynomial p-growth conditions,
namely (H2p ) and f = f (u , ∇u). The nonautonomous case f = f (x , ∇u) was also treated in [21]
under assumptions of qualified convexity at infinity, see below for the definition, a condition that
yields local Lipschitz regularity of minimizers of the corresponding relaxed problem. We wish to
emphasize that this hypothesis of qualified convexity may be dropped since the Baire’s category
method used in [21], as well as Gromov’s convex integration technique adopted here and in [5],
require the same regularity properties of minimizers: continuity and almost everywhere differentiability in the classical sense. As shown by Theorem 1.5, these properties are shared by all energies
satisfying p-growth conditions and the issue of the existence of solutions of nonconvex variational
problem in the general case f = f (x , u , ∇u) seems to be a mere technical problem.
By contrast, the interest for establishing the existence of solutions to nonconvex variational problems cannot be restricted to the case of standard growth. Many physical models arising from non
newtonian mechanics and from nonlinear elasticity, where in a natural way anisotropic phenomena take place, call for studying more general kind of growth. Commonly known as non standard
growth, they take into account different kinds of behaviour from above and below of the densities,
namely
(4.1)
c1 |ξ|p ≤ f (x , η , ξ) ≤ c2 (1 + |ξ|)q
with p < q. We wish to point out that the solution of the corresponding nonconvex variational
problem can be reduced to a regularity question when p ≤ N . Indeed, note that, when p > N ,
our attainment result is valid under only the second coercivity assumption of (H2p ), since in this
case minimizers are regular in the sense specified above. The same fact holds true also in the
nonautonomous case.
Unfortunately, the technique based on Gehring lemma, that yields the required regularity properties
in the standard framework, does not work here. Even in the simplest case of a convex function
f = f (ξ) satisfying (4.1), neither a higher integrability result, which is the first step toward the
smoothness of minimizers, has been proved so far.
ON THE EXISTENCE OF MINIMA FOR NONCONVEX VARIATIONAL PROBLEMS
15
The available regularity results in the non standard framework concern only energy densities of the
form f = f (x , ξ) that exhibit a qualified convexity, i.e. the integrand f has to be strictly convex
in the following sense:
p−2
1
ξ1 + ξ2
(4.2)
ν(1 + |ξ1 |2 + |ξ2 |2 ) 2 |ξ1 − ξ2 |2 + f x ,
≤ [f (x , ξ1 ) + f (x , ξ2 )],
2
2
for every (x , ξi ) ∈ Ω×RN and for some ν > 0. Other conditions need to be imposed, otherwise
the so called Lavrentev phenomenum may occur, i.e. the infimum of the relaxed problem may be
strictly smaller than the infimum of the original nonconvex problem.
This is a clear obstacle not only to regularity - minimizers may exhibit very wild behaviour, see
[22] - but also to the possibility of establishing the attainment result via relaxation. For a detailed
discussion on this issue, see [34] and the references therein. These requirements regard on one
side the smoothness properties of the integrand function with respect to the space variable, and
on the other one the relationship between the growth exponents p and q. To avoid pathological
phenomena, the bigger exponent q must not be too larger than p, i.e. q < c(N )p for a constant
c(N ) which depends only on the dimension and asymptotically tends to 1.
Therefore, a first step in the studying of nonconvex variational problems under non standard growth
hypotheses could be that of requiring that condition (4.2) be satisfied only at infinity, namely for
any x ∈ Ω and any segment with endpoints ξ1 and ξ2 contained in the complement of a sufficiently
large ball, a condition already considered in the framework of nonconvex minimum problems, see
[21] for instance.
References
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Parma – Firenze – Ferrara, January 20, 2004
Pietro Celada, Dipartimento di Matematica – Università degli Studi di Parma, Via M. D’Azeglio 85,
I-43100 Parma (Italy)
E-mail address: [email protected]
Giovanni Cupini, Dipartimento di Matematica “U. Dini” – Università degli Studi di Firenze, V.le Morgagni 67/A, I-50134 Firenze (Italy)
E-mail address: [email protected]
Marcello Guidorzi, Dipartimento di Matematica – Università degli Studi di Ferrara, Via Machiavelli 35, I-44100 Ferrara (Italy)
E-mail address: [email protected]
Stampato in proprio presso il Dipartimento di Matematica dell’Università degli Studi di Parma,
Via M. D’Azeglio 85/A, 43100 Parma. Sono stati adempiuti gli obblighi di cui all’art. 1 del
D.L. 660/1945.