1. Introduction
Let
be a bounded and open subset of
, and let
. As a model example, consider a functional
of the form
where
,
is continuous and
is a Carathéodory function such that
According to the well-known results (see [
1]), the functional
f is lower semicontinuous with respect to the weak topology of
. However, because of the lack of coercivity of the principal part, we cannot expect that the functional
f admits a minimum in
(see also the next Example 1). On the other hand, results on the minimization of functionals with a lack of coercivity can be found in [
2,
3,
4,
5,
6,
7], where it is proved, under suitable assumptions, that a minimum exists in a larger Sobolev space.
Our aim is to consider a case with the feature of being
diffeomorphism invariant. More precisely, denote by
the set of diffeomorphisms
of class
such that
. Then, if we set
, we formally have in the model case
where
Therefore, the structure of the functional is invariant, and also satisfy our assumptions if and only if do the same.
An important application of variational methods is the study of continuum mechanics, when a stored-energy function occurs (see [
8]). In the one-dimensional case (
), it is today standard to consider the case in which the target space of
u is a differentiable manifold. In several variables (
), this is not at all the case, particularly for existence theorems, and a first step, in view of this kind of application, is to consider scalar problems where the target space of
u (in fact
) is treated just as a differentiable manifold (see also Appendix of [
9]). This means that one cannot take advantage of the full structure of
, and the fact that the setting must be diffeomorphism invariant expresses such a restriction.
With respect to the mentioned papers, it is clear that a quantitative assumption like
already considered in [
2,
3,
4,
5,
6,
7], is not diffeomorphism invariant.
Remark 1. The question of invariant formulations has been already treated in [10] for quasilinear elliptic equations which are not, in general, the Euler–Lagrange equation of some functional. In such a case, also a uniform coercivity assumption can be considered. For instance, if we start from an equation of the formand we write , we obtainwhich can be written aswhere However, if we start from the Euler–Lagrange equation of some functional, e.g., so that (2) is the Euler–Lagrange equation of a functional of the formthen (3) is not, in general, the Euler–Lagrange equation of some functional. Different from [
2,
3,
4,
5,
6,
7], the setting of Sobolev spaces is not convenient for our purposes. First of all,
is not diffeomorphism invariant, unless
or
. Moreover, also in the case
, the space
is too small, as we have no estimate of the rate of degeneration of the principal part (see again Example 1 and also Remark 3). On the contrary, the space
, already considered in [
11,
12,
13] for the study of quasilinear elliptic equations with the right-hand-side measure, is much more suitable. First of all, it is easily seen that
if and only if
.
Let us state our main result, for the model case.
Theorem 1. Let be endowed with the topology of the convergence in measure and define according to (1). Then, f is lower semicontinuous with and the setis compact (possibly empty), for all . In particular, the functional f admits a minimum in . Moreover, if is any minimizing sequence, then there exist a subsequence and a minimum u such that is strongly convergent to in , for all .
Let us point out that it may happen that each minimum u of the functional f considered in Theorem 1 satisfies , for all , even when (see Example 1). On the other hand, under further (invariant) assumptions, each minimum of f belongs to (see Theorem 9).
In the end, the existence of a minimum follows in a direct way from a basic result (see (Proposition 1.2.2 of [
1]) or (Theorem of I.1.1 [
14]), while our task will be to prove some results on lower semicontinuity and coercivity in the setting of the spaces
. The strong convergence of
in
is related to the strict convexity of the function
In our setting, the main tool is provided by Theorem 3. Let us point out that results in this direction have been already obtained in [
15,
16,
17].
Actually, Theorem 1 is a particular case of Theorems 6 and 7, where more general functionals
of the form
with
, are considered.
It would be interesting, as a further development, to consider also critical points, not only minima, under diffeomorphism-invariant assumptions. Let us point out that some results in this direction have been already obtained in [
18].
In the next section, we recall some preliminary facts, while
Section 3 is devoted to a lower semicontinuity result in the space
and
Section 4 to the a.e. convergence of
under a strict convexity assumption.
Section 5 and
Section 6 are concerned with some coercivity results in our setting, while
Section 7 contains the main results. Finally, in
Section 8, we prove some regularity results for the minima of the functional, and in
Section 9 we prove that each minimum of the functional satisfies a suitable form of the Euler–Lagrange equation.
2. Notations and Preliminaries
In the following, will denote the -algebra of Lebesgue measurable subsets of , and the -algebra of Borel subsets of . With the terms “measurable” and “negligible”, we mean “Lebesgue measurable” and “Lebesgue negligible”, respectively. Moreover, we denote by the positive and negative parts of a real number s and by the usual -norm.
For every
, we define
by
Then, if
is an open subset of
and
, we denote by
the set of (classes of equivalence of) functions
such that
for all
and such that the set
is negligible. We also denote by
the set of
such that
for all
. Finally, we denote by
the set of
such that, for every
, there exists a sequence
in
converging to
in
with
converging to
in
(see [
11]).
According to [
13], each
with
admits a Borel and
-quasi continuous representative
, defined up to a set of null
p-capacity. Thus, the set
has a null measure but could have a strictly positive
p-capacity. Moreover, for every
, there exists one and only one measurable (class of equivalence)
such that
a.e. in
. If
and
, it turns out that
and
a.e. in
.
We also write if is an open subset of such that the closure is a compact subset of .
4. The Effect of Strict Convexity
Throughout this section, we assume that
is an open subset of
and that
satisfies (
L1), (
L3) and the following:
- (L′2)
There exists a negligible subset N of Ω such that we have the following:
For every , the function is continuous on ;
For every , the function is strictly convex on .
Again, it is clear that also satisfies (L′2), for all .
Theorem 3. Let and let be a sequence in such that is weakly convergent to in , for all and all . Assume also that Then, is strongly convergent to in and there exists a subsequence such that is convergent to a.e. in Ω.
For the proof, we need some elementary results.
Proposition 1. Let be a convex function and let be defined by Then ψ is nondecreasing.
Proposition 2. Letbe a continuous function such that is strictly convex, for all . Let and let be a sequence in such that Proof. For every
, let us set
If
and we apply Proposition 1 to the convex function
from
, we infer that
Of course, the inequality also holds if
, whence
Up to a subsequence,
is convergent to some
, whence
From the strict convexity of we infer that , so that is convergent to . Since it is either or , the assertion follows. □
Proof of Theorem 3. Without loss of generality, we may assume that in assumption (L3). Moreover, up to a first subsequence, we have that is convergent to u a.e. in . Let with for a.a. .
First of all, we claim that, for every
, there exists
such that
Actually, for every
, there exists
such that
On the other hand, by dominated convergence we also have that
Finally, if we set
then
satisfies (
L1)–(
L3), and we have that
From [
19] or (Theorem 2.3.1 of [
1]), we infer that
so that (
4) follows.
Now we claim that there exists a subsequence
such that
a.e. in
. Actually, from (
4), we infer that for every
there exists
such that
Then, for every
, there exists
such that
It follows that
and hence a.e. in
, up to a subsequence. Then, up to a subsequence, we infer that
a.e. in
and (
5) follows.
Thus, along a suitable subsequence
, we have that
a.e. in
.
From assumption (L′2) and Proposition 2, we infer that is convergent to a.e. in .
Since
from Fatou’s lemma, it follows that
is strongly convergent to
in
. As usual, since the limit is independent of the subsequence, we have the convergence of the full sequence in
. □
7. Existence of Minima
In this section, we prove the main results.
Theorem 6. Let Ω be an open subset of , let , and letbe a function satisfying (L1)–(L3), (L4,p) and (L5). Assume also that . Let be endowed with the distanceand define by Then f is lower semicontinuous with and the setis compact (possibly empty), for all . In particular, the functional f admits a minimum in .
Proof. By assumptions (
L1) and (
L3), the functional
f is well defined, and it is obvious that
. Let now
and let
be a sequence in
such that
. From Corollary 1, we infer that there exist
and a subsequence
such that
is convergent to
u a.e. in
, and
is convergent to
weakly in
, for all
and all
. In particular, we have
and, by Theorem 2,
so that
is sequentially compact. Since
is a metric space, the remaining assertions follow. □
Theorem 7. Let Ω be an open subset of , let , letbe a function satisfying (L1), (L′2), (L3), (L4,p) and (L5), and let be defined as before. Then, for every minimizing sequence , there exist a subsequence and a minimum u such that is convergent to u a.e. in Ω, and is strongly convergent to in , for all .
Proof. Without loss of generality, we may assume that
in assumption (
L3). Arguing as in the previous proof, we can find
and a subsequence
such that
is convergent to
weakly in
, for all
and all
. This time, we infer that
so that
u is a minimum of
f and, by Theorem 3,
is strongly convergent to
in
, while
is convergent to
a.e. in
, up to a further subsequence.
According to assumptions (
L3) and (
L4,p), for every
, we have
for some
and
. From the (generalized) Lebesgue’s theorem, we conclude that
is strongly convergent to
in
. □
Example 1. Let and let(see Figure 1). Then, we have whenever and for .
If is the convex function defined byit turns out thatNow, define first bywhere is the -function defined by It is easily seen thatand that is lower semicontinuous, strictly convex, proper and coercive. Therefore has one and only one minimum point, which is just , as Let now be a -function such thatand let be the primitive of such that . We have thatand that Therefore, if we set andit turns out that L is of class on and that the assumptions of Theorem 1 are satisfied. In particular, hypotheses (L1), (L′2), (L3), (L4,p) with and (L5) hold. On the other hand, the functionalhas one and only one minimum point , which is given by . Therefore, we have andwhencefor all . It follows that , for all (see, e.g., (Theorem 3.77 of [21])). Example 2. Let be a -function such that whenever and whenever . Let again and define first bywhere is the -function defined by It is easily seen thatand that is lower semicontinuous, strictly convex and coercive. Therefore has one and only one minimum point, which is just . Moreover, if we setit turns out that Let now again ν and ψ be defined as in Example 1. We have Moreover, if we set andit turns out that L is of class on and that assumptions (L1), (L′2), (L3), and (L4,p) with are satisfied. On the other hand, the functionalhas no minimum point in , becausebut , for all . In this case, assumption (L5) is not satisfied. Remark 2. As already observed in the Introduction, after proving the lower semicontinuity and the coercivity of the functional in a suitable functional setting, the existence of a minimum follows in a standard way. More precisely, (L1)–(L3) are the natural assumptions to ensure the lower semicontinuity, while (L4,p) and (L5) imply the coercivity. Assumption (L′2) is the typical stronger variant of (L2), designed to obtain the strong precompactness of the minimizing sequences.
Let us point out that (L5) is essential for the coercivity, according to Example 2, in which (L5) is not satisfied and the functional admits no minimum.
This is the basic set of assumptions for minimization. In the next sections, we will consider other assumptions either to ensure that each minimum is more regular or to prove that it satisfies a suitable form of the Euler–Lagrange equation.
Remark 3. Let us point out that, under the assumptions (L1), (L′2), (L3), (L4,p) and (L5), it may happen that the functional admits no minimum in . Actually, by Example 1, the situation is even worse. It may happen that each minimum u satisfies , for all . Of course, this implies that , but even that the minimization cannot be reduced to a Sobolev setting “up to diffeomorphism”.
9. Euler–Lagrange Equation
Throughout this section, we assume that
is an open subset of
, that
and that
is a function such that the following hold:
- (L6,p)
There exists a negligible subset N of Ω such that we have the following:
For every , the function is measurable on Ω;
For every , the function is of class on ;
For every , there exist , and such that for all with .
As usual, it follows that also satisfies (L6,p), for all .
Let
denote the set of
in
vanishing a.e. outside some compact subset of
. Then, following an idea of [
22], for every
, we denote by
the linear space of
in
such that
u is essentially bounded on
. For instance, if
and
, then
.
For every
, every
and every measurable function
v, we set also
The next assertions are easily proved.
Proposition 3. For every , and , the following facts hold:
- (a)
- (b)
We have and the linear map is a bijection of onto ;
- (c)
a.e. in Ω.
Theorem 10. Let be such that Proof. It easily follows from Lebesgue’s Theorem. □
Now, for every
and
, we define the linear space
We remark that
, for all
, and we set
Moreover, if
, it turns out that
Therefore, we have
for all
and the linear map
is a bijection of
onto
. As before, we also have
a.e. in
, for all
,
and
.
Theorem 11. Let be such that Then we have(both sides could be ). Proof. Let us first treat the case
, namely
. In this case the argument is an adaptation of the proof of (Theorem 4.7 of [
23]). Assume, for instance, that
. Let
be a
-function such that
whenever
, and
whenever
. Then,
and we have
Since
from Fatou’s Lemma, we infer that
whence
. Coming back to (
7), from Lebesgue’s Theorem, we conclude that
If , the argument is similar.
Consider now the general case. Let
and let
. If we set
, it follows that
, while
From the previous step, we infer that
and the assertion follows. □
Remark 5. If is positively homogeneous of degree p and we havefor some , then it follows that Therefore, the assumptionis implied by .