An integral representation result is obtained for the relaxation of a class of energy functionals depending on two vector fields with different behaviors which appear in the context of thermochemical equilibria and are related to image decomposition models and directors theory in nonlinear elasticity.
In this paper we consider energies depending on two vector fields with different behaviors: and , Ω being a bounded open subset of .
Let and for every define the functional
Minimization of energies depending on two independent vector fields have been introduced to model several phenomena. For instance the case of thermochemical equilibria among multiphase multicomponent solids and Cosserat theories in the context of elasticity: we refer to [7,9] and the references therein for a detailed explanation about this kind of applications.
In the present paper we are interested in studying the lower semicontinuity and relaxation of (1.1) with respect to the -strong -weak convergence (). Clearly, bounded sequences may converge in , up to a subsequence, to a function.
Here we allow v to be in , and f is not necessarily convex–quasiconvex. We provide an argument alternative to the one in [8, Section 4], devoted to clarify some points in the lower semicontinuity result therein.
We also emphasize that under specific restrictions on the density f, i.e. , the analysis in the case was considered already in  in order to describe image decomposition models. In  a general f was taken into account when the target u is in .
In this manuscript we consider , and the target .
We study separately the cases , and discuss briefly the case in the Appendix. Comparing the results in  (where the assumptions allow to invoke De La Vallé–Poussin Criterion) with the true linear growth setting as in .
To this end, we introduce for the functional
Since bounded sequences in converge in to a function u and bounded sequences in if (in if ), weakly converge to a function (weakly ∗ in ), the relaxed functionals and will be composed by an absolutely continuous part and a singular one with respect to the Lebesgue measure (see (2.12)). On the other hand, as already emphasized in , it is crucial to observe that v, regarded as a measure, is absolutely continuous with respect to the Lebesgue one, besides it is not defined on the singular sets of u, namely in those sets where the singular part with respect the Lebesgue measure of the distributional gradient of u, , is concentrated. Thus specific features of the density f will come into play to ensure a proper integral representation.
There exists such that
There exists , , such that
In order to characterize the functional introduced in (1.3) we will replace assumptions and by the following ones:
Given , there exists such that, if then
Given , there exist , , such that
Section 2 is devoted to notations, preliminaries about measure theory and some properties of the energy densities. In particular, we stress that a series of results is presented in order to show all the properties and relations among the relaxed energy densities involved in the integral representation and that can be of further use for the interested readers since they often appear in the integral representation context. Section 3 contains the arguments necessary to prove the main results stated below.
We denote by the convex–quasiconvex envelope of f in (2.5) and represents the recession function of , defined according to (1.4), which coincides, under suitable assumptions (see assumptions (2.6), (2.7), Proposition 2.12 and Remark 2.13), with the convex–quasiconvex envelope of , , and this allows us to remove the parenthesis.
For the case we have the following.
f is convex–quasiconvex.
2.Notations, preliminaries and properties of the energy densities
In this section, we start by establishing notations, recalling some preliminary results on measure theory that will be useful through the paper and finally we recall the space of functions of bounded variation.
Then we deduce the main properties of convex–quasiconvex functions, recession functions and related envelopes.
If and is an orthonormal basis of , denotes the unit cube centered at the origin with its faces either parallel or orthogonal to . If and , we set and , Q is the cube .
Let Ω be a generic open subset of , we denote by the space of all signed Radon measures in Ω with bounded total variation. By the Riesz Representation Theorem, can be identified to the dual of the separable space of continuous functions on Ω vanishing on the boundary . The N-dimensional Lebesgue measure in is designated as .
If and is a nonnegative Radon measure, we denote by the Radon–Nikodým derivative of μ with respect to λ. By a generalization of the Besicovich Differentiation Theorem (see [1, Proposition 2.2]), it can be proved that there exists a Borel set such that and
We recall that the exceptional set E above does not depend on C. An immediate corollary is the generalization of Lebesgue–Besicovitch Differentiation Theorem given below.
If μ is a nonnegative Radon measure and if then
A function is said to be of bounded variation, and we write , if all its first distributional derivatives, , belong to for and .
The matrix-valued measure whose entries are is denoted by and stands for its total variation. We observe that if then is lower semicontinuous in with respect to the topology.
By the Lebesgue Decomposition Theorem we can split into the sum of two mutually singular measures and , where is the absolutely continuous part and is the singular part of with respect to the Lebesgue measure . By we denote the Radon–Nikodým derivative of with respect to the Lebesgue measure so that we can write
If then for -a.e.
For more details regarding functions of bounded variation we refer to .
A Borel measurable function is said to be convex–quasiconvex if, for every , there exists a bounded open set D of such that
i) It can be easily seen that, if f is convex–quasiconvex then condition (2.2) is true for any bounded open set .
ii) A convex–quasiconvex function is separately convex.
iii) By [11, Proposition 3], the growth condition from above in , ii), entail that there exists such that
iv) By [11, Proposition 4]), under the growth assumptions in , ii) entails that, given there exists a constant such that
We introduce the notion of convex–quasiconvex envelope of a function, which is crucial to deal with the relaxation procedure.
Let be a Borel measurable function bounded from below. The convex–quasiconvex envelope is the largest convex–quasiconvex function below f, i.e.,
By Theorem 4.16 in , the convex–quasiconvex envelope coincides with the so called convex-quasiconvexification
Let be a continuous function satisfying . Then is continuous and satisfies . Consequently, satisfies (2.3).
In order to deal with and to compare with the result in , , one can consider a different setting of assumptions on the energy density f.
Namely, following [11, Proposition 6 and Remark 7], if is a convex and increasing function, such that and if is a continuous function satisfying
Analogously, one can assume that f satisfies the following variant of : there exist , , such that
2.2.The recession function
Let , and let be its recession function, defined in (1.4).
The following properties are an easy consequence of the definition of recession function and conditions , and , when .
Provided f satisfies , and , then
1. is convex–quasiconvex;
2. there exists such that
3. is constant with respect to b for every ;
4. is continuous.
We emphasize that not all the assumptions and in Proposition 2.8 are necessary to prove items above. In particular, one has that:
i) The proof of 2. uses only the fact that f satisfies .
ii) To prove 3. it is necessary to require that f satisfies only and . Indeed if it satisfies (2.3) one can avoid to require .
1. The convexity-quasiconvexity of can be proven exactly as in [8, Lemma 2.1], exploiting the growth condition and the estimate given by .
2. By definition (1.4) we may find a subsequence such that
4. The continuity is a consequence of the growth conditions and the convexity-quasiconvexity of .
i) is convex–quasiconvex;
ii) , for every b, with ;
iii) If , then is constant with respect to b.
We observe that, if is a continuous function satisfying and , then the function , obtained first taking the convex-quasiconvexification in (2.5) of f and then its recession through formula (1.4) applied to , satisfies the following properties:
1. is convex–quasiconvex;
2. there exists such that , for every ;
3. for every , is constant, i.e. is independent on b;
4. is Lipschitz continuous in ξ.
Assuming that f satisfies , one can prove that the convex-quasiconvexification of , , satisfies the following conditions:
5. is convex–quasiconvex;
6. there exists such that , for every ;
7. for every , and assuming that f satisfies (2.3), is constant, i.e. is independent on b;
8. is Lipschitz continuous in ξ.
On the other hand, Proposition 2.12 below entails that is independent on b, without requiring that f is Lipschitz continuous, but replacing this assumption with .
We also observe that and are only quasiconvex functions, since they are independent of b. In particular, in our setting, these functions coincide as it is stated below.
Let be a continuous function satisfying and .
The proof will be achieved by double inequality.
For every the inequality
In order to prove the opposite inequality, fix and, for every , take , with 0 average, and such that
Let L be the constant appearing in condition . We split the cube Q into the set and its complement in Q. Then we apply condition and (2.8) to get
Applying Hölder inequality and (2.10), we get
It is worth to observe that inequality
Furthermore, we emphasize that the proof of Proposition 2.12 cannot be performed in the same way in the case , with assumptions and replaced by and . Indeed, an bound on analogous to the one in cannot be obtained from . On the other hand, it is possible to deduce the equality between and , when f satisfies (2.6) and (2.7).
Here we prove that assumption on f is not necessary to provide an integral representation of as in (1.2). Indeed, we can assume that is a continuous function and satisfies assumptions and , (). First we extend, with an abuse of notation, the functional J in (1.1), to , , as
Clearly, it results that for every ,
Let and consider the functionals J and and their corresponding relaxed functionals and . If f satisfies conditions and if , and both f and satisfy and if , then
The proof is omitted since it can be performed as in [11, Lemma 8 and Remark 9]. In  it is not required that f satisfies , . Indeed, the equality holds independently on this assumption on f, but in order to remove hypothesis from the representation theorem we need to assume that inherits the same properties as f, which is the case as it has been observed in Proposition 2.7. It is also worth to observe that, when , (2.7) is equivalent to
The following result can be deduced in full analogy with [11, Theorem 13], where it has been proven for .
Let Ω be a bounded and open set of and let be a continuous function satisfying and , . Let J be the functional defined in (1.1), then in (1.2) (), (1.3) is a variational functional, namely it is lower semicontinuous with respect to the first arguments and for every , one can define , () (in analogy with (1.2) and (1.3)) as a set function on the open subsets of Ω, and it turns out to be the restriction of a Radon measure to these subsets of Ω.
By virtue of this result can be decomposed as the sum of two terms
Let be a continuous and convex–quasiconvex function, let J and be the functionals defined respectively by (1.1) and (1.2) when , respectively ((1.3), when ). Then the following scaling properties are satisfied
The following result will be exploited in the sequel. The proof is omitted since it develops along the lines of [2, Lemma 5.50], the only differences being the presence of v and the convexity-quasiconvexity of f.
This section is devoted to deduce the results stated in Theorems 1.1 and 1.2. We start by proving the lower bound in the case . For what concerns the upper bound we present, for the reader’s convenience, a self contained proof in Theorem 3.2. For the sake of completeness we observe that the upper bound, in the case , could be deduced as a corollary from the case (see Theorem 1.2), which, in turn, under slightly different assumptions, is contained in .
3.1.Lower semicontinuity in ,
Let Ω be a bounded open set of , let be a continuous function satisfying , and , and let be the functional defined in (1.2). Then
The proof will be achieved, in two steps, namely by showing that
For -a.e. it results that u is approximately differentiable (see (2.1)) and
Step 2. Here we present the proof of (3.3). To this end we exploit techniques very similar to  (see [2, Proposition 5.53]). Let be the polar decomposition of (see [2, Corollary 1.29]), for , and recall that for -a.e. , admits the representation , with and (see [2, Theorem 3.94]). In the following, we will denote the cube by Q.
To achieve (3.3) it is enough to show that
Setting , by the scaling properties of in Proposition 2.17 and by the growth conditions , we have
We start by observing that Theorem 1.2 is contained in  under a uniform coercivity assumption. We do not propose the proof in our setting, since it develops along the lines of Theorems 3.1 and 3.2.
On the other hand, several observations about Theorem 1.2 are mandatory:
i) If f satisfies and then for every .
ii) For the reader’s convenience we observe that the proof of the lower bound in Theorem 1.2 develops exactly as that of Theorem 3.1, using the bound on v to deduce (3.6) and the uniform bound on in (3.7), and (2.4) in order to estimate .
Regarding the upper bound, the bulk part follows from [11, Theorems 12 and 14], while for the singular part we can argue exactly as proposed in the proof of the upper bound in  just considering conditions and in place of and .
We are now in position to prove the upper bound for the case , for . We emphasize that an alternative proof could be obtained via a truncation argument from the case as the one presented in [11, Theorem 12], but we prefer the self contained argument below.
Let Ω be a bounded open set of and let be a continuous function. Then, assuming that f satisfies and ,
First we observe that Proposition 2.16 entails that is a variational functional. Thus the inequality can be proved analogously to [2, Proposition 5.49]. For what concerns the bulk part, it is enough to observe that given and , taking a sequence of standard mollifiers , where , it results that , where . The local Lipschitz behaviour of f in (2.3) gives
To achieve the result, it will be enough to show that
It is easily seen that g is -positively homogeneous, i.e. for every , , g is continuous and, since f satisfies (2.3), g inherits the same property. Moreover, the monotonicity property of difference quotients of convex functions ensures that, whenever rank , , where the latter is defined as
Then for every open set such that , defining for every , and where is a sequence of standard mollifiers and . Then in . Also [2, Theorem 2.2] entails that weakly ∗ in A and . Thus, since ,
Regarding the second term in the right hand side of (3.12), we have
Taking the Radon–Nikodým derivative, the last two terms disappear, since , is singular with respect to and the Hölder inequality can be applied, i.e.
It is worth to observe that an alternative argument to the one presented above, concerning the upper bound inequality for the singular part, can be provided by means of approximation. In fact, one can prove that for every , when and , and then a standard approximation argument via mollification allows to reach every .
For what concerns the case it is enough to consider the function , exploit its properties of positive 1-homogeneity in the second variable, i.e. , for every , , (2.4), and the fact that when rank , then is constant with respect to b and . To conclude it is enough to apply Reshetnyak continuity theorem.
The research of the authors has been partially supported by Fundação para a Ciência e Tecnologia (Portuguese Foundation for Science and Technology) through UTA-CMU/MAT/0005/2009 and CIMA-UE.
The second author is a member of INdAM-GNAMPA, whose support is gratefully acknowledged.
Consider the functional
There exists such that
There exists such that
Arguing as in [3, Lemma 4.7] one can prove that the set function , defined as in (A.1) for all the open subsets of Ω, is the trace on these latter sets of a Radon measure absolutely continuous with respect to .
Then the rest of the proof can be obtained as in  by providing a lower bound and an upper bound, that we will sketch in the sequel, just emphasizing the main differences.
For what concerns the lower bound, it is enough to prove that
All the inequalities can be proven following arguments analogous to the ones in [3, Lemma 5.1], taylored for thin structures. Indeed we observe that the density appearing therein coincides with , as proven in . Then the proofs in [3, Lemma 5.1] can be repeated line by line but with easier constructions, in particular it suffices to replace the unique sequence (and its average), by the couple with in , and in .
For what concerns the upper bound, i.e proving inequalities opposite to (A.3)–(A.5), one can argue as in the proof of Theorem 3.2, using as a recovery sequence the couple with and , where is a sequence of standard mollifiers. Then for the bulk term it suffices to exploit the standard Lipschitz property of f with respect to the couple (i.e. (2.3), when ), and the fact that , and finally taking the Radon–Nikodým derivative around points of absolute continuity with respect to the Lebesgue measure of and .
For what concerns the other two terms in the limiting energy they can be reached through the function , which is positively 1 homogeneous and coincides with , whenever . Finally the conclusion can be achieved differentiating with respect to or . □
We observe that in  it was considered for the the weak convergence in thus leading to a target function still in . In the present case the limiting function is a measure, which can be decomposed in three terms, the first two absolutely continuous with respect to and respectively, and the third possibly singular with respect to and this entails the presence of a third integrand in the energy above.
We also observe that an argument entirely similar to Proposition 3.6 warranties that for every .
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