Given an open and bounded set and a family of Lipschitz vector fields on Ω, with , we characterize three classes of local functionals defined on first-order X-Sobolev spaces, which admit an integral representation in terms of X, i.e.
with f being a Carathéodory integrand.
The representation of local functionals as integral functionals of the form
has a very long history and exhibits a natural application when dealing with relaxed functionals and related Γ-limits in a suitable topology. In the Euclidean setting, this problem is now very well understood, and we refer the interested reader to the papers [2, 6, 7, 8, 9] for a complete overview of the subject.
Recently, in , Franchi, Serapioni and Serra Cassano started the study of variational functionals driven by a family of Lipschitz vector fields. By a family of Lipschitz vector fields we mean an m-tuple , with , where each is a first-order differential operator with Lipschitz coefficients defined on a bounded open set , i.e.
Moreover, according to , we assume that the family satisfies the structure assumption (LIC), which roughly means that are linearly independent for a.e. as vectors of (cf. Definition 2.1). We stress that this point of view is pretty general and encompasses, among other things, the Euclidean setting and many interesting sub-Riemannian manifolds.
Since , the possibility to extend the classical results of the calculus of variations to the setting of variational functionals driven by vector fields has been the object of study of many papers. For example, the homogenization theory has been intensively studied so far in the setting of special sub-Riemannian manifolds, i.e. Carnot groups (see, for instance, [3, 18, 22]). More recently, in [20, 21] Maione, Pinamonti and Serra Cassano started the investigation of the Γ-convergence of translation-invariant local functionals , with being the class of all open subsets of Ω. In [20, Theorem 3.12], they found conditions under which F can be represented as
for any open and such that (cf. Definition 2.2 and ), and for a suitable . Finally, they applied this characterization to prove a Γ-compactness theorem for integral functionals of the form (1.1) when . Similar results have been proved in , under stronger conditions on the family . To conclude, we also point out that functional (1.1) was studied in  as far as its relaxation and in connection with the so-called Meyers–Serrin theorem for .
Inspired by the results proved in [7, 8], the aim of the present paper is to extend the results achieved in  when we drop the assumption of translation-invariance. We find some sufficient and necessary conditions under which a local functional
admits an integral representation of the form
for a suitable Carathéodory function . We point out that in this new framework, due to the lack of translation-invariance, a dependence of the integrand with respect to the function is expected. Let us observe that if F is defined on instead of , under reasonable improvements of some assumptions, it is easy to extend the integral representation to get
The main goal of this paper is to obtain a representation formula as in (1.2) for the following three different classes of functionals:
Convex functionals (Theorem 3.3).
weakly*-seq. l.s.c. functionals (Theorem 4.3).
None of the above (Theorem 5.6).
Unlike in Sobolev spaces, in this context no analogue of approximation results by a reasonable notion of piecewise X-affine function holds in general (cf. [20, Section 2.3]). To overcome this difficulty, we rely on the method employed in , consisting of three steps:
Find sufficient conditions on that guarantee the existence of a “non-Euclidean” Lagrangian f such that(1.3)
Extend the previous equality to the whole space .
The second step crucially exploits third-argument convexity of the Euclidean Lagrangian . Indeed, convexity of is sufficient to guarantee (1.3) (cf. Proposition 3.2). This is shown in , and the same ideas can be adapted to cases (i) and (ii) of convex and weakly*-seq. l.s.c. functionals, for which the convexity of is granted. On the contrary, due to the weaker assumptions on the functional, case (iii) is more demanding and requires a further step. In Section 5, we show that the convexity of is not necessary for (1.3). Thus, in order to find a more suitable notion of convexity, we define the weaker concept of X-convexity (cf. Definition 5.3), which strongly depends on the chosen family of vector fields. We show that, under a classical growth assumption on the functional, this new condition is equivalent to (1.3) (cf. Proposition 5.4). Finally, by slightly modifying a zig-zag argument due to Buttazzo and Dal Maso [8, Lemma 2.11], we show that X-convexity is a consequence of a reasonable lower semicontinuity assumption (cf. Lemma 5.5). This procedure allows to generalize the final case as well. Finally, for each of the previous results we show that our hypotheses are also necessary, in order to give a complete characterization of the classes of functionals studied.
The structure of the paper is the following. In Section 2, we briefly recall some basic facts about vector fields and X-Sobolev spaces. In Section 3, we get an integral representation result for a class of convex functionals. In Section 4, we deal with weakly*-sequentially l.s.c functionals. In Section 5, we drop both previous requirements, obtaining as well an integral representation result.
2 Vector fields and X-Sobolev spaces
Unless otherwise specified, we let and with , we denote by Ω an open and bounded subset of and by the family of all open subsets of Ω. Given two open sets A and B, we write whenever . We set to be the subfamily of of all open subsets A of Ω such that . For any , we denote by the Euclidean scalar product, and by the induced norm. We denote by the restriction to Ω of the n-th dimensional Lebesgue measure, and for any set we write . Given an integrable function , we write
Given and , we let , and given an integrable function we denote its integral average by
We usually omit the variable of integration when writing an integral: for instance, given two functions and such that is integrable over Ω, we write its integral as instead of . Finally, for , and we set
2.2 Basic definitions and properties
We will always identify a first order differential operator with the map
Let . We say that is a family of Lipschitz vector fields on Ω if for any and for any there exists a function such that
We will denote by the matrix defined by
We say that satisfies the linear independence condition (LIC) on Ω if the set
is such that . In this case, we set .
Let us point out that (LIC) embraces many relevant families of vector fields studied in the literature. In particular, neither the Hörmander condition for , that is, each vector field is smooth and the rank of the Lie algebra generated by equals n at any point of Ω, nor the (weaker) assumption that the X-gradient induces a Carnot–Carathéodory metric in Ω is requested. An exhaustive account of these topics can be found in .
Let , and , and let be a family of Lipschitz vector fields. We say that v is the X-gradient of u if for any it holds that
Whenever it exists, the X-gradient is shown to be unique a.e. In this case, we set .
If , we define the vector spaces
We refer to them as X-Sobolev spaces, and to their elements as X-Sobolev functions.
The next proposition can be found in .
Let . Then the vector space , endowed with the norm
is a Banach space. Moreover, if , it is a reflexive Banach space.
The following proposition tells us that X-Sobolev spaces are actually a generalization of the classical Sobolev spaces, both because each Sobolev function is in particular an X-Sobolev function, whatever we choose, and because, as expected, the choice of the “standard” family of vector fields
gives rise to the classical Sobolev spaces.
The following facts hold:
If and for every , then .
, the inclusion is continuous and
for every and a.e. .
Let us notice that, with Ω being bounded, we have that
for any family of Lipschitz vector fields. The following proposition tells us that the weak convergence in is weaker than the weak*-convergence in .
Let be a family of Lipschitz vector fields. Then, for any sequence and any , it follows that
This follows easily from [5, Theorem 3.10]. ∎
2.3 Approximation by regular functions
When dealing with representation theorems for local functionals defined on classical Sobolev spaces, a typical strategy is to exploit classical differentiation theorems for measures to get an integral representation of the form
for classes of “simple” functions, that is, for instance, linear or affine functions. Then one can combine some semicontinuity properties of the functional together with approximation results by means of piecewise affine functions (see, for instance, [13, Chapter X, Proposition 2.9]) in order to extend the integral representation to all Sobolev functions. In this context, one of the main difficulties is that an analogue of [13, Chapter X, Proposition 2.9] does not hold. We mean that, if we call a -function X-affine when Xu is constant, then there are choices of for which not all X-Sobolev functions can be approximated in by piecewise X-affine functions [20, Section 2.3]. So, as shown in Section 3, we have to adopt a different strategy. Anyway, we present some useful Meyers–Serrin-type results that are still true even in this non-Euclidean framework and that allow us to approximate X-Sobolev functions with smooth functions. For the following fundamental theorem, we refer to [17, Theorem 1.2.3].
Let Ω be an open subset of . For any , there exists a sequence
Given and let . Then there exists a function which coincides with u on .
Let φ be a smooth cut-off function between and Ω. It is straightforward to verify that the function satisfies the desired requirements. ∎
The previous proposition, together with Theorem 2.6, allows to prove the following result.
Consider a function and an open set . Then there exists a sequence such that
2.4 Failure of a Lusin-type theorem
When dealing with integral representation in classical Sobolev spaces, one might exploit the following Lusin-type result (cf. [10, Theorem 13]).
Let be open and bounded, let and let . Then, for any , there exist and such that and .
Under reasonable assumptions (cf. [8, Lemma 2.7]), this result allows to extend an integral representation result from to . The following counterexample shows that an analogue of Proposition 2.9 does not hold in a general X-Sobolev space.
In this example, we speak about approximate differentiability and approximate partial derivatives according to [14, Section 3.1.2]. Let us take , , and (which satisfies (LIC)). Let us consider a function which is bounded and continuous but not approximately differentiable for a.e. (see, for instance, [23, p. 297]), and define the function by
We have that , and it is constant with respect to x. Thus, for any , we have that
and so . Hence , and in particular we have that for any . If it was the case that u satisfies the desired property, then we would have, for a.e. in Ω, that u is approximately differentiable at (see [19, Theorem 1]). Thus, according to [23, Theorem 12.2] and to the fact that u is constant with respect to x, we would have that for any and for a.e. the function is approximately differentiable at y, but this last assertion is in contradiction with our choice of w.
2.5 Algebraic properties of
Here we present some algebraic properties of the coefficient matrix . The following results have been achieved in [20, Section 3.2].
Suppose is a family of Lipschitz vector fields. For any , we define the linear map by
From standard linear algebra, we know that , and so, for any and , there is a unique choice of and such that
Finally, we define as the projection .
These definitions make sense for a generic family of Lipschitz vector fields, but the following two propositions list some very useful invertibility and continuity properties that are typical of those families of vector fields satisfying (LIC).
Let be a family of Lipschitz vector fields satisfying (LIC) on Ω. Then the following facts hold:
for each and . In particular, is an isomorphism.
Then, for each , is a symmetric invertible matrix of order m . Moreover, the map
For each , the projection can be represented as
It is easy to see that . Hence, is closed in Ω.
Let be a family of Lipschitz vector fields satisfying (LIC) on Ω. Then the map is invertible and the map , defined by
belongs to .
2.6 Local functionals
We conclude this section by giving some definitions about increasing set functions, for which we refer to [12, Chapter 14], and local functionals defined on . From now on, we assume that is a family of Lipschitz vector fields satisfying (LIC) on Ω.
We say that is a locally integrable modulus of continuity if and only if
Let us consider a functional , where is a functional space such that . We make the following definitions:
F satisfies the strong condition if there exists a sequence of locally integrable moduli of continuity such that(2.2)
for any , , and such that
for all .
F satisfies the weak condition if there exists a sequence of locally integrable moduli of continuity such that
for any , , and such that
Let be a function. We make the following definitions:
α is increasing if it holds that for any such that .
α is inner regular if it is increasing and for any .
α is subadditive if it is increasing and, for any with ,
α is superadditive if it is increasing and, for any with and ,
α is a measure if it is increasing and the restriction to of a non-negative Borel measure.
Let us consider a functional
We make the following definitions:
F is a measure if, for any ,
is a measure.
F is local if, for any and ,
F is convex if, for any , the function is convex.
F is p-bounded if there exist and such that, for any and for any , it holds that
F is lower semicontinuous (resp. weakly sequentially lower semicontinuous) if, for any ,
is sequentially l.s.c. with respect to the strong (resp. weak) topology of .
F is weakly*-sequentially lower semicontinuous if, for any ,
is sequentially l.s.c. with respect to the weak*-topology of .
3 Integral representation of convex functionals
In this section, we completely characterize a class of convex local functionals defined on . As announced, we exploit [7, Lemma 4.1] to get an integral representation of the form
Finally, we extend the integral representation to the whole .
Let be a Carathéodory function. Define by
Then the following facts hold:
f is a Carathéodory function.
If is convex for a.e. , then is convex for a.e. .
If is convex for a.e. and for any , then is convex for a.e. and for any .
If we assume that(3.2)
then it follows that(3.3)
(i) First we want to show that, for any , the function is measurable. Let us fix then , define the function by and extend it to be zero on . By Proposition 2.14, is continuous, and so in particular Φ is measurable. Noticing that
with being a Carathéodory function, and recalling [11, Proposition 3.7], we conclude that is measurable. Let us define now the function
Since on we have that , for any fixed such that is continuous, is the composition of a continuous function and a linear function, and so it is continuous.
(ii) If is such that is convex, then is the composition of a convex function and a linear function, and so it is convex.
(iii) This follows as (ii).
Now, (3.3) follows by integrating over A. ∎
In the following result, we provide some sufficient conditions to guarantee (3.2).
Let be a Carathéodory function such that the following conditions hold:
is convex for a.e and any .
There exist and such that(3.4)
for a.e. and any .
Then satisfies (3.2).
This follows with some trivial modifications as in [20, Lemma 3.13]. ∎
Let us now state and prove the main result of this section.
Let be such that the following conditions hold:
F is a measure.
F is local.
F is convex.
F is p -bounded.
Then there exists a Carathéodory function such that
and the following representation formula holds:
We prove this theorem in five steps.
First step. Let
Then, from our assumptions on , it follows that . Let . By using (iv) and recalling that for all we have that , it follows that
Thus, we can apply [7, Lemma 4.1] to get a Carathéodory function such that
and fix , and . By (3.9), for any small enough to ensure that , we have that
and from (iv) we have that
where is as in (2.1). Combining these two facts and dividing by , we obtain that
Since the integrand on the right-hand side is in , and (3.11) holds indeed for all , the one on the left-hand side is in as well. Therefore, thanks to the Lebesgue theorem, we can find such that and
it holds that and
Since the map is continuous for any and is dense in , inequality (3.4) holds and the conclusion follows.
Third step. Thanks to the previous step, we can apply Proposition 3.1 (iv). Hence, we get
where is the function defined in (3.1). First of all, we can assume that f is finite up to modifying it on a set of measure zero. Moreover, thanks to (3.10) and Proposition 3.1 (ii), we have that f satisfies (3.5). Now we want to prove that f satisfies (3.6). Let us fix , and . By (iv), (3.9) and (3.12), we have that
and so, dividing by , we get that
Arguing as in the second step, we can conclude that
Finally, by recalling that for the map is surjective, inequality (3.6) follows.
Fourth step. Here we want to prove that (3.7) holds. Let us fix and , and consider the two functionals
and so we assert that
and so we can conclude that (3.7) holds.
Fifth step. Let us show the uniqueness of the Lagrangian. Fix then , and : since (3.7) holds both for and , for any small enough, we have that
Since both integrand functions satisfy (3.6), they are both in . Again, thanks to the Lebesgue theorem, there exists such that and
If we set
clearly we have , and it holds that
Since and are continuous for any , and by recalling again that is surjective for any , we infer (3.8). ∎
The following theorem tells us that all hypotheses of Theorem 3.3 are also necessary.
Let be a Carathéodory function such that
for some and . If we set the functional as
then F satisfies hypotheses (i)–(iv) of Theorem 3.3.
Let us fix . Our aim is to prove that is a measure. Notice that, with , α is increasing, and of course . Then, according to [12, Theorem 14.23], it suffices to show that α is subadditive, superadditive and inner regular. The first two properties are trivial, so let us focus on the third one. Let us fix and define the sequence of sets by . We have that , and . Thus by the monotone convergence theorem, we conclude that
4 Integral representation of weakly*-sequentially lower semicontinuous functionals
In this section, we characterize a class of local functionals defined on for which we require neither translation-invariance nor convexity, but which are weakly*-sequentially lower semicontinuous in . It is well known (cf. ) that, for an integral functional of the form
the weak*-lower semicontinuity is equivalent to the convexity in the third entry of . Therefore, we can adopt the same strategy employed in the previous section, exploiting [8, Theorem 1.10] to get a Euclidean integral representation of the form
We now start by proving a useful continuity result in , whose classical version is usually known as Carathéodory continuity theorem.
Let be a Carathéodory function such that there exist and such that