We prove a compactness result in which also provides a new proof of the compactness theorem in , due to Chambolle and Crismale. Our proof is based on a Fréchet–Kolmogorov compactness criterion and does not rely on Korn or Poincaré–Korn inequalities.
In this paper, we prove a compactness result in , which in particular provides an alternative proof of the compactness theorem in obtained by Chambolle and Crismale in [5, Theorem 1.1]. Referring to Section 2 for the notation used below, the theorem reads as follows.
Let be an open bounded subset of and let be such that
Then there exists a subsequence, still denoted by , such that the set
has finite perimeter, i.e. a.e. in for some function with in A. Furthermore,
We notice that the main difference to  is that we do not request equi-integrability of the approximate symmetric gradient and boundedness of the measure of the jump sets , but only boundedness of , which is the natural assumption for sequences in . Hence, when passing to the limit, the absolutely continuous and the singular parts of could interact. For this reason, it is not possible to get weak -convergence of the approximate symmetric gradients or lower-semicontinuity of the measure of the jump.
Nevertheless, we are able to recover the lower-semicontinuity (1.2) for the set A where . In particular, formula (1.2) highlights that the emergence of the singular set A results from an uncontrolled jump discontinuity along the sequence . Hence, an equi-boundedness of the measure of the super-level sets , i.e.
for a positive function ϕ with superlinear growth at infinity. The novelty of our proof, presented in Section 3, concerns the compactness part of Theorem 1.1. It is based on the Fréchet–Kolmogorov criterion and makes no use of Korn or Poincaré–Korn-type of inequalities  (see also [2, 7, 8]), which are instead the key tools of . The remaining lower-semicontinuity results of [5, Theorem 1.1] can be obtained by standard arguments.
2 Preliminaries and notation
We briefly recall here the notation used throughout the paper. For , we denote by and the Lebesgue and the k-dimensional Hausdorff measure in , respectively. Given , we indicate with the Hausdorff dimension of F. For all compact subsets and of , stands for the Hausdorff distance between and . We denote by the characteristic function of a set . For every measurable set and every measurable function , we further set to be the set of approximate discontinuity points of u and
where with being the unilateral approximate limit of u at x.
For we denote by the space of matrices with real coefficients, and set . The symbol (resp. ) indicates the subspace of of squared symmetric (resp. skew-symmetric) matrices of order m. We further denote by the set of rotation matrices.
Let us now fix . For every , stands for the projection over the subspace orthogonal to ξ. For every measurable set , every , and every , we set
For measurable, , and , we define
For every open bounded subset U of , the space of generalized functions of bounded deformation  is defined as the set of measurable functions which admit a positive Radon measure such that for every one of the following two equivalent conditions is satisfied [6, Theorem 3.5]:
For every such that , the partial derivative is a Radon measure in U and
for every Borel subset B of U.
For -a.e. , the function belongs to and(2.1)
for every Borel subset B of U.
A function u belongs to if and (2.1) holds. Every function admits an approximate symmetric gradient . The jump set is countably -rectifiable with approximate unit normal vector . We will also use measures defined in [6, Definitions 4.10 and 4.16] for and . We further refer to  for an exhaustive discussion on the fine properties of functions in .
3 Proof of Theorem 1.1
This section is devoted to the presentation of an alternative proof of Theorem 1.1, based on the Fréchet–Kolmogorov compactness criterion. We start by giving two definitions.
Let denote an orthonormal basis of . We define
Given , we define the δ-neighborhood of by
In order to simplify the notation, given a family and a positive natural number m, we denote by the set consisting of all subsets of containing exactly m-elements of , i.e.
In order to prove Theorem 1.1, we need the following two lemmas, which allow us to construct a suitable orthonormal basis of that will be used to test the Fréchet–Kolmogorov compactness criterium.
Let be such that and consider a family of orthonormal bases of such that for every ,
Then there exists a further orthonormal basis such that for every ,
First of all, notice that whenever is such that
then we have
Indeed, let us suppose by contradiction that (3.3) does not hold for some . Since for we have that each is a finite union of -dimensional subspaces of intersected with , the equality
As a consequence, we get
Hence, if we denote by the basis contained in , then, by using Grassmann’s formula,
which is valid for each couple of vector subspaces of , we deduce
which is a contradiction to the assumption (3.1).
Fix an orthonormal basis of and let be the group of special orthogonal matrices. It can be endowed with the structure of an -dimensional submanifold of . We can identify an element with an -matrix whose columns are the vectors of an orthonormal basis Ξ written with respect to and vice versa.
In order to show the existence of Σ satisfying (3.2), we prove the following stronger condition: given , for -a.e. choice of Σ we have that
where denotes the j-th column vector of the matrix representing Σ. In order to show (3.4), we claim that it is enough to prove that for every we have
is the canonical projection map. Indeed, if Σ does not belong to for every and for every , then, by using the definition of the map , we deduce immediately that Σ satisfies
Therefore, if (3.5) holds, then the set (remember that is a discrete set)
To prove (3.5) it is enough to show that the differential of has full rank at every point
Indeed, this implies that is an -dimensional submanifold for every , which ensures the validity of (3.5) since
Notice that is the restriction to of the map defined by
where is the j-th column vector of the matrix . To show that the differential of has full rank everywhere, it is enough to check that for every the differential of restricted to has rank equal to . By using the relation
valid for every , we can reduce ourselves to the case , where denotes the identity matrix and is such that . It is well known that
where denotes the space of skew symmetric matrices. Using that , we identify a point as
A direct computation shows that the differential of at the point acting on the vector Z is given by
It is better to introduce the matrix defined by
Roughly speaking, given , the product is the matrix in obtained by removing from X the i-th column, while given , the product is the matrix in obtained by adding a new row made of zero entries at the i-th position. With this definition, the linear map can be rewritten more compactly as
Given such that (where denotes the reference orthonormal basis of ), we can rewrite the system as
Hence, by the well-known relation
valid for every linear map and all finite-dimensional vector spaces V and W, if we want to prove that has full rank, i.e.
(where the first inequality comes from ), it is enough to show that
Again by relation (3.6), we can reduce ourselves to find the dimension of the kernel of the map
But this dimension can easily be computed to be
which immediately implies (3.7). ∎
By a standard argument from linear algebra, it is possible to construct n orthonormal bases of , say , satisfying
Moreover, given open, can be chosen in such a way that
Let be a measurable set with , let be measurable subsets of A, and let be measurable functions . Then, given a sequence , there exist a sequence with , a map , and an orthonormal basis Ξ of such that, up to passing through a subsequence on k,
We claim that for every natural number , for every , for every , and for every open set there exist and a family of orthonormal bases such that, up to subsequences on k, we have
Clearly, the pair depends on , but we do not emphasize this fact. We proceed by induction on j. The case : given , , and any open set , we can make use of Lemma 3.3 and Remark 3.4 to find N orthonormal bases such that
Since the are closed sets, there exists such that
We want to prove the same for . For this purpose, we fix a natural number , a parameter , and an open set . By using the induction hypothesis, we may suppose that (3.8) and (3.9) hold true for . This means that, given and (to be chosen later), we find and orthonormal bases such that (3.8) and (3.9) hold true for . Choose and consider the set
which is the union of all possible -intersections of sets of the form for .
We recall the following identity valid for any finite family of subsets of A, say :
Now we partition into disjoint subsets (without loss of generality, we may choose N to be an integer multiple of M) each of which belongs to . We call this partition . By construction, any l-intersection of sets of the form with can be written as the union of sets each of which, thanks to the fact that (we use that is a partition)
is the intersection of at least different sets of the form with . Taking this last fact into account, if we replace the sets by and in identity (3.11), we obtain
Now suppose that for every it holds true for some k that
Then inequality (3.12) implies
Therefore, if we choose N sufficiently large in such a way that
and such that
where we have used that , the domain of , is contained in A. Since is a finite family, we may suppose that, up to subsequences on k, we find a common for which
Now we prove the lemma. For , the claim says in particular that we find an orthonormal basis and such that, up to passing to a subsequence on k, we have
Notice that, by using a continuity argument, we find a neighborhood of in such that
By applying again the claim, we find an orthonormal basis and such that, up to passing to a further subsequence on k, we have
Hence if we set , we obtain as well
Proceeding again by induction, we find for every an orthonormal basis , , and a subsequence such that
If we denote with abuse of notation the diagonal sequence simply as k, then we can find a map such that
Since the family is made of compact subsets of , then it is relatively compact with respect to the Hausdorff distance. This means that, up to a subsequence on h, we find an orthonormal basis Ξ such that
By using (3.17) and the fact that are relatively open subsets of , this last convergence tells us that for every h the compact inclusion holds true. But this implies that, up to defining suitable with , we can write
Finally, with abuse of notation, we set for every h. Then (3.16) implies
This gives the desired result. ∎
Given , , and , we have that
Indeed, given , one can consider a partition of into a finite family of measurable sets such that for every there exists an orthonormal basis with for every and for every and . Consider then the partition of given by , where
We then have
where we have used that for every for -a.e. with .
Let and . Given and , if we introduce the map as
then we have . More precisely, for -a.e. we have
(notice that for -a.e. y the right-hand side is a finite measure thanks to Remark 3.6). By using the inclusion , valid for every for every , and for -a.e. , we deduce
We are now in a position to prove Theorem 1.1.
Proof of Theorem 1.1.
Let . We claim that for every the family is relatively compact in , where denotes a suitable orthonormal basis of . Now given , by using Lemma 3.5, there exists such that if we define and by , then
for a suitable orthonormal basis Ξ and a suitable map .
In order to simplify the notation, let us denote . Fix and set
for every and . Notice that
We define . Since we want to apply the Fréchet–Kolmogorov theorem, we have to estimate for ,
Now notice that, by the definition of (see Definition 3.1), there exists a positive constant such that for every and every ,
Moreover, by taking into account (3.19), we deduce the existence of a dimensional parameter such that
Analogously, if and , we have
Moreover, setting , we can write
By a mollification argument, we have that
so that we obtain from (3.25) that
Summarizing, we have shown that if is such that and
then for every we have for every ,
As a consequence, there exists a positive constant such that
Since the index i chosen at the beginning was arbitrary, this means also that if we consider the diffeomorphism defined by , then
By the Fréchet–Kolmogorov theorem, this last inequality implies that the sequence is relatively compact in . Hence, we can pass to another subsequence, still denoted by , such that as strongly in . By eventually passing through another subsequence, we may suppose