Abstract
In this paper we prove an integral representation formula for a general class of energies defined on the space of generalized special functions of bounded deformation (
1 Introduction
Integral representation results are a fundamental tool in the abstract theory of variational limits by Γ-convergence or in relaxation problems (see [32]). The topic has attracted widespread attention in the mathematical community over the last decades, with applications in various contexts, such as homogenization, dimension reduction, or atomistic-to-continuum approximations. In this paper we contribute to this topic by proving an integral representation result for a general class of energies arising in the modeling of linear elastic solids with surface discontinuities.
Integral representation theorems have been provided with increasing generality, ranging from functionals defined on Sobolev spaces [1, 17, 18, 19, 33, 49] to those defined on spaces of functions of bounded variation [12, 22, 30, 14], in particular on the subspace
When coming to the variational description of rupture phenomena in general linearly elastic materials, however, the functional setting to be considered becomes weaker. Indeed, problems need to be formulated in suitable subspaces of functions of bounded deformation (
In the mathematical description of linear elasticity, the elastic properties are determined by the elastic strain. For a solid in a (bounded) reference configuration
Prototypical examples of functionals described above are energies which are controlled from above and below by suitable multiples of
where
Whereas in case (1) the energy space is be given by
The goal of the present article is twofold: we generalize the results of [28] for energies with control of type (1) to arbitrary space dimensions and, more importantly, we extend the theory to encompass also problems of the form (2), which are most relevant from an applicative viewpoint. Indeed, already in dimension two, the extension of [28] to the case where only a control of type (2) is available is no straightforward task. This is a fundamental difference with respect to the
Let us also remark that, while in the majority of integral representation results in
In our main result (Theorem 2.1), we prove an integral representation for variational functionals
(
Let us comment on the proof strategy. We follow the general approach of the global method for relaxation provided in [13, 14] for variational functionals in
one first shows that, for fixed
one then proves that the Radon–Nikodym derivative
When dealing with all of the abovementioned issues, a key ingredient is given by
a Korn-type inequality for special functions of bounded deformation, established recently by Cagnetti, Chambolle, and Scardia [20], which generalizes a two-dimensional result in [28] (see also [39]) to arbitrary dimension. It provides a control of the full gradient in terms of the symmetrized gradient, up to an exceptional set whose perimeter has a surface measure comparable to that of the discontinuity set. In particular, this estimate is used to approximate the function u with functions
In contrast to [28], the Korn inequality is also used in the proof of Lemma 4.1: at this point, one needs to show that functions of the form
approximate u in the topology of the convergence in measure, where
We also point out that, if instead a control of the type (1) is assumed, the arguments leading to Theorem 2.1 can be successfully adapted to extend the result for functionals on
For a related purpose, in Section 8 we discuss how our arguments can also provide a direct proof for integral representation results on
is given, see Theorem 8.1. In particular, no perturbation or truncation arguments are needed in the proof. Therefore, we believe that this provides a new perspective and a slightly simpler approach to integral representation results in
We close the introduction by mentioning that in a subsequent work [42] we use the present result to obtain integral representation of Γ-limits for sequences of energies in linear elasticity with surface discontinuities. There, we additionally
characterize the bulk and surface densities as blow-up limits of cell minimization formulas where the minimization is not performed on
The paper is organized as follows. In Section 2 we present our main integral representation result in
2 The integral representation result
In this section we present our main result. We start with some basic notation. Let
For definition and properties of the space
We consider functionals
there exist
We now formulate the main result of this article addressing integral representation of functionals
For
Moreover, for
In this paper, we will prove the following result.
Theorem 2.1 (Integral representation in
GSBD
p
).
Let
for all
for all
for all
Remark 2.2.
We proceed with some remarks on the result.
As
If the functional
where
An analogous result holds on the space
We will additionally discuss the minor modifications needed in order to deal with functionals
there exist
In this case,
3 Preliminaries
We start this preliminary section by introducing some further notation. For
The diameter of E is indicated by
3.1 BD and GBD functions
Let
where
The space
The spaces
Definition 3.1.
Let
for fixed
Then
Moreover, the function v belongs to
We recall that every
The notation for
for every
Any function
Proposition 3.2.
If
If U has Lipschitz boundary, for each
3.2 Korn’s inequality and fundamental estimate
In this subsection we discuss two important tools which will be instrumental for the proof of Theorem 2.1. We start by the following Korn and Korn–Poincaré inequalities in
Theorem 3.3 (Korn inequality for functions with small jump set).
Let
and an infinitesimal rigid motion a such that
Moreover, there exists
Note that the result is indeed only relevant for functions with sufficiently small jump set, as otherwise one can choose
Remark 3.4 (Almost Sobolev regularity, constants, and scaling invariance).
(i) More precisely, in [20] it is proved that there exists
for an infinitesimal rigid motion a. This directly implies (3.5), see [20, Theorem 4.1, Theorem 4.4].
(ii) Given a collection of bounded Lipschitz domains
(iii) Recall definition (3.1). Consider a bounded Lipschitz domain Ω,
and
where
From Theorem 3.3, one can also deduce that for
Lemma 3.5 (Approximate gradient).
Let
We point out that the result in Lemma 3.5 has already been obtained in [40] for
To control the affine mappings appearing in Theorem 3.3, we will make use of the following elementary lemma on affine mappings, see, e.g., [44, Lemma 3.4] or [29, Lemmas 4.3] for similar statements. (It is obtained by the equivalence of norms in finite dimensions and by standard rescaling arguments.)
Lemma 3.6.
Let
We now proceed with another consequence of Theorem 3.3.
Corollary 3.7.
Let
such that
Proof.
We start by choosing a bounded Lipschitz domain
where
for a sufficiently large constant
Now, consider a function u satisfying (3.8). We apply Theorem 3.3 on
In particular,
By (3.4) and (3.8) we get
and, in a similar fashion,
We conclude this subsection with another important tool in the proof of the integral representation, namely a fundamental estimate in
Lemma 3.8 (Fundamental estimate in
GSBD
p
).
Let
where
where we used the notation introduced in (3.1).
The same statement holds if
In the statement above, we intend that
Proof.
The proof follows the lines of [16, Proposition 3.1]. Choose
Let
For
Consider
For the last term, we compute using (H4) (
Notice that we can obtain the same estimate also if
where
4 The global method
This section is devoted to the proof of Theorem 2.1 which is based on three ingredients. First, we show that
Lemma 4.1.
Suppose that
We prove this lemma in the final part of this section.
The second ingredient is that, asymptotically as
Lemma 4.2.
Suppose that
We defer the proof of Lemma 4.2 to Section 5. The third ingredient is that, asymptotically as
Lemma 4.3.
Suppose that
We defer the proof of Lemma 4.3 to Section 6, and now proceed to prove Theorem 2.1.
Proof of Theorem 2.1.
In view of (H4) on
where f was defined in (2.4), and that for
where g was defined in (2.5).
By Lemma 4.1 and the fact that
for
for
for
In the remaining part of the section we prove Lemma 4.1. We basically follow the lines of [13, 14, 28], with the difference that the required compactness results are more delicate due to the weaker growth condition from below (see H4) compared to [13, 14, 28]. We start with some notation. For
where, as before,
In the following lemma, we prove that
Lemma 4.4.
Suppose that
Proof.
We follow the lines of the proof of [28, Lemma 4.1] focusing on the necessary adaptions due to the weaker growth condition from below (see H4) compared to [28]. For each ball
We now address the reverse inequality. We fix
By the definition of
We define
where
where we also used the fact that
We now claim that
for a constant
By (4.11) (ii) and the fact that the balls
As
Now, combining (4.12)–(4.14) and using (4.10), we find
Proof of Lemma 4.1.
The statement follows by repeating exactly the arguments in [28, proofs of Lemma 4.2 and Lemma 4.3]. We report a sketch for the reader’s convenience.
The definition of
The converse inequality follows by proving that, for every
satisfies
and we show that
satisfies
Then
We now confirm (4.15). The inclusion
By recalling also the definition of
Passing to the limit in δ, (4.5) and Lemma 4.4 imply
so that
To conclude the proof of Theorem 2.1, it remains to prove Lemmas 4.2 and 4.3. This is the subject of the following two sections.
5 The bulk density
This section is devoted to the proof of Lemma 4.2. We start by analyzing the blow-up at points with approximate gradient. The latter exists for
Lemma 5.1 (Blow-up at points with approximate gradient).
Let
Proof.
Let
These properties hold for
Fix
where
We define
and proceed by confirming the properties stated in (5.1). Notice that, by construction,
Since, by a direct computation,
To this end, fix
By the definition of
where
Therefore, by using also (5.2) (i), we derive
Now, (5.2) (i) and (5.3) (ii) give that
As
We are now in a position to prove Lemma 4.2.
Proof of Lemma 4.2.
It suffices to prove (4.1) for points
Step 1 (Inequality “
We extend
By (3.11)–(3.12) there exist functions
where
This along with (5.10) shows that there exists a sequence
On the one hand, by using that
where in the last step we substituted
By (5.1) (iii)–(iv) this implies
Recall that
Passing to
Step 2 (Inequality “
We consider
We extend
We apply Lemma 3.8 on
By (5.17) and the choice of
Let us now estimate the terms in (5.18). We get by (H1), (H4), (5.16)–(5.17), and the choice of
Therefore, by (5.14), (5.15), and the fact that
Estimating
Passing to
This shows inequality “
6 The surface density
This section is devoted to the proof of Lemma 4.3. We start by analyzing the blow-up at jump points.
In the following, for any
Lemma 6.1 (Blow-up at jump points).
Let
where
Proof.
We start by using the fact that
In particular, (ii) follows from the fact that
where
For any
where
where, as before, the constant is independent of ε. (Here, we used again the properties of the functions
where
We now show (6.1) (iii). Concerning the ”
for any
by (6.2) (i), while the last three terms are estimated by (6.5) (ii) and (6.4).
To see the converse, we first apply [25, Theorem 1.1]
to the functions
It remains to prove (6.1) (ii). We notice that this easily follows from
In fact, (6.2) (ii) and (6.6) give that
Then (6.8), (6.9), the triangle inequality,
Therefore, let us now confirm (6.8). We only address the “
satisfy
In view of (6.5) (i), (iii), we have that
Then, by (6.11), the definition of
By (6.2) (i), (6.3), (6.5) (ii), and (6.10) we obtain
Let us consider
(6.14)
So we are left to prove (6.14) which corresponds to [28, equations (3.18)–(3.19)]. The proof goes in the same way with slight modifications that we indicate below. For fixed
For
In fact, the first inequality follows from (6.5) (ii)
and Lemma 3.6, and the second one from (6.11), (6.15), and the triangle inequality.
Similarly, employing (6.5) (i), (iv) in place of (6.11), and recalling
At this stage, (6.14) follow exactly as in [28]: for
The right-hand side
vanishes as
for all
for
Remark 6.2 (Construction of
u
ε
).
We point out that our definition of
Then, instead of (6.8), one needs to check
This, however, is in general false if
see [13, equation (24)] and [28, Lemma 3.4 (i)]. It is not clear to us if it is possible that for
We now proceed with the proof of Lemma 4.3.
Proof of Lemma 4.3.
The proof follows the same strategy of the proof of Lemma 4.2. We fix
Step 1 (Inequality “
We extend
Recalling notation (3.1) and (3.11)–(3.12), we find a function
where
Inserting this in (6.21), we find that, for a suitable sequence
We evaluate the first terms on the right-hand side of (6.22): since
where, as in Lemma 6.1, we denote by
Collecting (6.22), (6.23), (6.24), and recalling
Passing to the limit as
Step 2 (Inequality “
We consider
We extend
We apply Lemma 3.8 for
We observe that
We estimate the first terms in (6.28). We get by (H1), (H4), (6.26)–(6.27), and the choice of
By (6.24), (6.25), and the fact that
and, similarly to (6.23),
Collecting (6.28), (6.30), (6.31) and using
Finally, recalling that
This shows “
7 The
SBD
p
case
This section is devoted to the analysis of the integral representation result for
We start by pointing out that, under (H4’), only competitors in
Then the following integral representation result holds.
Theorem 7.1 (Integral representation in
SBD
p
).
Let
for all
for all
for all
The remainder of this section is devoted to the proof of Theorem 7.1 which follows along the lines of the proof of Theorem 2.1 devised in Section 4. First, the analogue of Lemma 4.1 holds essentially with the same proof.
Lemma 7.2.
Suppose that
Proof.
One can follow the same argument used to prove Lemma 4.1 through Lemma 4.4.
First, we remark that [28, Lemmas 4.2 and 4.3] is proved under the assumptions (H4’) and
Concerning Lemma 4.4, as the lower bound of (H4’) is stronger than the one of (H4), the
We now address the adaptions necessary for the bulk density. When
Lemma 7.3.
Let
Proof.
Since
and (see [4, Theorem 7.4]) that
where for brevity we again let
We can now perform the same construction as in (5.5) with
To see (7.2) (ii), observe that, since
Hence, by the trace inequality and by scaling back to
With this, (7.5), and the fact that
Hence, we get by the construction of
which concludes the proof. ∎
With the above lemma at our disposal, we can deduce the asymptotic equivalence of the minimization problems (7.1) for u and
Lemma 7.4.
Suppose that
Proof.
We argue as in the proof of Lemma 4.2.
For the ”
Thus, using (5.1) (iii), (iv) and (7.2) (ii), we still get (5.14), and may deduce inequality “
For the “
where
so that,
using (5.14), (5.15), (7.7), and the fact that
Similar changes have to be performed also for the surface density. We first deduce the analogue of Lemma 6.1. We again set
Lemma 7.5.
Let
where
Proof.
Since
Hence, by Fubini’s Theorem we find
and
We can now perform the same construction as in (6.7) and derive (6.1). In particular, this in combination with Hölder’s inequality,
To see (7.8) (ii), observe that, since
Hence, by the trace inequality and by scaling back to
Since
Given a Borel set
By (7.10) and (7.11) we also have
where
By combining with (7.12), this concludes the proof of (7.8) (ii). ∎
With this lemma at hand, we can address the equivalence of minimization problems for the surface scaling.
Lemma 7.6.
Suppose that
Proof.
We argue as in the proof of Lemma 4.3.
For the “
which is the analogue of (6.24). This is enough to derive inequality “
For the reverse one, take again
where
so that, using (6.25), (7.14), (7.15), and the fact that
Then one concludes exactly in the same way, upon replacing β in equation (6.31) with the larger constant
8 The
GSBV
p
case
In this section we briefly remark that the strategy devised in this paper allows also to establish an integral representation result in the space
with the following general assumptions:
there exist
In this setting, we replace definition (2.1) by
Moreover, as in (2.2)–(2.3), we define the functions
Theorem 8.1 (Integral representation in
GSBV
p
).
Let
for all
for all
for all
We point out that integral representation results in
restricted to
For the proof of the theorem we need the following Poincaré-type inequality, which can be directly deduced from Theorem 3.3.
Theorem 8.2 (Poincaré inequality for functions with small jump set).
Assume
and
In particular, for all
Proof.
It suffices to consider the case
Proof of Theorem 8.1.
We follow the proof of Theorem 2.1 and only indicate briefly the necessary adaptions. First, we observe that a version of the fundamental estimate in Lemma 3.8 holds true in
Lemma 4.1: The result is proved via (4.15) and Lemma 4.4. Notice indeed that the derivation of (4.15), as well as the argument ensuing therefrom are the same, up to using the growth condition ($\mathbb{H}$4) instead of (H4). (We also refer to [13, Lemma 6] for the corresponding argument in
Lemma 4.2: With the fundamental estimate in
for
With (5.5), and using again (8.2), we deduce that (5.1) (iii) can be improved to
This adaption is enough to redo the proof of Lemma 4.2 in the present situation.
Lemma 4.3: Here, we can follow the proof of Lemma 4.3 for each
First, we use (6.7) to see that
This along with
Funding source: Horizon 2020 Framework Programme
Award Identifier / Grant number: 793018
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: FR 4083/1-1
Award Identifier / Grant number: EXC 2044-390685587
Funding source: Ministero dell’Istruzione, dell’Università e della Ricerca
Award Identifier / Grant number: PRIN 2017
Funding statement: For this project Vito Crismale has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 793018. Manuel Friedrich is supported by the DFG project FR 4083/1-1 and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044-390685587, Mathematics Münster: Dynamics–Geometry–Structure. The work of Francesco Solombrino is part of the project “Variational methods for stationary and evolution problems with singularities and interfaces” PRIN 2017 financed by the Italian Ministry of Education, University, and Research.
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