Abstract
Measure-valued structured deformations are introduced to present a unified theory of deformations of continua. The energy associated with a measure-valued structured deformation is defined via relaxation departing either from energies associated with classical deformations or from energies associated with structured deformations. A concise integral representation of the energy functional is provided both in the unconstrained case and under Dirichlet conditions on a part of the boundary.
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1 Introduction
The primary objective of continuum mechanics in solids is to articulate how a solid body will alter its shape when subjected to specified external forces or boundary conditions. A crucial initial step toward achieving this objective involves selecting a category of deformations for the continuum. In describing numerous continua, certain widely accepted criteria for the chosen category of deformations have been established: these deformations should be invertible, with differentiable mappings and inverses, and the combination of two deformations within this category should result in another deformation within the same category. However, classical deformations may not always suffice for describing all continua, requiring alternative selections in many cases. One approach involves introducing additional kinematic variables, such as the director fields in a polar continuum. An alternative approach entails incorporating supplementary fields that, while connected to the deformation, function as internal variables. For instance, in theories concerning plasticity, the plastic deformation tensor follows an evolutionary law outlined in the constitutive equations of the continuum.
Del Piero and Owen (1993) proposed an alternative approach that identifies classes of deformations called structured deformations, suited for continua featuring supplementary kinematical variables, as well as for continua featuring internal variables (we refer the reader to Matias et al. (2023) for a comprehensive survey on this topic). In the theory of structured deformations, if \(\Omega \subset {\mathbb {R}}^{N}\) is the continuum body, the role usually played by the deformation field \(u:\Omega \rightarrow {\mathbb {R}}^{d}\) and by its gradient \(\nabla u:\Omega \rightarrow {\mathbb {R}}^{d\times N}\) is now played by a triple \((\kappa ,g,G)\), where the piecewise differentiable field \(g:\Omega \setminus \kappa \rightarrow {\mathbb {R}}^{d}\) is the macroscopic deformation and the piecewise continuous matrix-valued field \(G:\Omega \setminus \kappa \rightarrow {\mathbb {R}}^{d\times N}\) captures the contribution at the macroscopic level of smooth submacroscopic changes. The (possibly empty) discontinuity set \(\kappa \subset \Omega \) of g and G can be regarded as the crack set of the material. The main result obtained by Del Piero and Owen is the Approximation Theorem (Del Piero and Owen 1993, Theorem 5.8) stating that any structured deformation \((\kappa ,g, G)\) can be approximated (in the \(L^\infty \) convergence) by a sequence of simple deformations \(\{(\kappa _n,u_n)\}\). The matrix-valued field \(\nabla g-G\) captures the effects of submacroscopic disarrangements, which are slips and separations that occur at the submacroscopic level. The spirit with which structured deformations were introduced was that of enriching the existing class of energies suitable for the variational treatment of physical phenomena without having to commit at the outset to a specific mechanical theory such as elasticity, plasticity, or fracture. Ideally, the regime of the deformation is energetically chosen by the body depending on the applied external loads: if these are small, then the deformation will most likely be elastic, whereas if these are large, a plastic regime or even fracture may occur.
The natural mathematical context to study problems similar to those mentioned above is that of calculus of variations, in which equilibrium configurations of a deforming body subject to external forces are obtained as minimizers of a suitable energy functional. In the classical theories where the mechanics is described by the gradient of the deformation field u, a typical expression of the energy is
where \(W:{\mathbb {R}}^{d\times N}\rightarrow [0,+\infty )\) and \(\psi :{\mathbb {R}}^{d}\times {\mathbb {S}}^{N-1}\rightarrow [0,+\infty )\) are continuous functions satisfying suitable structural assumptions and model the bulk and interfacial energy densities, respectively. In the context of Del Piero and Owen, it is not clear how to assign energy to a structured deformation \((\kappa ,g,G)\); the issue was solved by Choksi and Fonseca who, providing a suitable version of the approximation theorem (Choksi and Fonseca 1997, Theorem 2.12), use the technique of relaxation to assign the energy \(I(g,G;\Omega )\) as the minimal energy along sequences \(\{u_n\}\subset SBV(\Omega ;{\mathbb {R}}^{d})\) converging to \((g,G)\in SBV(\Omega ;{\mathbb {R}}^{d})\times L^1(\Omega ;{\mathbb {R}}^{d\times N})=:SD(\Omega ;{\mathbb {R}}^{d}\times {\mathbb {R}}^{d\times N})\) in the following sense:
where \(\nabla u_n\) denotes the absolutely continuous part of the distributional gradient Du. More precisely, the relaxation process reads
and is accompanied by integral representation theorems in \(SD(\Omega ;{\mathbb {R}}^{d}\times {\mathbb {R}}^{d\times N})\) for the relaxed energy \(I(g,G;\Omega )\) (see (Choksi and Fonseca 1997, Theorems 2.16 and 2.17) and (Owen and Paroni 2015, Theorem 3)). The reader might have noticed that the crack set \(\kappa \) has been identified with the jump set \(S_{u_n}\) of the field \(u_n\in SBV(\Omega ;{\mathbb {R}}^{d})\). The variational setting introduced in Choksi and Fonseca (1997) gave rise to numerous applications of structured deformations in various contexts, see Amar et al. (2022), Barroso et al. (2017a), Barroso et al. (2022), Barroso et al. (2024), Carita et al. (2018), Matias et al. (2021), and Matias et al. (2017), in which an explicit form of the energy \(I(g,G;\Omega )\) could be provided.
We stress that, although we look at targets (g, G) belonging to \(SBV(\Omega ;{\mathbb {R}}^{d})\times L^1(\Omega ;{\mathbb {R}}^{d\times N})\), in general, the convergence (1.2) might lead to limits that are in \(BV(\Omega ;{\mathbb {R}}^{d})\times {\mathcal {M}}(\Omega ;{\mathbb {R}}^{d\times N})\) and that, in assigning the energy (1.3), Choksi and Fonseca make the explicit choice to represent the relaxed energy only in \(SBV(\Omega ;{\mathbb {R}}^{d})\times L^1(\Omega ;{\mathbb {R}}^{d\times N})\). Moreover, from the mechanical point of view, one cannot, in principle, exclude that \(\{\nabla u_n\}\) develop singularities in the limit, which would reflect on a weaker regularity of the field G, possibly not even of the same type as those of the singular part \(D^sg\) of the distributional derivative Dg, as is the case in Baía et al. (2012), Barroso et al. (2017b), and Matias and Santos (2014). Both these mathematical and mechanical reasons suggest that the definition of structured deformations should be extended from \(SD(\Omega ;{\mathbb {R}}^{d}\times {\mathbb {R}}^{d\times N})\) to the larger set
which we call measure-valued structured deformations, and which we abbreviate here with mSD.
In this paper, we generalize the results of Choksi and Fonseca (1997) to mSD. In particular, denoting with \(u_n{\mathop {\rightharpoonup }\limits ^{*}}(g,G)\) in mSD the convergence in (1.2), we prove the Approximation Theorem 2.3: given any measure-valued structured deformation \((g,G)\in mSD\), there exists a sequence \(\{u_n\}\subset SBV(\Omega ;{\mathbb {R}}^{d})\) such that \(u_n{\mathop {\rightharpoonup }\limits ^{*}}(g,G)\) in mSD. This serves to define the energy \(I:mSD\rightarrow [0,+\infty )\) via the relaxation (1.3) in the larger space mSD, see (2.2), for which we prove the integral representation result, Theorem 2.4. This is one of the main results of the paper, in which we recover the same structure of Choksi and Fonseca (1997, Theorems 2.16 and 2.17) and Owen and Paroni (2015, Theorem 3), with the presence of an additional diffuse part. One of the novelties of our setting is that we manage to obtain a concise form of the relaxed energy functional involving only a bulk contribution H and its recession function at infinity \(H^\infty \)
where \(D^sg\) and \(G^s\) are the singular parts of the measures Dg and G, respectively, see (2.13), in the typical form of Goffman and Serrin (1964) for functionals defined on measures for a density, which is a particular case of those treated in Arroyo-Rabasa et al. (2020). The relaxed bulk energy density H turns out to be quasiconvex-convex; see Proposition 4.3. It is interesting to notice that not every quasiconvex-convex function can be obtained as the bulk energy density associated with a structured deformation: ours retains the memory of the specific relaxation process (2.2) (see also the counterexample in Proposition 6.3). In Theorem 6.1 we prove that the energy \(I(g,G;\Omega )\) can be obtained by relaxing from \(SD(\Omega ;{\mathbb {R}}^{d}\times {\mathbb {R}}^{d\times N})\) to \(mSD(\Omega ;{\mathbb {R}}^{d}\times {\mathbb {R}}^{d\times N})\) the energy (1.1) with the addition of a term penalizing the structuredness \(\nabla g-G\)
see (6.1). Another relevant result is the possibility of performing the relaxation under trace constraints, see Theorem 5.1, which has the far-reaching potential of studying minimization problems in \(mSD(\Omega ;{\mathbb {R}}^{d}\times {\mathbb {R}}^{d\times N})\) with the addition of boundary data.
From the point of view of continuum mechanics, measure-valued structured deformations have the potential of extending the multiscale theory of structured deformations to include the contributions to macroscopic deformations of submacroscopic deformations that concentrate on subsets of \({\mathbb {R}}^{N}\) of dimension lower than \(N-1\). We envisage that allowing G to be a measure-valued field would allow the theory to include concentration of strain in the crease of a folded sheet of paper (where the resistance to bending, for instance, is different in the flat part of the folded paper and in the crease), as well as to include the presence of dislocation lines in three-dimensional solids. Considering the deformations of a material domain in the shape of a three-quarter cylinder would allow one to consider concentrations located at the axis of the cylinder and possibly lead one to a description of disclinations (see, for reference, Cesana et al. 2024; Olbermann 2018; Seung and Nelson 1988; Volterra 1907) in terms of measure-valued structured deformations. Similarly, the inclusion of a third object \(\Gamma \in {\mathcal {M}}(\Omega ;{\mathbb {R}}^{d\times N\times N})\) that plays the role of a second-order gradient, could lead to a definition of measure-valued second-order structured deformations \((g,G,\Gamma )\), in the spirit of Owen and Paroni (2000) and Barroso et al. (2017b), and also in the spirit of Fonseca et al. (2019) in case the structuredness is only at the level of the second-order gradient. This investigation will be the object of future work.
2 Setting and the Definition of the Energy in mSD
We assume that the main results about functions of bounded variations are known, otherwise we refer the reader to the monograph (Ambrosio et al. 2000) for a thorough introduction; likewise, we refer the reader to Dal Maso (1993) for an introduction to relaxation (see also Braides 2002).
We consider an initial energy as in (1.1), which, since we take \(\kappa =S_u\), now can be written as \(E:SBV(\Omega ;{\mathbb {R}}^{d})\rightarrow [0,+\infty )\)
where \(W:{\mathbb {R}}^{d\times N}\rightarrow [0,+\infty )\) and \(\psi :{\mathbb {R}}^{d}\times {\mathbb {S}}^{N-1}\rightarrow [0,+\infty )\) are continuous functions satisfying the following assumptions for \(A\in {\mathbb {R}}^{d\times N}\), \(\lambda ,\lambda _1,\lambda _2\in {\mathbb {R}}^{d}\) and \(\nu \in {\mathbb {S}}^{N-1}\):
We consider measure-valued structured deformations, that is, pairs \((g,G)\in mSD\), see (1.4); we endow the space mSD with the norm
the latter term denoting the total variation of the measure G. We are interested in assigning an energy \(I:mSD\rightarrow [0,+\infty )\) by means of the relaxation
where, for every open set \(U\subset \Omega \),
is the set of admissible sequences. Our main result is a representation theorem for this energy, namely that \(I=J\) with the explicit representation of the limit functional given by
where \(H:{\mathbb {R}}^{d\times N}\times {\mathbb {R}}^{d\times N}\rightarrow [0,+\infty )\), \(h^j:{\mathbb {R}}^{d}\times {\mathbb {R}}^{d\times N}\times {\mathbb {S}}^{N-1}\rightarrow [0,+\infty )\), and \(h^c:{\mathbb {R}}^{d\times N}\times {\mathbb {R}}^{d\times N}\rightarrow [0,+\infty )\) are suitable bulk, surface, and Cantor-type relaxed energy densities. In (2.4), we have the following objects: since \(g\in BV(\Omega ;{\mathbb {R}}^{d})\), we know that, by De Giorgi’s structure theorem,
and we can decompose
where
Here, in case of \(G^a\) and other measures absolutely continuous with respect to the Lebesgue measure, our notation does not distinguish between the measure and its density with respect to \({\mathcal {L}}^N\). Also, notice that \(G_g^s\) is singular with respect to \({\mathcal {L}}^N+|Dg|\).
To carry out our program, we will use the following results.
Theorem 2.1
(Alberti 1991, Theorem 3; Choksi and Fonseca 1997, Theorem 2.8) Let \(G \in L^1(\Omega ; {\mathbb {R}}^{d{\times } N})\). Then there exist a function \(f \in SBV(\Omega ; {\mathbb {R}}^{d})\), a Borel function \(\beta :\Omega \rightarrow {\mathbb {R}}^{d{\times } N}\), and a constant \(C_N>0\) depending only on N such that
Lemma 2.2
(Choksi and Fonseca 1997, Lemma 2.9) Let \(u \in BV(\Omega ; {\mathbb {R}}^{d})\). Then there exist piecewise constant functions \({\bar{u}}_n\in SBV(\Omega ;{\mathbb {R}}^{d})\) such that \({\bar{u}}_n \rightarrow u\) in \(L^1(\Omega ; {\mathbb {R}}^{d})\) and
The following approximation theorem generalizes the one obtained in Silhavý (2015).
Theorem 2.3
(approximation theorem) Let \(\Omega \subset {\mathbb {R}}^{N}\) be a bounded, open set with Lipschitz boundary. For each \((g,G)\in mSD\) there exists a sequence \(\{u_n\}\subset SBV(\Omega ;{\mathbb {R}}^{d})\) such that \(u_n{\mathop {\rightharpoonup }\limits ^{*}}(g,G)\) in mSD according to (1.2). In addition, we have that
and
for constants \(C_1=C_1(N)>0\) and \(C_2(\Omega )=C_2(N,\Omega )>0\) independent of \(\{u_n\}\) and (g, G).
Proof
Let \(\{G^k\}\subset L^1(\Omega ;{\mathbb {R}}^{d\times N})\) be a sequence of functions such that \(G^k{\mathop {\rightharpoonup }\limits ^{*}}G\) as \(k\rightarrow \infty \) and \(\sup _{k\in {\mathbb {N}}}\left\Vert G^k\right\Vert _{L^1(\Omega ;{\mathbb {R}}^{d\times N})}\leqslant \left| G\right| (\Omega )\) (see Krömer et al. 2023), and consider the corresponding pairs \((g,G^k)\in BV(\Omega ;{\mathbb {R}}^{d})\times L^1(\Omega ;{\mathbb {R}}^{d\times N})\). By Theorem 2.1, for each \(k\in {\mathbb {N}}\), there exists \(f^k\in SBV(\Omega ;{\mathbb {R}}^{d})\) such that \(\nabla f^k=G^k\) and, by the estimate in (2.5),
Since \(\Omega \) is a Lipschitz set, we can use the Poincaré inequality and obtain that
By Lemma 2.2, for each \(k\in {\mathbb {N}}\) there exists a sequence \(\{{\bar{v}}^k_n\}\subset SBV(\Omega ;{\mathbb {R}}^{d})\) of piecewise constant functions such that \({\bar{v}}^k_n\rightarrow g-f^k\) in \(L^1(\Omega ;{\mathbb {R}}^{d})\) and, by (2.6), \(\left| D{\bar{v}}^k_n\right| (\Omega )\rightarrow \left| D(g-f^k)\right| (\Omega )\) as \(n\rightarrow \infty \). Now, the sequence of functions \(v^k_n:={\bar{v}}^k_n+f^k\) is such that \(v^k_n\rightarrow g\) in \(L^1\) and \(\nabla v^k_n=G^k\), as \(n\rightarrow \infty \), for every \(k\in {\mathbb {N}}\). The convergences in (2.7) and the estimates in (2.7) now follow from estimates (2.8) by a diagonal argument, by defining \(u_n:=v^{k(n)}_n\), with \(k(n)\rightarrow \infty \) slowly enough. \(\square \)
Before stating our integral representation result, we define the following classes of competitors for the characterization of the relaxed energy densities below. We let \(Q\subset {\mathbb {R}}^{N}\) be the unit cube centered at the origin with faces perpendicular to the coordinate axes, and for \(\nu \in {\mathbb {S}}^{N-1}\), we let \(Q_\nu \subset {\mathbb {R}}^{N}\) be the rotated unit cube so that two faces are perpendicular to \(\nu \). For \(A,B,\Lambda \in {\mathbb {R}}^{d\times N}\) and \(\lambda \in {\mathbb {R}}^{d}\), we define
where \(s_{\lambda ,\nu }(x):=\frac{1}{2}\lambda ({\text {sgn}}(x\cdot \nu )+1)\). Moreover, for any open set \(U\subset {\mathbb {R}}^{N}\) and \(v\in SBV(U;{\mathbb {R}}^{d})\), we let
Theorem 2.4
(integral representation) Let \(\Omega \subset {\mathbb {R}}^{N}\) be a bounded Lipschitz domain, and assume that (W:1)–(W:3) and (\(\psi \):1)–(\(\psi \):3) hold true. Then
where I and J are defined in (2.2) and (2.4), respectively, and the densities in J are given by
The proof is given in Sect. 4.1 (upper bound: \(I\leqslant J\)) and Sect. 4.2 (lower bound: \(I\geqslant J\)).
Remark 2.5
For the special case \((g,G)\in SD(\Omega ;{\mathbb {R}}^{d}\times {\mathbb {R}}^{d\times N})\), Theorem 2.4 reduces to Choksi and Fonseca (1997, Theorem 2.16) (for the functional \(I_1\) in the notation of Choksi and Fonseca (1997)). Unlike Choksi and Fonseca (1997), we assumed coercivity of W in (W:1), but only to avoid additional technicalities.
Remark 2.6
As shown in Proposition 3.1 below, \(h^c\) coincides with the recession function \(H^\infty \) of H, and \(h^j\) can be replaced by \(h^c=H^\infty \), more precisely,
This allows for another, much more elegant representation of J:
see Definition 3.3 and Proposition 4.1 below.
Remark 2.7
-
(i)
As a consequence of (\(\psi \):1) and (\(\psi \):3), \(\psi \) is also globally Lipschitz in \(\lambda \):
$$\begin{aligned} \left| \psi (\lambda _1,\nu )-\psi (\lambda _2,\nu )\right| \leqslant C_\psi \left| \lambda _1-\lambda _2\right| . \end{aligned}$$(2.14) -
(ii)
We will never use the symmetry condition in (\(\psi \):2) directly, but it is necessary to make E well-defined in SBV, as jump direction and jump normal are only uniquely defined up to a simultaneous change of sign.
Remark 2.8
(Instability of the contribution of \(G^s_g\) in \(I=J\)) As in the case of typical integral functionals in BV with \(G=0\), the individual contributions in J handling each of the four components of the measure decomposition
are not continuous with respect to strict or area-strict convergence; for instance, Lebesgue-absolutely continuous contributions can generate Cantor or jump contributions in the limit. The last contribution in J of the singular rest \(G_g^s\) is even worse than the others, though, because it is not even continuous in the norm topology of \(BV(\Omega ;{\mathbb {R}}^{d})\times {\mathcal {M}}(\Omega ;{\mathbb {R}}^{d\times M})\).
Take, for instance, \(N=1\),
In particular, \(h^c(0,B)=\left| B\right| \) for all \(B\in {\mathbb {R}} \). Then \((g_k,G)\rightarrow (g,G)=(0,\delta _0)\) strongly in \(BV\times {\mathcal {M}}\), but \(G_{g_k}^s=0\) for all k while \(G_{g}^s=\delta _0\) (since \(D^jg_k=\frac{1}{k} \delta _0\), the whole singular contribution of G with respect to \({\mathcal {L}}^N+|Dg_k|\) is captured by \(\frac{\mathrm dG}{\mathrm d|D^jg_k|}|D^jg_k|=k\frac{1}{k}\delta _0=\delta _0\), while \(g=0\) so that \(G^s_g=\delta _0=G\)). As a consequence, the contribution of \(G_{g_k}^s\) in J jumps in the limit as \(k\rightarrow \infty \):
3 Auxiliary Results
In this section, we present some auxiliary results that are pivotal for the proof of Theorem 2.4. In particular, we show that all three densities H, \(h^j\), and \(h^c\) are linked (Proposition 3.1) and we present a sequential characterization for them (Proposition 3.2). In Sect. 3.2, functionals depending on measures are introduced, as well as the notion of area-strict convergence.
3.1 Equivalent Characterizations of the Relaxed Energy Densities
Proposition 3.1
Assume that (W:3) and (\(\psi \):2) hold true and H, \(h^j\), and \(h^c\) are defined as in Theorem 2.4. Then the strong recession function of H,
exists. Moreover, we have that
and for all \(B\in {\mathbb {R}}^{d\times N}\), \(\lambda \in {\mathbb {R}}^{d}\), and \(\nu \in {\mathbb {S}}^{N-1}\),
where \(h^c_\nu \) is obtained from \(h^c\) by replacing the standard unit cube Q by the unit cube \(Q_\nu \) oriented according to the normal \(\nu \), i.e.,
Proof
We define \(H^\infty (A,B):=\limsup _{t\rightarrow \infty }\frac{1}{t}H(tA,tB)\). With this definition, we obtain (3.1) as a consequence of (W:3) and (\(\psi \):2). Moreover, this even holds if t is replaced by an arbitrary subsequence. The \(\limsup \) above is thus independent of subsequences and, therefore, a limit. It remains to show (3.2).
First equality in (3.2): We claim that \(h^c(A,B)=h^c_\nu (A,B)\) for arbitrary \(A,B\in {\mathbb {R}}^{d\times N}\). We will first show that \(h^c_\nu (A,B)\leqslant h^c(A,B)\). Let \(\varepsilon >0\) and choose an \(\varepsilon \)-almost minimizer \(u\in SBV(Q;{\mathbb {R}}^{d})\) for the infimum in the definition of \(h^c(A,B)\):
Up to a set of measure zero, \(Q_\nu \) can be covered with countably many shifted and rescaled, pairwise disjoint copies of Q:
with suitable \(x_i\in Q\), \(0<\delta _i\leqslant 1\). Defining
we obtain \({\tilde{u}}\in SBV(Q_\nu ;{\mathbb {R}}^{d})\) with
and \({\tilde{u}}=Ax\) on \(\partial {Q_\nu }\) (as well as on \(x_i+\delta _i \partial Q\)). Observe that by the definition of \(E^\infty \) in (2.10), the positive one-homogeneity of \(W^\infty \) and \(\psi \) and a change of variables,
the latter due to (3.4). In addition, (3.5) gives that \(\sum _{i\in {\mathbb {N}}} \delta _i^N=\sum _{i\in {\mathbb {N}}} {\mathcal {L}}^N(x_i+\delta _i Q)={\mathcal {L}}^N(Q_\nu )=1\). Using the additivity of the integrals in E, (3.5) and (3.7), we can sum (3.8) over i to conclude that
Similarly, we can also check that 
. Since \(\varepsilon >0\) was arbitrary and \({\tilde{u}}\) is admissible for the infimum in the definition of \(h^c_\nu \), this implies that \(h^c_\nu (A,B)\leqslant h^c(A,B)\). The opposite inequality follows in exactly the same way, with exchanged roles of Q and \(Q_\nu \).
Second equality in (3.2): We have to show that \(h^c_\nu (\lambda \otimes \nu ,B)=h^j(\lambda ,B,\nu )\). For \(k\in {\mathbb {N}}\), define the laterally extended cuboid
where \(\nu _j^\perp \), \(j=1,\ldots ,N-1\), are the pairwise orthogonal unit vectors perpendicular to \(\nu \) corresponding to the lateral faces of \(Q_\nu \). Notice that up to a set of measure zero formed by overlapping boundaries, \(R_\nu (k)\) can be written as a pairwise disjoint union of \((2k+1)^{N-1}\) shifted copies of \(Q_\nu \):
Now let \(\varepsilon >0\) and choose an \(\varepsilon \)-almost minimizer \(u\in SBV(Q_\nu ;{\mathbb {R}}^{d})\) for the infimum in the definition of \(h^c_\nu (\lambda \otimes \nu ,B)\):
with \(E^\infty \) defined in (2.10). Since \(\nu _j^\perp \cdot \nu =0\), the affine function \(x\mapsto (\lambda \otimes \nu )x\) determining the boundary values of u is constant direction \(\nu _j^\perp \) for each \(j=1,\ldots ,N-1\). We can therefore extend u periodically in the \((N-1)\) directions \(\nu _j^\perp \) to a function \(u_k\in SBV(R_\nu (k);{\mathbb {R}}^{d})\), without creating jumps at the interfaces between elementary cells of periodicity: \(u_k|_{Q_\nu }=u\), \(u_k(x+\nu _j^\perp )=u_k(x)\) whenever \(x,x+\nu _j^\perp \in R_\nu (k)\), \(u_k=(\lambda \otimes \nu )x\) on \(\xi + \partial Q_\nu \) for each \(\xi \in Z(k)\) (in the sense of traces), and \(|Du_k|(\xi + \partial Q_\nu )=0\) for each \(\xi \in Z(k)\). As a consequence, (3.10) is equivalent to
for all \(k\in {\mathbb {N}}\). Analogously, we can also extend the elementary jump function \(s_{\lambda ,\nu }\) used in the definition of \(h^j\) periodically to \(s_{\lambda ,\nu ,k}\in SBV(R_\nu (k);{\mathbb {R}}^{d})\), again without creating jumps at the interfaces since \(s_{\lambda ,\nu }\) is constant in directions perpendicular to \(\nu \).
Now choose functions \(\varphi _k\in C_c^\infty (R_\nu (k);[0,1])\) such that
Defining
we obtain that \({\tilde{u}}_k=s_{\lambda ,\nu ,k}\) on \(\partial R_\nu (k)\), \({\tilde{u}}_k=u_k\) on \(R_\nu (k-1)\) and
Since \(\# Z(k)=(2k+1)^{N-1}\), we conclude that \(\frac{1}{\# Z(k)}|D{\tilde{u}}_k|(R_\nu (k){\setminus } R_\nu (k-1))|=O(1/k)\rightarrow 0\) as \(k\rightarrow \infty \). Using the Lipschitz properties of W (W:2) and \(\psi \) (2.14), we can thus replace \(u_k\) with \({\tilde{u}}_k\) in (3.11), with an error that converges to zero as \(k\rightarrow \infty \):
Since \({\tilde{u}}_k=s_{\lambda ,\nu }\) on \(\partial R_\nu (k)\), we can define
without creating a jump at the interface between \(\frac{1}{2k+1}R_\nu (k)\) and the rest. As defined, \({\hat{u}}_k\) is now admissible for the infimum defining \(h^j\), and by a change of variables on the right-hand side of (3.12), we see that
As \(\varepsilon >0\) and \(k\in {\mathbb {N}}\) were arbitrary, (3.13) implies that \(h^c_\nu (\lambda \otimes \nu ,B)\geqslant h^j(\lambda ,B,\nu )\). The reverse inequality can be shown analogously. \(\square \)
In the following proposition, we prove a sequential characterization of the relaxed energy densities defined in (2.11). To do so, we define the classes of sequences of competitors (see (2.9))
Proposition 3.2
Suppose that (W:1)–(W:3) and (\(\psi \):1)–(\(\psi \):3) hold true. Then
Proof
The formulae (3.15a) and (3.15c) are obtained in the same way as in Choksi and Fonseca (1997, Proposition 3.1) (for the latter, notice that \(W=W^\infty \) is an admissible choice in (3.15a)); formula (3.15b) is obtained in the same way as in Choksi and Fonseca (1997, Proposition 4.1), i.e., by applying the technique of Choksi and Fonseca (1997, Proposition 3.1) in a vanishing strip around the jumps of \(s_{\lambda ,\nu }\).. \(\square \)
3.2 Nonlinear Transformation of Measures and Area-Strict Convergence
The following shorthand notation will prove useful below.
Definition 3.3
(nonlinear transformation of measures) For any Borel set \(U\subset {\mathbb {R}}^{N}\), any Borel function \(h:{\mathbb {R}}^{m}\rightarrow {\mathbb {R}} \) with strong recession function \(h^\infty \) and any Radon measure \(\mu \in {\mathcal {M}}(U;{\mathbb {R}}^{m})\), we define
where \(\mu ^s\) denotes the singular part of the Radon-Nikodym decomposition of \(\mu \) with respect to the Lebesgue measure \({\mathcal {L}}^N\): \(\mu =\frac{\mathrm d\mu }{\mathrm d{\mathcal {L}}^N}{\mathcal {L}}^N+\mu ^s\).
Definition 3.4
(area-strict convergence, cf. Kristensen and Rindler 2010) For a Borel set V, a sequence \((G_k)\subset {\mathcal {M}}(V;{\mathbb {R}}^{d\times N})\) and \(G\in {\mathcal {M}}(V;{\mathbb {R}}^{d\times N})\), we say that \(G_k\rightarrow G\) area-strictly if \(G_k{\mathop {\rightharpoonup }\limits ^{*}}G\) in \({\mathcal {M}}\) and
Analogously, if V is open, for a sequence \((g_k)\subset BV(V;{\mathbb {R}}^{d})\) and \(g\in BV(V;{\mathbb {R}}^{d})\), we say that \(g_k\rightarrow g\) area-strictly if \(g_k{\mathop {\rightharpoonup }\limits ^{*}}g\) in BV and \(\int _V \mathrm da(Dg_k)\rightarrow \int _V \mathrm da(Dg)\).
The following lemma is a generalized Reshetnyak continuity theorem; see Kristensen and Rindler (2010, Theorem 4) or Rindler (2018, Theorem 10.3).
Lemma 3.5
If \(H:{\mathbb {R}}^{d\times N}\times {\mathbb {R}}^{d\times N}\rightarrow {\mathbb {R}} \) is continuous and has a recession function in the strong uniform sense (see Proposition 4.2), then the functional defined on \({\mathcal {M}}(\Omega ;{\mathbb {R}}^{d\times N})\times {\mathcal {M}}(\Omega ;{\mathbb {R}}^{d\times N})\) by
is sequentially continuous with respect to the area-strict convergence of measures.
We also need the following well-known lemma combining area-strict approximations in BV and \({\mathcal {M}}\).
Lemma 3.6
Let \((g,G)\in mSD\). Then there exists a sequence \(\{(g_k,G_k)\}\subset W^{1,1}(\Omega ;{\mathbb {R}}^{d})\times L^1(\Omega ;{\mathbb {R}}^{d\times N})\) such that \(g_k\rightarrow g\) area-strictly in BV and \(G_k\rightarrow G\) area-strictly in \({\mathcal {M}}\).
Proof
The sequences \(\{g_k\}\) and \(\{G_k\}\) can be defined separately, essentially by mollification. As to \(\{g_k\}\), see, for instance, Rindler (2018, Lemma 11.1), while the case of \(\{G_k\}\) is simpler. \(\square \)
4 Proof of Theorem 2.4
The proof of Theorem 2.4 is divided into two parts, each of which is carried out in the following section.
4.1 Upper Bound
Proof
We have to show that \(I(g,G;\Omega )\leqslant J(g,G;\Omega )\), which is equivalent to the existence of a “recovery” sequence \(\{u_n\}\) admissible in the definition of I, i.e., such that \(u_n{\mathop {\rightharpoonup }\limits ^{*}}(g,G)\) in mSD and \(E(u_n;\Omega )\rightarrow J(g,G;\Omega )\). The proof here is presented using a series of auxiliary results collected below.
First observe that based on Proposition 3.1, our candidate J for the limit functional, introduced in (2.4) using H, \(h^j\), and \(h^c\), can be expressed as a standard integral functional of the measure variable (Dg, G) using only H and its recession function \(H^\infty \) as integrands (Proposition 4.1). As H is continuous and its recession function exists in a strong enough sense (cf. Proposition 4.2), J is sequentially continuous with respect to the area-strict convergence of measures (Lemma 3.5). Since any \((g,G)\in mSD\) can be approximated area-strictly by sequences in \(W^{1,1}\times L^1\) (Lemma 3.6), a diagonalization argument allows us to reduce the construction of the recovery sequence to the case \((g,G)\in W^{1,1}\times L^1\subset SBV\times L^1\). This special case was already obtained in Choksi and Fonseca (1997), see Remark 2.5. \(\square \)
Proposition 4.1
Suppose that (W:1)–(W:3) and (\(\psi \):1)–(\(\psi \):3) hold. Then with the notation of Definition 3.3,
where J is defined in (2.4) and H is given by (2.11a).
Proof
Clearly, \(H\Big (\frac{\mathrm d(Dg,G)}{\mathrm d{\mathcal {L}}^N}\Big )=H\Big (\nabla g,\frac{\mathrm dG}{\mathrm d{\mathcal {L}}^N}\Big )\). In addition,
since \(|G^s_g|\) and \(|Dg^s|+{\mathcal {L}}^N\) are mutually orthogonal by definition of \(G^s_g\), and the Cantor and jump parts of Dg are mutually orthogonal as well. Since \(H^\infty \) is positively 1-homogeneous, the definition of J implies the asserted representation once we use Proposition 3.1 to replace \(H^\infty \) by \(h^j\) and \(h^c\), respectively. \(\square \)
We need the following regularity properties of H, in particular at infinity.
Proposition 4.2
Suppose that (W:1), (W:3) and (\(\psi \):1)–(\(\psi \):3) hold. Then H is globally Lipschitz and the recession function \(H^\infty \) exists in the strong uniform sense, i.e., the limit
exists for all \((A,B)\in {\mathbb {R}}^{d\times N}\times {\mathbb {R}}^{d\times N} {\setminus } \{(0,0)\}\).
Proof
The Lipschitz property of H was proved in Barroso et al. (2022, Theorem 2.10) (the case \(p=1\)). Concerning (4.1), first observe that since H is Lipschitz with some constant \(L>0\),
Here, \(H^\infty (A,B)=\limsup _{t\rightarrow \infty } \frac{1}{t}H(tA,tB)\) as before. It, therefore suffices to show that
We claim that in fact, we even have that
where \(C(A,B)>0\) is a constant independent of t and \(H^\infty (A,B)=\limsup _{t\rightarrow +\infty } \frac{1}{t}H(tA,tB)\).
For a proof of 4.3, first fix \(\varepsilon >0\) and choose \(\varepsilon \)-almost optimal sequence \(\{u_{t,n}\}_n\) for the sequential characterization of H(tA, tB) in Proposition 3.2, dependent on \(t>0\) (and A,B). This choice yields that
The sequence \(v_{t,n}:=t^{-1}u_{t,n}\) then is also in the class of admissible sequences for the sequential characterization of \(h^c(A,B)\) in Proposition 3.2, and since \(h^c(A,B)=H^\infty (A,B)\) by Proposition 3.1, this entails that
where we exploited that \(W^\infty \) and \(\psi (\cdot ,\nu )\) are positively 1-homogeneous. Multiplying (4.4) by \(t^{-1}\) and combining it with (4.5) yields
Analogously, we can also choose and \(\varepsilon \)-almost optimal sequence \({\tilde{v}}_h\) for the sequential characterization of \(h^c(A,B)=H^\infty (A,B)\), which makes \({\tilde{u}}_{t,h}:=t{\tilde{v}}_h\) admissible for the sequential characterization of H(tA, Bt). With this, get that
The right hands sides of (4.6) and (4.7) can now be estimated in the same fashion: by (W:1) and the homogeneity of \(W^\infty \) we have that
This is exactly the case excluded in (W:3), so that together with (W:3), we obtain that
since \(0<\alpha <1\). Moreover, (4.4) implies that \(\left\| \nabla v_{t,n} \right\| _{L^1}=t^{-1}\left\| \nabla u_{t,n} \right\| _{L^1}\) is equi-bounded for \(t\geqslant 1\) since H is globally Lipschitz, \(\psi \geqslant 0\) and W is coercive by (W:1). Similarly, \(\left\| \nabla {\tilde{v}}_{t,n} \right\| _{L^1}\) is equi-bounded. Thus,
Now we can use (4.8) to obtain upper bounds for the right-hand sides of (4.6) and (4.7) and combine them. By (4.9), this yields that
for all \(t\geqslant 1\). Since \(\varepsilon >0\) was arbitrary, (4.10) implies (4.3). \(\square \)
4.2 Lower Bound
Our proof of the lower bound relies on the following lower semicontinuity property.
Proposition 4.3
Assume that (W:1)–(W:3) and (\(\psi \):1)–(\(\psi \):3) hold. Then the integrand H defined in (2.11a) is quasiconvex-convex in the sense that for all \(A,B\in {\mathbb {R}}^{d\times N}\),
Moreover, the functional \((g,G)\mapsto \int _\Omega \mathrm dH(Dg,G)\) is sequentially lower semi-continuous with respect to the convergence in (1.2).
Proof
We will first show that \((g,G)\mapsto \int _\Omega H(\nabla g,G)\,\mathrm dx\) is sequentially lower semi-continuous with respect to weak convergence in \(W^{1,1}\times L^1\). Take \((g,G)\in W^{1,1}(\Omega ;{\mathbb {R}}^{d})\times L^1(\Omega ;{\mathbb {R}}^{d\times N})\) and \(\{(g_n,G_n)\}\subset W^{1,1}(\Omega ;{\mathbb {R}}^{d})\times L^1(\Omega ;{\mathbb {R}}^{d\times N})\) with \((g_n,G_n)\rightharpoonup (g,G)\) weakly in \(W^{1,1}\times L^1\). By Choksi and Fonseca (1997, Theorem 2.16) (recovery sequence for the case of \(I_1\) therein), for each n there exists a sequence \(\{u_{n,k}\}_k\subset SBV(\Omega ;{\mathbb {R}}^{d})\) such that as \(k\rightarrow \infty \),
and
In addition, we may assume that up to a (not relabeled) subsequence
Since E is coercive as a consequence of (W:1) and (\(\psi \):1), the latter implies that \(\{Du_{n,k}\}\) is equi-bounded in \({\mathcal {M}}(\Omega ;{\mathbb {R}}^{d\times N})\). We can therefore find a diagonal subsequence \(u_{n,k(n)}\) with \(k(n)\rightarrow \infty \) fast enough, such that
Since the sequence \(\{u_{n,k(n)}\}_n\) is admissible for the lower bound in Choksi and Fonseca (1997, Theorem 2.16) (for the functional \(I_1\)), we conclude that
i.e., the functional with integrand H is weakly lower semi-continuous in \(W^{1,1}\times L^1\). Since H also has at most linear growth by Proposition 4.2 and is non-negative as a consequence of (W:1), Carita et al. (2010, Theorem 1.1) (see also Carita et al. 2011) implies that H is quasiconvex-convex.
It remains to show that \((g,G)\mapsto \int _\Omega \mathrm dH(Dg,G)\) is sequentially lower semi-continuous with respect to the convergence (1.2). This follows from Arroyo-Rabasa et al. (2020, Theorem 1.7). Here, notice that with \(\mathcal {A}:={\text {diag}}({\text {Curl}},0)\), i.e., with \(\mathcal {A}(\nabla g^\top ,G)^\top :=( ({\text {Curl}} \nabla g)^\top ,0)^\top \), \(\mathcal {A}(\nabla g_k^\top ,G_k)^\top =0\) in the sense of distributions, and the \({\mathcal {A}}\)-quasiconvexity of H for this special case is equivalent to quasiconvexity-convexity of H. The latter can equivalently be tested with periodic functions on the simply connected U where all curl-free fields are gradients.
\(\square \)
Proof of Theorem 2.4, lower bound
Let H be the integrand in (2.11a). Moreover, let \((g,G) \in mSD\) and \(\{v_n\} \subset SBV(\Omega ;{\mathbb {R}}^d)\) be such that \(v_n {\mathop {\rightharpoonup }\limits ^{*}}(g,G)\) in the sense of (1.2). Observing that for each n, \(v_n\) can be interpreted as a constant sequence converging to itself in mSD, by Proposition 3.1 and Choksi and Fonseca (1997, Theorem 2.16) (its lower bound for the case of \(I_1\) therein), we have that
In addition, \((u,G)\mapsto \int _\Omega \mathrm dH(Du,G)\) is weak\(^*\)-sequentially lower semi-continuous in mSD by Proposition 4.3. In particular,
Taking Proposition 4.1 into account, the lower bound inequality now follows from (4.12) and (4.11):
\(\square \)
5 Relaxation Under Trace Constraints
Let \(\Omega '\) be a bounded Lipschitz domain such that \(\Omega \subset \Omega '\), and let
Let \(u_0 \in W^{1,1}(\Omega ';{\mathbb {R}}^d)\) and let \((g,G) \in mSD\). The relaxed functional subject to the Dirichlet condition \(u=u_0\) on \(\Gamma \) is defined as
where, for every open subset A of \(\Omega ' \), \(E(\cdot ; A)\) is the functional given by (2.1), with W and \(\psi \) satisfying (W:1)–(W:3) and (\(\psi \):1)–(\(\psi \):3),
We have the following integral representation for \(I_\Gamma \).
Theorem 5.1
Let \(\Omega \subset {\mathbb {R}}^{N}\) be a bounded Lipschitz domain and assume that (W:1)–(W:3) and (\(\psi \):1)–(\(\psi \):3) hold. Moreover, let \(\Omega '\supset \Omega \) be a bounded domain and \(u_0 \in W^{1,1}(\Omega ';{\mathbb {R}}^d)\). In addition, for \(\Gamma :=\Omega ' \cap \partial \Omega \) assume that \({\mathcal {H}}^{N-1}({\overline{\Gamma }}\setminus \Gamma )=0\). Then,
where
and H is the function defined in (2.11a).
The proof will be given in two parts. We immediately start with the lower bound, and the proof of the upper bound will follow after an auxiliary result needed there.
Proof of Theorem 5.1, the lower bound
We have to show that \(I_\Gamma (g,G;\Omega )\geqslant J_\Gamma (g,G;\Omega )\). For every \(k \in {\mathbb {N}}\), let \(\Omega _k:=\{x \in {\mathbb {R}}^N: \textrm{dist}(x, {{\overline{\Omega }}}) \leqslant \tfrac{1}{k}\}\), and consider
Thus \(\Gamma = \Omega '_k\cap \partial \Omega \), for every k and \(\Omega '_k\) shrinks to \(\Omega \cup \Gamma \) as \(k \rightarrow \infty \). As for \(I_\Gamma (g,G;\Omega )\), define for every \(k \in {\mathbb {N}}\)
Thus
On the other hand,
where
In particular,
Clearly, for every \(\Omega '_k\),
where \(I({{\hat{g}}},{{\hat{G}}};\Omega '_k)\) is the functional introduced in (2.2), and in the equality we have exploited Theorem 2.4 and Remark 2.6. The proof is concluded by letting \(k\rightarrow \infty \), in the above inequality, taking into account (5.2) and the fact that \(\lim _{k \rightarrow \infty } \int _{\Omega '_k{\setminus } \Omega } W(\nabla u_0) \, \mathrm dx =0\). \(\square \)
Below, we will reduce the construction of the recovery sequence needed for the upper bound to that of Theorem 2.4. This relies on the following lemma.
Lemma 5.2
(domain shrinking (Krömer and Valdman 2023, Lemma 3.1)) Let \(\Omega \subset {\mathbb {R}}^{N}\) be a bounded Lipschitz domain. Then there exists an open neighborhood \(U\supset {\overline{\Omega }}\) and a sequence of maps \(\{\Psi _j\}\subset C^\infty (\overline{U};{\mathbb {R}}^{N})\) such that for every \(j\in {\mathbb {N}}\),
In addition, \(\Psi _j\rightarrow {\text {id}}\) in \(C^m(\overline{U};{\mathbb {R}}^{N})\) as \(j\rightarrow \infty \), for all \(m\in {\mathbb {N}}\cup \{0\}\).
Proof
This is the case \(\Gamma =\emptyset \) in Krömer and Valdman (2023). The statement there has \(\Psi _j\) only defined on \({{\overline{\Omega }}}\), but the proof also provides the extension to \(\overline{U}\) (as long as \(\overline{U}\) is still fully covered by the union of \(\Omega \) and the open cuboids covering \(\partial \Omega \) in which \(\partial \Omega \) can be seen as a Lipschitz graph). \(\square \)
Remark 5.3
If \(\Omega \) is strictly star-shaped with respect to some \(x_0\in \Omega \), Lemma 5.2 is easy to show with \(\Psi _j(x):=x_0+\frac{j}{j+1}(x-x_0)\). The proof of Krömer and Valdman (2023, Lemma 3.1) for the general case glues local constructions near the boundary using a decomposition of unity, exploiting that everything happens uniformly \(C^1\)-close to the identity to preserve invertibility.
Proof of Theorem 5.1, the upper bound
We have to show that \(I_\Gamma (g,G;\Omega )\leqslant J_\Gamma (g,G;\Omega )\), for each \((g,G)\in BV(\Omega ;{\mathbb {R}}^{d})\times {\mathcal {M}}(\Omega \cup \Gamma ;{\mathbb {R}}^{d\times N})\). For this, it suffices to find a recovery sequence, i.e., a sequence \((u_n)\) admissible in the definition of \(I_\Gamma (g,G;\Omega )\) such that \(E(u_n;\Omega )\rightarrow J_\Gamma (g,G;\Omega )\). In particular, we must have \(u_n=u_0\) on \(\Gamma \) in the sense of traces in BV. The proof is divided into three steps. In the first two steps, we define a suitable approximating sequence of limit states \(({\hat{g}}_j,{\hat{G}}_j)\) such that \(({\hat{g}}_j,{\hat{G}}_j){\mathop {\rightharpoonup }\limits ^{*}}(g,G)\) in mSD, \(J_\Gamma ({\hat{g}}_j,{\hat{G}}_j;\Omega )\rightarrow J_\Gamma (g,G;\Omega )\) and \({\hat{g}}_j=u_0\) on \(\Gamma \). In the final step, we will then use the upper bound in Theorem 2.4, which for each j gives a “free” recovery sequence \(\{u_{j,n}\}_n\subset BV\) for \(I({\hat{g}}_j,{\hat{G}}_j;\Omega )\) that again can be modified to match the trace of its weak\(^*\) limit \({\hat{g}}_j\) on \(\Gamma \). The assertion then follows by a diagonal subsequence argument.
Step 1: Approximating limit states \((g_j,G_j)\) with values “close” to \(u_0\) near \(\Gamma \). Choose a bounded neighborhood U of \(\Omega \) according to Lemma 5.2 and an extension
With this, we define
In particular, with the outer normal \(\nu _\Gamma \) to \(\partial \Omega \) on \(\Gamma \),
, \(g_0|_\Omega =g\), and \(g_0\) jumps at \(\Gamma \) from (the trace of) g to \(u_0\) and at \(U\cap (\partial \Omega '\setminus \Omega )\) from \({\tilde{g}}\) to \(u_0\).
With the maps \(\Psi _j\) from Lemma 5.2, we define
Here, in the definition of \(G_j\), \(G_0\circ \Phi _j\) is the measure defined as \((G_0\circ \Phi _j)(A):=G_0(\Phi _j(A))\) for all Borel sets \(A\subset \Psi _j(U)\), and \(\nabla \Phi _j\in C^0(\overline{U};{\mathbb {R}}^{N\times N})\) is interpreted as a continuous density function attached to it by matrix multiplication from the right. Altogether, \(G_j\) is the measure satisfying \(\mathrm dG_j(z)=\mathrm d(G_0\circ \Phi _j)(z)\nabla \Phi _j(z)\), similar to \(Dg_j\) which satisfies \(\mathrm dDg_j(z)=(Dg\circ \Phi _j)(z) \nabla \Phi _j(z)\) by the chain rule. Also notice that as a consequence of Lemma 5.2 (where we only need the case \(m=1\)), for all j big enough, \(\nabla \Psi _j(x)\) is an invertible matrix for all \(x\in U\), \({\overline{\Omega }}\subset \Psi _j(U)\) and \(\Phi _j(\partial \Omega )\cap {{\overline{\Omega }}}=\emptyset \). Passing to a subsequence (not relabeled), we thus may assume that
We claim that the sequence \(\{(g_j,G_j)\}_j\) has the following properties:
where \(T_\Omega :BV(\Omega ;{\mathbb {R}}^{d})\rightarrow L^1(\partial \Omega ;{\mathbb {R}}^{d})\) denotes the trace operator,
and
The second part of (5.6) follows from the definition of \(G_j\) because \(\Phi _j(\partial \Omega )\subset U{\setminus } {\overline{\Omega }}\) and \(|G_0|(U{\setminus } {\overline{\Omega }})=0\). As to the first part of (5.6), first notice that since \(u_0\in W^{1,1}(U;{\mathbb {R}}^{d})\), we do not have to distinguish between the inner and outer traces \(T_\Omega u_0\) and \(T_{U{\setminus }\Omega } u_0\) of \(u_0\) on \(\partial \Omega \). Moreover,
and \(u_0\circ \Phi _j\rightarrow u_0\) in \(W^{1,1}(\Omega ;{\mathbb {R}}^{d})\), so that the asserted convergence of traces follows from the continuity of the trace operator in \(W^{1,1}\) once we see that \((g_0-u_0)\circ \Phi _j=0\) in some neighborhood of \(\Gamma \) (which may depend on j). The latter is trivial by definition of \(g_0\) if
here, we already have that \(\Phi _j({\overline{\Gamma }})\cap {\overline{\Omega }}=\emptyset \). We can therefore assume (5.9) without loss of generality: otherwise, if \(\Phi _j({\overline{\Gamma }})\not \subset \Omega '\), we can define \(r(j):=\frac{1}{2}{\text {Dist}}(\Phi _j({\overline{\Gamma }}),\Omega )>0\) and take
instead of \(\Omega '\). Here, recall that \(\Omega '\) is just an auxiliary object to define \(\Gamma \) (and \(g_0\) above, outside of \({{\overline{\Omega }}}\)), and by construction, \({\tilde{\Omega }}'\) still has all the properties we required for \(\Omega '\): \({\tilde{\Omega }}'\supset \Omega \) is a bounded domain and \({\tilde{\Omega }}'\cap \partial \Omega = \Gamma =\Omega '\cap \partial \Omega \).
For the proof of (5.7) and (5.8), fix \(\varphi \in C({{\overline{\Omega }}})\), continuously extended to \(\varphi \in C({\mathbb {R}}^{N})\). By the definition of \((g_j,G_j)\) and the change of variables \(x=\Phi _j(z)\), we get that for every Borel set \(V\subset U\),
As to the second term \(T_j(\varphi ;V)\) (integrals on \(\Phi _j(\Omega )\setminus V\)), we exploit that \((\nabla \Psi _j)^{-1}\) is uniformly bounded and H has at most linear growth. Hence, there is a constant \(C>0\) such that with \(C_\varphi :=C \left\| \varphi \right\| _{L^\infty (U)}\),
by dominated convergence and the fact that \(\Phi _j\rightarrow {\text {id}}\) in \(C^1\). Here, we also used that
by definition of \(g_0\), \(G_0\) and our assumption that \({\mathcal {H}}^{N-1}({\overline{\Gamma }}\setminus \Gamma )=0\).
For the term \(S_j(\varphi ;V)\) (integrals on \(\Phi _j(\Omega )\cap V\)) on the right-hand side of (5.10), we again use that \(\Phi _j\rightarrow {\text {id}}\) in \(C^1\); in particular, \((\nabla \Psi _j)^{-1}\rightarrow I\) (identity matrix) uniformly. In addition, \(H\geqslant 0\) is Lipschitz and \(\varphi \) is uniformly continuous. Consequently, for all \(\varphi \geqslant 0\),
Here, to handle the limit in the domain of integration \(\Phi _j(\Omega )\cap V\), for the lower bound we used monotonicity and the fact that \({{\overline{\Omega }}}\subset \Phi _j(\Omega )\) for all j by (5.5), while for the upper bound we used that \(\Phi _j(\Omega )\searrow {\overline{\Omega }}\) and dominated convergence.
By splitting a general \(\varphi \) into positive and negative parts, (5.12) immediately implies that
for all \(\varphi \in C(\overline{U})\). Combining (5.10), (5.11) and (5.13) for the case \(V=\Omega \cup \Gamma \), we infer that
In particular, (5.14) yields (5.8) when we choose \(\varphi \equiv 1\).
In addition, we can analogously obtain (5.14) for other functions instead H (globally Lipschitz with a uniform strong recession function in the sense of (4.1); if needed, H can be temporarily split into a positive and a negative part for the proof of (5.13), just like \(\varphi \)). With the choices
for \(i=1,\ldots ,d\) and \(j=1,\ldots ,N\), (5.14) implies the second part of (5.7), in particular that 
Finally, it is not hard to see that \(g_j\rightarrow g\) in \(L^1(\Omega ;{\mathbb {R}}^{d})\). We conclude that \(g_j{\mathop {\rightharpoonup }\limits ^{*}}g\) in \(BV(\Omega ;{\mathbb {R}}^{d})\), which completes the proof of (5.7).
Step 2: Approximating limit states \(({\hat{g}}_j,{\hat{G}}_j)\) with \({\hat{g}}_j=u_0\) on \(\Gamma \).
The functions \(g_j\) defined in the previous step do not yet satisfy \(g_j=u_0\) on \(\Gamma \), although their traces converge to \(u_0\) by (5.6). We can correct this using the trace extension theorem: Choose \(\{v_j\}\subset W^{1,1}(\Omega ;{\mathbb {R}}^{d})\) such that
By (5.6), we infer that \(\left\| v_j \right\| _{W^{1,1}(\Omega ;{\mathbb {R}}^{d})}\rightarrow 0\). Consequently, for
instead of \((g_j,G_j)\) we still have (5.6), (5.7) and (5.8), and in addition, \({\hat{g}}_j=u_0\)on \(\Gamma \). Namely, defining
so that \(Dg_0= Dg + \Theta \) on \(\Omega \cup \Gamma \), we have that
and
Step 3: Recovery by diagonalizing free recovery sequences for \(({\hat{g}}_j,{\hat{G}}_j)\)
We first observe that I in (2.2) admits the following equivalent representation
for every \((g,G) \in mSD\).
Clearly \(I(g,G;\Omega )\leqslant I_g(g,G;\Omega )\). The opposite one can be obtained following an argument of Bouchitté et al. (2002). The details are provided below for the reader’s convenience.
For any \( SBV(\Omega ;{\mathbb {R}}^d) \ni u_n {\mathop {\rightharpoonup }\limits ^{*}}(g, G)\) in the sense of (2.7), almost optimal for \(I(g,G;\Omega )\), i.e., for every \(\varepsilon >0\),
Without loss of generality, assume that the above lower limit is indeed a limit and consider the sequence of measures \(\nu _n:={\mathcal {L}}^N+ |Du_n|+ |D g|\), which converges weakly* to some Radon measure \(\nu \).
Denoting, for every \(t>0\), \(\Omega _t :=\{x\in \Omega |\ \textrm{dist} (x,\partial \Omega ) > t\}\), we fix some \(\eta >0\) and for every \(0<\delta <\eta \) we define the subsets \(L_\delta :=\Omega _{\eta -2\delta }{\setminus }\overline{ \Omega _{\eta +\delta }}\). Consider a smooth cut-off function \(\varphi _\delta \in {\mathcal {C}}_0^\infty (\Omega _{\eta -\delta };[0,1])\) such that \(\varphi _\delta =1\) on \(\Omega _{\eta }\). As the thickness of the strip \(L_\delta \) is of order \(\delta \), we have an upper bound of the form \(\Vert \nabla \varphi _\delta \Vert _{L^\infty (\Omega _{\eta -\delta })}\le C/\delta \). Define
Clearly this sequence converges to g in \(L^1 (\Omega ;{\mathbb {R}}^d)\) and satisfies \(T_\Omega w_n^\delta =T_\Omega g\) on \(\partial \Omega \).
Moreover
as \(n \rightarrow \infty \) and then \(\delta \rightarrow 0\). Indeed
Concerning the energies, we have
where \(C_{W,\psi }\) is any bigger constant which bounds from above the constants appearing in(W:1), (\(\psi \):1) and in \(L^\infty \) bound of \(\nabla \psi _\delta \) on \(L_\delta \). Taking the limit as \(n\rightarrow \infty \) we have
Letting \(\delta \rightarrow 0\) we obtain
Choose a subsequence \(\{\eta _n\}\) such that \(\eta _n\rightarrow 0^+\) and \( \nu (\partial A_{\eta _n})=0\). By letting first \(n\rightarrow \infty \) and then \( \varepsilon \rightarrow 0^+\) we conclude that \(I_g(g,G;\Omega )\leqslant I(g,G;\Omega )\).
Then, for any \(({\hat{g}}_j,{\hat{G}}_j) {\mathop {\rightharpoonup }\limits ^{*}}(g,G)\) as in Step 2, satisfying (5.16), (5.17), and (5.18), we can apply Theorem 2.4 and find a recovery sequence for \(I_{{\hat{g}}_j}({\hat{g}}_j,{\hat{G}}_j;\Omega )=I({\hat{g}}_j,{\hat{G}}_j;\Omega )\) for each j, i.e., \(\{u_n^j\}_n\subset SBV(\Omega ;{\mathbb {R}}^d)\) such that \( u_n^j {\mathop {\rightharpoonup }\limits ^{*}}({{\hat{g}}}_j, {{\hat{G}}}_j)\) in the sense of (2.7), \(T_\Omega u_n^j= T_\Omega {{\hat{g}}}_j\) on \(\partial \Omega \), in particular \(T_\Omega u_n^j= u_0\) on \(\Gamma \), and
Since \(|{{\hat{G}}}_j|(\partial \Omega )=0\), we also have that \(\nabla u_n^j {\mathop {\rightharpoonup }\limits ^{*}}{{\hat{G}}}_j \) in \({\mathcal {M}}(\Omega \cup \Gamma ;\mathbb R^{d \times N})\).
A standard diagonalization argument, exploiting the coercivity of E given by (W:1) and (\(\psi \):1) to obtain bounds uniform in n and j, now concludes the proof. \(\square \)
6 Further Properties and Examples
As shown below, we also have an alternative way of interpreting I, as a more classic relaxation problem of a functional on \(SBV\times L^1\) in \(BV\times {\mathcal {M}}\).
Theorem 6.1
Assume (W:1)–(W:3) and (\(\psi \):1)–(\(\psi \):3). For \((g,G)\in SBV(\Omega ;{\mathbb {R}}^{d})\times L^1(\Omega ;{\mathbb {R}}^{d\times N})\) and \(R>0\), we define
and its relaxation
for \((g,G)\in mSD\). Then there exists \(R_0=R_0(N,W,\psi )>0\) such that
where \(I(\cdot ,\cdot ;\Omega )\) is the relaxation of \(E(\cdot ;\Omega )\) defined in (2.2).
Remark 6.2
Theorem 6.1 in principle opens another route to proving Theorem 2.4, our representation theorem for I, via a relaxation theorem characterizing \({\hat{I}}_R\). However, the closest available results in this direction seem to be Arroyo-Rabasa et al. (2020) and Baía et al. (2013) (for the case \(\mathcal {A}={\text {diag}}({\text {Curl}},0)\), cf. the proof of Proposition 4.3) and Barroso et al. (1996). However, the former does not allow us to choose \(\psi \) freely, and the latter does not allow us to include G.
Proof of Theorem 6.1
We first observe that, for every \(R >0\), \({{\hat{E}}}_R(u, \nabla u, \Omega )= E(u;\Omega )\) for every \(u \in SBV(\Omega ;{\mathbb {R}}^d)\). Let \((g, G) \in mSD\) and let \(SBV(\Omega ;{\mathbb {R}}^d) \ni g_n {\mathop {\rightharpoonup }\limits ^{*}}(g, G)\) according to (1.2). Since \(\{(g_n, \nabla g_n)\}\) is an admissible sequence for \({{\hat{E}}}_R(g, G;\Omega )\),
Hence, passing to the infimum over all the admissible sequences \(\{g_n\}\), we have
To prove the opposite inequality for \(R \geqslant R_0\) with a suitable \(R_0\) to be chosen later, take \(\{(g_n, G_n)\}\) admissible for \({{\hat{I}}}_R(g,G;\Omega )\), so that \(g_n {\mathop {\rightharpoonup }\limits ^{*}}g\) in BV, \(G_n {\mathop {\rightharpoonup }\limits ^{*}}G\) in \({\mathcal {M}}\). We choose a sequence \(\{v_n\}\) given by S̆ilhavý (2015, Theorem 1.1) such that
and
In particular, the sequence \(u_n :=g_n+ v_n\) is admissible for \(I(g, G;\Omega )\).
Taking into account that
also using (W:1), (\(\psi \):1) and (2.14) we obtain that
as long as \(R\geqslant R_0:=L+ C_\psi C(N)\). Here, L, \(C_\psi \), and C(N) denote the Lipschitz constant of W, the Lipschitz and growth constant of \(\psi \) in (2.14) and (\(\psi \):1), and the constant appearing in (6.2), respectively. Passing to the limit as \(n\rightarrow \infty \), we conclude that
for all \(R\geqslant R_0\). As this holds for all sequences \(\{(g_n,G_n)\}\) that are admissible for \({{\hat{I}}}_R(g,G;\Omega )\), the thesis follows. \(\square \)
In view of Theorem 6.1, it is a natural question to what degree our relaxed functional I is influenced by its origin from E, defined on structured deformations. The following example shows that this special background is still present in the relaxed I at least in the sense that not all quasiconvex-convex densities H (that could be obtained by general relaxation in \(BV\times {\mathcal {M}}\)) can be obtained in I.
Proposition 6.3
For all W and \(\psi \) satisfying the assumptions of Theorem 2.4, there exists \(B_0\in {\mathbb {R}}^{d\times N}\) and \(\xi \in {\mathbb {R}}^{d}\), \(\nu \in {\mathbb {R}}^{N}\) with \(\left| \xi \right| =\left| \nu \right| =1\) so that for the function H defined in (2.11a),
In particular, for any possible choice of W and \(\psi \),
because the function \((0,+\infty )\ni t\mapsto H_0(B_0+t\xi \otimes \nu ,B_0)\) is not affine.
Proof
To see “\(\leqslant \)” in (6.3), it suffices to choose a suitable admissible sequence in (3.15a), the sequential characterization of H: On Q, we have \(\nabla u_n{\mathop {\rightharpoonup }\limits ^{*}}B_0\) in \({\mathcal {M}}\) and \(u_n{\mathop {\rightharpoonup }\limits ^{*}}A_0 x\) for \(A_0:=B_0+t\xi \otimes \nu \) for
where \([s]:=\min \{z\in {\mathbb {Z}} :|z-s|={\text {dist}}(s,{\mathbb {Z}} )\}\) denotes rounding of s to the closest integer. An upper bound for \(H(A_0,B_0)\) is therefore given by \(\lim _n E(u_n;Q)=W(B_0)+\psi (t\xi ,\nu )\) (by 1-homogeneity of \(\psi \)), for any possible choice of \(B_0\), \(\xi \) and \(\nu \).
To obtain “\(\geqslant \)” in (6.3), we use a particular choice: Since both W and \(\psi \) are continuous and W is coercive, there always exists global minima \(B_0\) of W on \({\mathbb {R}}^{d\times N}\) and \((\xi ,\nu )\) of \(\psi \) on the compact set \({\mathbb {S}}^{d-1}\times {\mathbb {S}}^{N-1}:=\big \{(\xi ,\nu )\in {\mathbb {R}}^{d}\times {\mathbb {R}}^{N}\mid \left| \xi \right| =\left| \nu \right| =1\big \}\). As a consequence,
Here, \(W^{**}\) denotes the convex hull of W. For any u admissible in the definition (2.11a) of \(H(A_0,B_0)\) with \(A_0:=B_0+t\xi \otimes \nu \), we now have that
by Jensen’s inequality, since 
for all admissible u in (2.11a). Moreover, since \(u\in SBV\) and \(u=A_0x\) on \(\partial Q\), we have that
Multiplied with the fixed unit vector \(\nu \) from the right, this reduces to
By the positive 1-homogeneity of \(\psi \), the minimality property of \((\xi ,\nu )\) in (6.4) and another application of Jensen’s inequality with (6.6) to the convex function \(|\cdot |\), we infer that
for all \(t>0\). Combining (6.5) and (6.7), we conclude that \(E(u;Q)\geqslant W(B_0)+\psi (t\xi ,\nu )\) for all u admissible in (2.11a) with \((A,B)=(A_0,B_0)\). This implies the asserted lower bound for \(H(A_0,B_0)\). \(\square \)
Data Availability Statement
Data sharing is not applicable to this article as no new data were created or analyzed in this study.
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Acknowledgements
The authors are grateful to David R. Owen for inspiring discussions on this topic. Part of this work has been developed during the visits at UTIA in Prague by MM and EZ who are very grateful for the hospitality and the support. SK acknowledges the support of INdAM-GNAMPA through the project Programma Professori Visitatori 2023, CUP E53C22001930001, and of Dipartimento di Scienze di Base ed Applicate per l’Ingegneria di Sapienza, Università di Roma. MK, SK, and EZ are grateful to Dipartimento di Matematica at Politecnico di Torino, where also part of this work has been undertaken. SK and MK were supported by the GAČR project 21-06569K. The research of EZ is part of the projects GNAMPA-INdAM 2023 Prospettive nelle scienze dei materiali: modelli variazionali, analisi asintotica e omogeneizzazione CUP E53C22001930001, and PRIN Mathematical Modeling of Heterogeneous Systems CUP 853D23009360006. MM and EZ are members of GNAMPA (INdAM). MM acknowledges support from the PRIN2020 grant Mathematics for Industry 4.0 (2020F3NCPX). This study was carried out within the Geometric-Analytic Methods for PDEs and Applications project (2022SLTHCE), funded by European Union—Next Generation EU within the PRIN 2022 program (D.D. 104 - 02/02/2022 Ministero dell’Universitàe della Ricerca). This manuscript reflects only the authors’ views and opinions and the Ministry cannot be considered responsible for them.
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Krömer, S., Kružík, M., Morandotti, M. et al. Measure-Valued Structured Deformations. J Nonlinear Sci 34, 100 (2024). https://doi.org/10.1007/s00332-024-10076-w
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DOI: https://doi.org/10.1007/s00332-024-10076-w
Keywords
- Structured deformations
- Energy minimization
- Relaxation
- Functionals depending on measures
- Integral representation