Abstract
We present a way to study a wide class of optimal design problems with a perimeter penalization. More precisely, we address existence and regularity properties of saddle points of energies of the form
where \(\Omega \) is a bounded Lipschitz domain, \(A\subset \mathbb {R}^N\) is a Borel set, \(u:\Omega \subset \mathbb {R}^N \rightarrow \mathbb {R}^d\), \(\mathscr {A}\) is an operator of gradient form, and \(\sigma _1, \sigma _2\) are two not necessarily well-ordered symmetric tensors. The class of operators of gradient form includes scalar- and vector-valued gradients, symmetrized gradients, and higher order gradients. Therefore, our results may be applied to a wide range of problems in elasticity, conductivity or plasticity models. In this context and under mild assumptions on f, we show for a solution (w, A), that the topological boundary of \(A \cap \Omega \) is locally a \(\mathrm {C}^1\)-hypersurface up to a closed set of zero \(\mathscr {H}^{N-1}\)-measure.
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1 Introduction
The problem of finding optimal designs involving two materials goes back to the work of Hashin and Shtrikman. In [1], the authors made the first successful attempt to derive the optimal bounds of a composite material. It was later on, in the series of papers [2,3,4], that Kohn and Strang described the connection between composite materials, the method of relaxation, and the homogenization theory developed by Murat and Tartar [5, 6]. In the context of homogenization, better designs tend to develop finer and finer geometries; a process which results in the creation of non-classical designs. One way to avoid the mathematical abstract of infinitely fine mixtures is to add a cost on the interfacial energy. In this regard, there is a large amount of optimal design problems that involve an interfacial energy and a Dirichlet energy. The study of regularity properties in this setting has been mostly devoted to problems where the Dirichlet energy is related to a scalar elliptic equation; see [7,8,9,10,11,12], where partial \(\mathrm {C}^1\)-regularity on the interface is shown for an optimization problem oriented to find dielectric materials of maximal conductivity. We shall study regularity properties of similar problems in a rather general framework. Our results extend the aforementioned results to linear elasticity and linear plate theory models.
Before turning to a precise mathematical statement of the problem let us first present the model in linear plate theory that motivated our results. Let \(\Omega = \omega \times [-h,h]\) be the reference configuration of a plate of thickness 2h and cross section \(\omega \subset \mathbb {R}^2\). The linear equations governing a clamped plate \(\Omega \) as h tends to zero for the Kirchhoff model are
where \(u : \omega \rightarrow \mathbb {R}\) represents the displacement of the plate with respect to a vertical load \(f \in \mathrm {L}^\infty (\omega )\), and the design of the plate is described by a symmetric positive definite fourth-order tensor \(\sigma \) (up to a cubic dependence on the constant h). Here, we denote the second gradient by
Consider the physical problem of a thin plate \(\Omega \) made-up of two elastic materials. More precisely, for a given set \(A \subset \omega \subset \mathbb {R}^2\) we define the symmetric positive tensor
where \(\sigma _1, \sigma _2 \in \text { Sym }(\mathbb {R}^{2 \times 2},\mathbb {R}^{2 \times 2})\). In this way, to each Borel subset \(A \subset \omega \), there corresponds a displacement \(u_A :\omega \rightarrow \mathbb {R}\) solving Eq. (1) with \(\sigma = \sigma _A\). One measure of the rigidity of the plate is the so-called compliance, i.e., the work done by the loading. The smaller the compliance, the stiffer the plate is. A reasonable optimal design model consists in finding the most rigid design A under the aforementioned costs. One seeks to minimize an energy of the form
Optimality conditions for a stiffest plate can be derived by taking local variations on the design. For such analysis to be meaningful, one has to ensure first that the variational equations of optimality have a suitable meaning in the interface. Hence, it is natural to ask for the maximal possible regularity of \(\partial A\) and \(\nabla ^2 u_A\).
We will introduce a more general setting where one can replace the second gradient \(\nabla ^2\) by an operator \(\mathscr {A}\) of gradient type (see Definition 2.1 and the subsequent examples in the next section for a precise description of this class).
1.1 Statement of the problem
Let \(N \ge 2\), and let d, k be positive integers. We shall work in \(\Omega \subset \mathbb {R}^N\); a nonempty, open, and bounded Lipschitz domain. We also fix a function \(f \in \mathrm {L}^\infty (\Omega ;\mathbb {R}^d)\) and let \(\sigma _1\) and \(\sigma _2\) be two positive definite tensors in \(\mathrm{Sym}(\mathbb {R}^{dN^k} \otimes \mathbb {R}^{dN^k})\) satisfying a strong pointwise Gårding inequality: there exists a positive constant M such that
For a fixed Borel set \(A \subset \mathbb {R}^N\), define the two-point valued tensor
We consider a k’th-order homogeneous linear differential operator \(\mathscr {A}: \mathrm {L}^2\left( \Omega ;\mathbb {R}^d\right) \rightarrow \mathrm {W}^{-k,2}(\Omega ;\mathbb {R}^{dN^k})\) of gradient form (see Definition 2.1 in Sect. 2). As a consequence of the definition of operators of gradient form, the following equation
has a unique solution (cf. Theorem 1.1). We will refer to Eq. (4) as the state constraint and we will denote by \(w_A\) its unique solution.
It is a physically relevant question to ask which designs have the least dissipated energy. To this end, consider the energy defined asFootnote 1
We will be interested in the optimal design problem with Dirichlet boundary conditions on sets:
where \(A_0 \subset \mathbb {R}^N\) is a set of locally finite perimeter.Footnote 2
Most attention has been drawn to the case where designs are mixtures of two well-ordered materials. The presentation given here places no comparability hypotheses on \(\sigma _1\) and \(\sigma _2\). Instead, we introduce a weaker condition on the decay of generalized minimizers of a double-well problem. Our technique also holds under various constraints other than Dirichlet boundary conditions; in particular, any additional cost that scales as \(O(r^{N -1 + \varepsilon })\). For example, a constraint on the volume occupied by a particular material (cf. [8, 12, 13]). Lastly, we remark that our technique is robust enough to treat models involving the maximization of dissipated energy.
1.2 Main results and background of the problem
Existence of a minimizer of (5) can be established by standard methods. We are interested in proving that a solution pair \((w_A,A)\) enjoys better regularity properties than the ones needed for existence. The notion of regularity for a set A will be understood as the local regularity of \(\partial A\) seen as a submanifold of \(\mathbb {R}^N\), whereas the notion of regularity for \(w_A\) will refer to its differentiability and integrability properties.
It can be seen from the energy, that the deviation from being a perimeter minimizer for a solution A of problem (5) is bounded by the dissipated energy. Therefore, one may not expect better regularity properties for A than the ones for perimeter minimizers; and thus, one may only expect regularity up to singular set (we refer the reader to [14, 15] for classic results, see also [12] for a partial regularity result in a similar setting to ours).
Since a constrained problem may be difficult to treat, we will instead consider an equivalent variational unconstrained problem by introducing a multiplier as follows. Consider the saddle point problem
where
Our first result shows the equivalence between problem (P) and the minimization problem (5) under the state constraint (4):
Theorem 1.1
(Existence) There exists a solution (w, A) of problem (P). Furthermore, there is a one to one correspondence
between solutions of the problem (P) and solutions of the minimization problem (5) under the state constraint (4).
We now turn to the question of regularity. Let us depict an outline of the key steps and results obtained in this regard. The Morrey space \(\mathrm {L}^{p,\lambda }(\Omega ;\mathbb {R}^d)\) is the subspace of \(\mathrm {L}^p(\Omega ;\mathbb {R}^d)\) for which the semi-norm
is finite.
The first step in proving regularity for solutions (w, A) consists in proving a critical \(\mathrm {L}^{2,\,N-1}\) local estimate for \(\mathscr {A}w\). This estimate arises naturally since we expect a kind of balance between \(\int _{B_r(x)} \sigma _A \mathscr {A}w \cdot \mathscr {A}w \,\mathrm {d}y\) and the perimeter part \(\mathrm {Per}(A;B_r(x))\) that scales as \(r^{ N-1}\) in balls of radius r.
To do so, let us recall a related relaxed problem. As part of the assumptions on \(\mathscr {A}\) there must exist an m’th-order diferential operator \(\mathscr {B}: \mathrm {L}^2(\Omega ;Z) \rightarrow \mathrm {W}^{-m,2}(\Omega ;\mathbb {R}^n)\) with constant rank and \(\text { Ker }(\mathscr {B}) = \mathscr {A}[\mathrm {W}^\mathscr {A}(\Omega )]\).Footnote 3 It has been shown by Fonseca and Müller [16], that a necessary and sufficient condition for the lower semi-continuity of integral energies with superlinear growth under a constant rank differential constraint \(\mathscr {B}v = 0\) is the \(\mathscr {B}\)-quasiconvexity of the integrand. In this context, the \(\mathscr {B}\)-free quasiconvex envelope of the double-well \(W(P):=\min \{\sigma _1 \, P \cdot P,\,\sigma _2\, P \cdot P\}\), at a point \(P\in Z \subset \mathbb {R}^{dN^k}\), is given by
The idea is to get an \(\mathrm {L}^{2,\,N-1}\) estimate by transferring the regularizing effects from generalized minimizers of the energy \(u \mapsto \int _{B_1} W(\mathscr {A}u)\) onto our original problem. In order to achieve this, we use a \(\Gamma \)-convergence argument with respect to a perturbation in the interfacial energy from which the next result follows:
Theorem 1.2
(Upper bound) Let (w, A) be a variational solution of problem (P). Assume that, for some \(\delta \in [0,1)\) and some positive constant c, the higher integrability condition
holds for local minimizers of the energy \(u \mapsto \int _{B_1} Q_{\mathscr {B}} W(\mathscr {A}u)\), where \(u \in \mathrm {W}^\mathscr {A}(\Omega )\). Then, for every compactly contained set \(K \subset \subset \Omega \), there exists a positive constant \( \Lambda _K\) such that
Remark 1.3
(Well-ordering assumption) If \(\sigma _1, \sigma _2\) are well-ordered, say \(\sigma _2 - \sigma _1\) is positive definite, then \(Q_{\mathscr {B}}W\) is precisely the quadratic form \(\sigma _2\, P \cdot P\). Due to standard elliptic regularity results (cf. Lemma 2.6), estimate (Reg) holds for \(\delta =0\); therefore, assuming that the materials are well-ordered is a sufficient condition for the higher integrability assumption (Reg) to hold.
Remark 1.4
(Non-comparable materials) In dimensions \(N = 2, 3\) and restricted to the setting \(\mathscr {A}= \nabla \), \(d = 1\), condition (Reg) is strictly weaker than assuming the materials to be well-ordered. Indeed, one can argue by a Moser type iteration as in [17] to lift the regularity of minimizers. For higher-order gradients or in the case of systems it is not clear to us whether assumption (Reg) is equivalent to the well-ordering of the materials.
The second step, consists of proving a discrete monotonicity for the excess of the Dirichlet energy on balls under a low perimeter density assumption. More precisely, on the function that assigns
The discrete monotonicity of the map above, together with the upper bound estimate (6), will allow us to prove a local lower bound \(\lambda _K\) on the density of the perimeter:
As usual, the lower bound on the density of the perimeter is the cornerstone to prove regularity of almost perimeter minimizers. In fact, once the estimate (LB) is proved we simply apply the excess improvement results of [8, Sections 4 and 5] to obtain our main result:
Theorem 1.5
(Partial regularity) Let (w, A) be a saddle point of problem (P) in \(\Omega \). Assume that the operator \(P_Hu = \mathscr {A}^*(\sigma _H \mathscr {A}u)\) is hypoelliptic and regularizing for the half-space problem (see properties (60), (61)), and that the higher integrability (Reg) holds. Then there exists a positive constant \(\eta \in (0,1]\) depending only on N such that
Moreover if \(\mathscr {A}\) is a first-order partial differential operator, then \(\mathscr {A}w \in \mathrm {C}_\mathrm{{loc}}^{0,\eta /8}(\Omega {\setminus } (\partial A {\setminus } \partial ^* A))\); and hence, the trace of \(\mathscr {A}w\) exists on either side of \(\partial ^* A\).
Let us make a quick account of previous results. To our knowledge, only optimal design problems modeling the maximal dissipation of energy have been treated.
In [7] Ambrosio and Buttazzo considered the case where \(\mathscr {A}= \nabla \) is the gradient operator for scalar-valued (\(d = 1\)) functions and where \(\sigma _2 \ge \sigma _1\) in the sense of quadratic forms. The authors proved existence of solutions and showed that, up to choosing a good representative, the topological boundary is the closure of the reduced boundary and \(\mathscr {H}^{N-1}(\partial A {\setminus } \partial ^*A) = 0\). Soon after, Lin [8], and Kohn and Lin [9] proved, in the same case, that \(\partial ^* A\) is an open \(\mathrm {C}^1\)-hypersurface. From this point on, there have been several contributions aiming to discuss the optimal regularity of the interface for this particular case. In this regard and in dimension \(N =2\), Larsen [10] proved that connected components of A are \(\mathrm {C}^1\) away from the boundary. In arbitrary dimensions, Larsen’s argument cannot be further generalized because it relies on the fact that convexity and positive curvature are equivalent in dimension \(N =2\). During the time this project was developed, we have learned that Fusco and Julin [11] found a different proof for the same results as stated in [8]; besides this, De Philippis and Figalli [12] recently obtained an improvement on the dimension of the singular set (\(\partial ^*A {\setminus } \partial A\)).
The paper is organized as follows. In the beginning of Sect. 2 we fix notation and discuss some facts of linear operators, Young measures and sets of finite perimeter. We also give the precise definition of gradient type operators and include a compensated compactness result that will be employed throughout the paper. In Sect. 3 we show the equivalence of the constrained problem (4), (5) and the unconstrained problem (P) (Theorem 1.1). In the first part of Sect. 4 we shortly discuss how the higher integrability assumption (Reg) holds for various operators of gradient form. The rest of the section is devoted to the proof of the Upper bound (6). Section 5 is devoted to the proof of the Lower bound estimate (LB). Finally, in Sect. 6 we recall the flatness excess improvement [8] from which Theorem 1.5 easily follows.
2 Notation and preliminaries
We will write \(\Omega \) to represent a non-empty, open, bounded subset of \(\mathbb {R}^N\) with Lipschitz boundary \(\partial \Omega \). The use of capital letters \(A, B, \dots ,\) will be reserved to denote Borel subsets of \(\mathbb {R}^N\) and we will write \(\mathfrak B(\mathbb {R}^N)\) to denote the Borel \(\sigma \)-algebra of \(\mathbb {R}^N\).
The letters x, y will denote points in \(\Omega \); while \(z \in \mathbb {R}^d\) and \(P \in \mathbb {R}^{dN^k}\) will be reserved for vectors and arrays in Euclidean space. The Greek letters \(\varepsilon , \delta , \rho \) and \(\gamma \) shall be used for general smallness or scaling constants. We follow Lin’s convention in [8], bounding constants will be generally denoted by \(c_1\ge c_2\ge \cdots \), while smallness and decay constants will be usually denoted by \(\varepsilon _1\ge \varepsilon _2\ge \cdots \), and \(\theta _1\ge \theta _2\ge \cdots ,\) respectively. Let us mention that in proving regularity results one may often find it impractical to keep track of numerical constants due to the large amount of parameters; to illustrate better their uses and dependencies we have included a glossary of constants at the end of the paper.
It will often be useful to write a point \(x \in \mathbb {R}^{N} = \mathbb {R}^{N-1} \times \mathbb {R}\) as \(x = (x',x_{N})\), in the same fashion we will also write \(\nabla = (\nabla ',\partial _{N})\) to decompose the gradient operator. The bilinear form \(\mathbb {R}^p \times \mathbb {R}^p \rightarrow \mathbb {R}: (x,y) \mapsto x \cdot y\) will stand for the standard inner product between two points while we will use the notation \(| x | := \sqrt{x \cdot x}\) to represent the standard p-dimensional Euclidean norm. To denote open balls in \({\mathbb {R}}^N\) centered at a point x with radius r we will simply write \(B_r(x)\). Similarly, \(B'_r :=\{x' \in \mathbb R^{N-1} : (x',0) \in B_r \}\).
We keep the standard notation for \(\mathrm {L}^p\) and \(\mathrm {W}^{l,p}\) spaces. We write \(\mathrm {C}^l(\Omega ;Z)\), and \(\mathrm {C}^l_\mathrm{c}(\Omega ;\mathbb {R}^d)\) to denote the spaces of functions with values in \(\mathbb {R}^d\) and with continuous l-th derivative, and its subspace of functions compact support respectively. Similar notation stands for \(\mathscr {M}(\Omega ;\mathbb {R}^d)\) the space of bounded Radon measures in \(\Omega \), and \(\mathscr {D}(\Omega ; \mathbb {R}^d)\) the space of smooth functions in \(\Omega \) with compact support. For X and Y Banach spaces, the standard pairing between X and Y will be denoted by \(\langle \cdot ,\cdot \rangle : X \times Y \rightarrow \mathbb {R}: (u,v) \mapsto \langle u, v \rangle \).
2.1 Operators of gradient form
We introduce an abstract class of linear differential operators \(\mathscr {A}: \mathrm {L}^2(\Omega ;\mathbb {R}^d) \rightarrow \mathrm {W}^{-k,2}(\Omega ;\mathbb {R}^{dN^k})\).
This class contains scalar- and vector-valued gradients, higher gradients, and symmetrized gradients among its elements. The motivation behind it is that we may treat different models by employing a general and neat abstract setting. At a first glance this framework may appear too sterile, however, this definition is only meant to capture some of the essential regularity and rigidity properties of gradients.
Let \(\mathscr {A}: \mathrm {L}^2(\Omega ;\mathbb {R}^p) \rightarrow \mathrm {W}^{-k,2}(\Omega ;\mathbb {R}^q)\) be a k’-th order homogeneous partial differential operator of the form
where \(A_\alpha \in \text {Lin }(\mathbb {R}^p;\mathbb {R}^{q})\), and \(\partial ^\alpha = \partial _1^{\alpha _1}\dots \partial _N^{\alpha _N}\) for every multi-index \(\alpha = (\alpha _1,\dots ,\alpha _N) \in (\mathbb N \cup \{0\})^N\) with \(|\alpha |:=|\alpha _1| + \cdots |\alpha _N|\). We define the \(\mathscr {A}\)-Sobolev space of \(\Omega \) as
endowed with the norm \(\Vert u\Vert ^2_{\mathrm {W}^\mathscr {A}(\Omega )} :=\Vert u \Vert ^2_{\mathrm {L}^2(\Omega )} + \Vert \mathscr {A}u \Vert ^2_{\mathrm {L}^2(\Omega )}\). We also define the \(\mathscr {A}\)-Sobolev space with zero boundary values in \(\partial \Omega \) by letting
The principal symbol of \(\mathscr {A}\) is the positively k-homogeneous map defined as
where \(\xi ^\alpha = \xi _1^{\alpha _1}\cdots \xi _N^{\alpha _N}\). One says that \(\mathscr {A}\) has the constant rank property if there exists a positive integer r such that
Definition 2.1
(Operators of gradient form) Let \(\mathscr {A}\) a homogeneous partial differential operator as in (7) with \(p = d\) and \(q = dN^k\). We say that \(\mathscr {A}\) is an operator of gradient form if the following properties hold:
-
1.
Compactness. There exists a positive constant \(C(\Omega )\) for which
$$\begin{aligned} \Vert \varphi \Vert ^2_{\mathrm {W}^{k,2}(\Omega )} \le C(\Omega )\bigg (\Vert \varphi \Vert _{\mathrm {L}^2(\Omega )}^2 + \Vert \mathscr {A}\varphi \Vert ^2_{\mathrm {L}^2(\Omega )}\bigg ) \end{aligned}$$(8)for all \(\varphi \in \mathrm {C}^\infty (\overline{\Omega };\mathbb {R}^d)\). Even more, for every \(u \in W^{\mathscr {A}}(\Omega )\) the following Poincaré inequality holds:
$$\begin{aligned} \inf \big \{\;\Vert u - v\Vert ^2_{\mathrm {W}^{k,2}(\Omega )} \; :\; v \in \mathrm {W}^\mathscr {A}(\Omega ), \mathscr {A}v = 0\; \big \} \le C(\Omega )\Vert \mathscr {A}u \Vert ^2_{\mathrm {L}^2(\Omega )}. \end{aligned}$$(9) -
2.
Exactness. There exists an m’-th homogeneous partial differential operator
$$\begin{aligned} \mathscr {B}:=\sum _{|\alpha | = m} B_\alpha \partial ^\alpha , \end{aligned}$$(10)with coefficients \(B_\alpha \in \text {Lin }(Z;\mathbb {R}^n)\) for some positive integer n and a subspace Z of \(\mathbb {R}^{dN^k}\), such that for every open and simply connected subset \(\omega \subset \Omega \) we have the property
$$\begin{aligned} \big \{\mathscr {A}u : u \in \mathrm {W}^\mathscr {A}(\omega ) \big \}= \big \{ v \in \mathrm {L}^2(\omega ;Z) : \mathscr {B}v = 0 \; \text {in} \; \mathscr {D}'(\omega ;\mathbb {R}^n)\big \}. \end{aligned}$$
We write \(\mathscr {A}^*\) to denote the \(\mathrm {L}^2\)-adjoint of \(\mathscr {A}\), which is given by
Remark 2.2
(Constant rank) Let \(\mathscr {A}\) and \(\mathscr {B}\) be two linear differential operators satisfying an exactness property as in Definition 2.1. Then both operators \(\mathscr {A}\) and \(\mathscr {B}\) have the constant rank property (†). This follows from the lower semi-continuity of the rank in any subspace of matrices.
Remark 2.3
(Rigidity) The wave cone of an operator \(\mathscr {A}\) of the form (7) which is defined as
contains the admissible amplitudes in Fourier space for which concentration and oscillation behavior is allowed under the constraint \(\mathscr {A}u = 0\). As in the case of gradients, it can be seen from the compactness assumption in Definition 2.1 that the wave cone \(\Lambda _\mathscr {A}\) of a gradient operator \(\mathscr {A}\) is the zero space. In particular, there exists a positive constant \(\lambda \) (depending only on the coefficients of \(\mathscr {A}\)) such that
Remark 2.4
(Poincaré inequality II) It follows from the definition of \(\mathrm {W}_0^\mathscr {A}(\Omega )\) and the compactness assumption of \(\mathscr {A}\) that \(\mathrm {W}^\mathscr {A}_0(\Omega )\subset \mathrm {W}^{k,2}_0(\Omega ;\mathbb {R}^d)\). In particular, \(\text {Ker}(\mathscr {A}) \cap \mathrm {W}^\mathscr {A}_0(\Omega ) = \{0\} \subset \mathrm {L}^2(\Omega ;\mathbb {R}^d)\) and \(\mathscr {A}[\mathrm {W}^\mathscr {A}_0(\Omega )]\) is closed in the \(\mathrm {L}^2\) norm. Thus, by [18, Theorem 2.21], there exists a constantFootnote 4 \(C(\Omega )\) such that
2.1.1 Elliptic regularity
Let \(\mathscr {A}\) be an operator of gradient form as in Definition 2.1 and let \(\mathbf \sigma \in \mathrm {L}^\infty (\Omega ;\mathbb {R}^{dN^k})\) be a tensor of variable coefficients satisfying the strong pointwise Gårding inequality (see (2))
If we define
then we may write
It is easy to verify, using the compactness assumption of \(\mathscr {A}\), that \(\mathbf C :=(\mathbf A^T \sigma \;\mathbf A)\) satisfies the weak Gårding inequality
where \(C = C(\Omega )\) is the constant in the compactness assumption of Definition 2.1; for all smooth, \(\mathbb {R}^d\)-valued functions \(\varphi \) in \(\overline{\Omega }\).
Lemma 2.5
(Caccioppoli inequality) Let \(\mathbf \sigma \in \mathrm {L}^\infty (\Omega ;\mathbb {R}^{dN^k})\) satisfy the strong pointwise Gårding inequality (13) and let \(w \in \mathrm {W}^{\mathscr {A}}(\Omega )\) be a solution of the state equation
Then there exists a positive constant C depending only on \(M,N, \sigma \) and \(\mathscr {A}\) such that
Proof
We may re-write \(\mathscr {A}^* (\sigma \mathscr {A}u)\) as the elliptic operator in divergence form
for coefficients \(\mathbf C = (\mathbf A^T \sigma \mathbf A)\) satisfying a weak Gårding inequality as in (15). The assertion then follows from Corollary 22 in [19]. \(\square \)
Using Lemma 2.5 one can show, by classical methods, the following lemma on the regularizing properties of elliptic operators with constant coefficients:
Lemma 2.6
(Constant coefficients) Let \(\mathscr {A}\) be an operator of gradient form and let \(\sigma _0 \in \mathrm{Lin }(\mathbb {R}^{dN^k};\mathbb {R}^{dN^k})\) be a tensor satisfying the strong Gårding inequality (13). Then the operator
is hypoelliptic in the sense that if \(\Omega \) is open and connected, and \(w \in \mathrm {L}^2(\Omega ;\mathbb {R}^d)\), then
Furthermore, there exists a constant \(c = c(M,N) \ge 2^N\) such that
for every \(B_r(x) \subset \Omega \).
2.1.2 Examples
Next, we gather some well-known differential structures that fit into the definition of operators of gradient form.
-
(i)
Gradients. Let \(\mathscr {A}: \mathrm {L}^2(\Omega ;\mathbb {R}^d) \rightarrow \mathrm {W}^{-1,2}(\Omega ;\mathbb {R}^{dN}) : u \mapsto (\partial _j u^i)\) for \(1 \le i \le d\) and \(1 \le j \le N\). In this case
$$\begin{aligned} A_j \, z = z \otimes \mathbf e_j \quad \text {for every }z \in \mathbb {R}^d. \end{aligned}$$Hence, \(\mathrm {W}^\mathscr {A}(\Omega ) = \mathrm {W}^{1,2}(\Omega ;\mathbb {R}^d)\) and the compactness property is a consequence of the classical Poincaré inequality on \(\Omega \).
The exactness assumption is the result of the characterization of gradients via curl-free vector fields.
Let \(\mathscr {B}: \mathrm {L}^2(\Omega ;\mathbb {R}^{dN}) \rightarrow \mathrm {W}^{-1,2}(\Omega ;\mathbb {R}^{dN^2})\) be the curl operator
$$\begin{aligned} \mathscr {B}v = (\mathrm {curl}(v^i))_i:=\left( \partial _l v_{ir} - \partial _r v_{il} \right) _{ilr} \quad 1 \le i \le d, \quad 1 \le l,r \le N, \end{aligned}$$then condition (10) is fulfilled for \(\mathscr {B}= \sum _{j=1}^N B_j \partial _i\) with coefficients
$$\begin{aligned} (B_j)_{ilr,pq} = \delta _{ip}(\delta _{jl}\delta _{rq} - \delta _{jr}\delta _{lq}) \quad 1\le l,r,q \le N, \quad 1 \le i,p \le d. \end{aligned}$$Observe that \(\mathscr {B}v = 0\) if and only if curl \(v^i = 0\), for every \(1 \le i \le d\); or equivalently, \( v^i = \nabla u^i\) for some function \( u^i : \Omega \subset \mathbb {R}^N \rightarrow \mathbb {R}\), for every \(1 \le p \le d\) (as long as \(\Omega \) is simply connected). Hence,
$$\begin{aligned} \big \{ \nabla u : u \in \mathrm {W}^{1,2}(\omega ;{\mathbb {R}}^d)\big \} = \big \{v \in \mathrm {L}^2(\omega ;\mathbb {R}^{dN}) : \mathscr {B}v = 0\big \}, \end{aligned}$$for all Lipschitz, and simply connected \(\omega \subset \subset \Omega \).
-
(ii)
Higher gradients. Let \(\mathscr {A}: \mathrm {L}^2(\Omega ) \rightarrow \mathrm {W}^{-k,2}(\Omega ;\mathbb {R}^{N^k})\) be the linear operator given by
$$\begin{aligned} u \mapsto \partial ^\alpha u, \quad \text {where }|\alpha | = k. \end{aligned}$$Compactness is similar to the case of gradients.
We focus on the exactness condition: Let \(\mathscr {B}^k : \mathrm {L}^2(\Omega ;\text {Sym}(\mathbb {R}^{N^k})) \rightarrow \mathrm {W}^{-1,2}(\Omega ;\mathbb {R}^{N^{k+1}})\) be the curl operator on symmetric functions defined by the coefficients
$$\begin{aligned}&(B^k_j)_{pq\beta _2\dots \beta _k,\alpha _1\dots \alpha _k} := \left( \delta _{jp}\delta _{\alpha _1 q}\prod _{h= 2}^k\delta _{\alpha _h\beta _h} - \delta _{jq}\delta _{\alpha _1 p}\prod _{h= 2}^k\delta _{\alpha _h\beta _h}\right) ,\quad \\&\quad 1 \le p,q,\beta _h,\alpha _h \le N, \quad h\in \{2,\cdots ,k\}. \end{aligned}$$We write
$$\begin{aligned} \mathscr {B}^k v :=\sum _{i = 1}^N B^k_j \, \partial _j v, \quad v : \Omega \subset \mathbb {R}^N \rightarrow \text {Sym }(\mathbb {R}^{N^k}). \end{aligned}$$It easy to verify that \(\mathscr {B}^k v = 0\) if and only if
$$\begin{aligned} \mathrm {curl}((v_{p\alpha '})_p) = 0 \quad \text {for all } |\alpha '| = k-1. \end{aligned}$$If \(\Omega \) is simply connected, then there exists a function \(u^{\alpha '} : \Omega \rightarrow \mathbb {R}\) such that \(v_{p\alpha '} = \partial _p u^{\alpha '}\) for every \(|\alpha '|= k-1\). Using the symmetry of v under the permutation of its coordinates one can further deduce the existence of a function \(u_k : \Omega \rightarrow \text {Sym }(\mathbb {R}^{N^{k-1}})\) with
$$\begin{aligned} v = \nabla u_k \quad \text {and }\quad (u_k)_{\alpha '} = u^{\alpha '}. \end{aligned}$$Moreover, \(\mathscr {B}^{k-1} u_k = 0\). By induction one obtains that
$$\begin{aligned} v = \nabla ^{k}u_0 \quad \text {for some function } u_0:\Omega \subset \mathbb {R}^N \rightarrow \mathbb {R}. \end{aligned}$$ -
(iii)
Symmetrized gradients. Let \(\mathscr {E} : \mathrm {L}^2(\Omega ;\mathbb {R}^N) \rightarrow \mathrm {W}^{-1,2}(\Omega ;\text {Sym }(\mathbb {R}^{N^2}))\) be the linear operator given by
$$\begin{aligned} u \mapsto \mathscr {E} u:=\frac{1}{2}(\partial _j u^i + \partial _i u^j)_{ij}, \quad \text {for }\; 1\le i,j \le N. \end{aligned}$$The compactness property is a direct consequence of Korn’s inequality. Consider the second-order homogeneous differential operator \(\mathscr {B}: \mathrm {L}^2(\Omega ;\text {Sym }(\mathbb {R}^{N^2})) \rightarrow \mathrm {W}^{-2,2}(\Omega ;\mathbb {R}^{N^3})\) defined in the following way
$$\begin{aligned} \mathscr {B}v = \mathrm {curl}\, (\mathrm {curl}(v)) = \left( \frac{\partial ^2 v_{ij}}{\partial x_i \partial x_l} + \frac{\partial ^2 v_{il}}{\partial x_i \partial x_j} -\frac{\partial ^2 v_{ii}}{\partial x_j \partial x_l} -\frac{\partial ^2 v_{jl}}{\partial x_i \partial x_i} \right) _{1 \le i,j,l\le N}. \end{aligned}$$Then \(\mathscr {B}v = 0\), if and only if \(v = \mathscr {E}u\) for some \(u \in \mathrm {W}^{1,2}(\Omega ;\mathbb {R}^N) = \mathrm {W}^{\mathscr {E}}(\Omega )\).Footnote 5
Remark 2.7
In the previous examples, we have omitted the characterization of higher gradients of vector-valued functions; however, the ideas remain the same as in the examples (i) and (ii).
Remark 2.8
(Two-dimensional elasticity) In dimension \(N = 2\) and provided that \(\Omega \) is simply connected, the fourth-order equation for pure bending of a thin plate given by
is equivalent to the in-plane elasticity equation
for some tensor \(\mathbf S\) such that \(\mathbf D = (\mathbf R_\perp \mathbf S^{-1} \, \mathbf R_\perp )\), and where \( \mathbf R_\perp \) is the fourth-order tensor whose action is to rotate a second-order tensor by \(90^\circ \) (see, e.g., [20, Chapter 2.3]). Furthermore,
For this reason, when working with the linear equations for pure bending of a thin plate we may indistinctly use regularizing properties of any of the equations above in the portions where \(\mathbf D\) is regular.
2.2 Compensated compactness
The following theorem is a generalized version of the well-known div-curl Lemma.
Lemma 2.9
Let \(\mathscr {A}\) be a k’-th order operator of gradient form and let \(\{\sigma _h\}\subset \mathrm {L}^2(\Omega ; Sym (\mathbb R^{dN^k} \otimes \mathbb R^{dN^k}))\) be a sequence of strongly elliptic tensors as in (13). Assume also that \(\{u_h\} \subset \mathrm {W}^\mathscr {A}(\Omega )\) and \( \{f_h\} \subset \mathrm {W}^{-k,2}(\Omega ;\mathbb {R}^d)\) are sequences for which
Further assume there exist \(\sigma \in \mathrm {L}^2(\Omega ; Sym (\mathbb R^{dN^k} \otimes \mathbb R^{dN^k}))\), \(u \in \mathrm {W}^\mathscr {A}(\Omega )\), and \(f \in \mathrm {W}^{-k,2}(\Omega ;\mathbb {R}^d)\) for which
Then,
In particular,
Proof
For simplicity we denote \(\tau _h:= \sigma _h\mathscr {A}u_h, \tau := \sigma \mathscr {A}u\). It suffices to observe that \(\tau _h\rightharpoonup \tau \) in \(\mathrm {L}^2\) to prove that
The strong convergence on compact subsets of \(\Omega \) requires a little bit more effort. Considering that \(\mathscr {A}\) is a k’-th order linear differential operator, we may find constants \(c_{\alpha \beta }\) with \(|\alpha | + |\beta | \le k, |\beta | \ge 1\) such that
Hence,
By the compactness assumption on \(\mathscr {A}\) we may assume without loss of generality that \(u_h \rightharpoonup u\) in \(\mathrm {W}^{k,2}(\Omega ;\mathbb {R}^d)\). Thus, passing to the limit we obtain
for every \(\varphi \in \mathscr {D}(\Omega )\). One concludes that
Fix \(\omega \subset \subset \Omega \) and let \(0 \le \varphi \in \mathscr {D}(\Omega )\) with \(\varphi \equiv 1\) on \(\omega \). Using the convergence in (16), the uniform ellipticity (2) and the symmetry of \(\{\sigma _h\}\), one gets
\(\square \)
2.3 Young measures and lower semi-continuity of integral energies
In this section \(\mathscr {B}: \mathrm {L}^2(\Omega ;Z) \rightarrow \mathrm {W}^{-m,2}(\Omega ;\mathbb {R}^n)\) is assumed to be a an m’-th order homogeneous partial differential operator of the form
satisfying the constant rank condition (†).
Next, we recall some facts about \(\mathscr {B}\)-quasiconvexity, lower semi-continuity and Young measures. The results in this section hold for differential operators with coefficients \(B_\alpha \) in arbitrary spaces \(\text {Lin }(\mathbb {R}^{p};\mathbb {R}^{q})\) for p, q a pair of positive integers; however, we only present versions where the dimensions match our current setting. We start by stating a version of the Fundamental theorem for Young measures due to Ball [21].
Theorem 2.10
(Fundamental theorem for Young measures) Let \(\Omega \subset \mathbb {R}^N\) be a measurable set with finite measure and let \(\{v_j\}\) be a sequence of measurable functions \(v_j : \Omega \rightarrow Z\). Then there exists a subsequence \(\{v_{h(j)}\}\) and a weak\(^*\) measurable map \(\mu : \Omega \rightarrow \mathscr {M}(Z)\) with the following properties:
-
1.
We denote \(\mu _x := \mu (x)\) for simplicity, then \(\mu _x \ge 0\) in the sense of measures and \(|\mu _x|(Z) \le 1\) for a.e. \(x \in \Omega \).
-
2.
If one additionally assumes that \(\{v_{h(j)}\}\) is uniformly bounded in \(\mathrm {L}^1(\Omega ;Z)\), then \(|\mu _x|(Z) = 1\) for a.e. \(x \in \Omega \).
-
3.
If \(F : \mathbb {R}^{dN^k} \rightarrow \mathbb {R}\) is a Borel and lower semi-continuous function, and is also bounded from below, then
$$\begin{aligned} \int _\Omega \langle \mu _x, F\rangle \, \,\mathrm {d}x \le \liminf _{j \rightarrow \infty } \int _\Omega F(v_{h(j)}) \, \,\mathrm {d}x. \end{aligned}$$ -
4.
If \(\{v_{h(j)}\}\) is uniformly bounded in \(\mathrm {L}^1(\Omega ;Z)\) and \(F: \mathbb {R}^{dN^k} \rightarrow \mathbb {R}\) is a continuous function, and bounded from below, then
$$\begin{aligned} \int _\Omega \langle \mu _x, F\rangle \, \,\mathrm {d}x = \liminf _{j \rightarrow \infty } \int _\Omega F(v_{h(j)}) \, \,\mathrm {d}x \end{aligned}$$if and only if \(\{F\circ v_{h(j)}\}\) is equi-integrable. In this case,
$$\begin{aligned} F \circ v_{h(j)} \rightharpoonup \langle \mu _x,F\rangle \quad \text {in } \mathrm {L}^1(\Omega ). \end{aligned}$$
In the sense of Theorem 2.10, we say that the sequence \(\{v_{h(j)}\}\) generates the Young measure \(\mu \).
The following proposition tells us that a uniformly bounded sequence in the \(\mathrm {L}^p\) norm, which is also sufficiently close to \(\text {Ker}(\mathscr {B})\), may be approximated by a p-equi-integrable sequence in \(\text {Ker}(\mathscr {B})\) in a weaker \(\mathrm {L}^q\) norm. We remark that this rigidity result is the only one where Murat’s constant rank condition (†) is used.
Proposition 2.11
[16, Lemma 2.15] Let \(1< p < \infty \). Let \(\{v_h\}\) be a bounded sequence in \(\mathrm {L}^p(\Omega ;Z)\) generating a Young measure \(\mu \), with \(v_h \rightharpoonup v\) in \(\mathrm {L}^p(\Omega ;Z)\) and \(\mathscr {B}v_h \rightarrow 0\) in \(\mathrm {W}^{-m,p}(\Omega ;\mathbb {R}^n)\). Then there exists a p-equi-integrable sequence \(\{u_h\}\) in \(\mathrm {L}^p(\Omega ;Z) \cap \text {Ker}(\mathscr {B})\) that generates the same Young measure \(\mu \) and is such that
\(\square \)
Let \(F : \mathbb {R}^{dN^k} \rightarrow \mathbb {R}\) be a lower semi-continuous function with \(0 \le F(P) \le C(1 + |P|^p)\) for some positive constant C. The \(\mathscr {B}\)-quasiconvex envelope of F at \(P \in Z \subset \mathbb {R}^{dN^k}\) is defined as
The most relevant feature of \(Q_{\mathscr {B}}F\) is that, for \(p > 1\), the lower semi-continuous envelope with respect to the weak-\(\text {L}^p\) topology of the functional
is given by the functional
If \(\mu \) is a Young measure generated by a sequence \(\{v_h\}\) in \(\mathrm {L}^p(\Omega ;Z)\) such that \(\mathscr {B}v_h = 0\) for every \(h \in \mathbb N\), then we say that \(\mu \) is a \(\mathscr {B}\)-free Young measure.
We recall the following Jensen inequality for \(\mathscr {B}\)-free Young measures [16, Theorem 4.1]:
Theorem 2.12
Let \(1< p < \infty \). Let \(\mu \) be a \(\mathscr {B}\)-free Young measure in \(\Omega \). Then for a.e. \(x \in \Omega \) and all lower semi-continuous functions that satisfy \(|F(P)| \le C(1 + |P|^p)\) for some positive constant C and all \(P \in \mathbb {R}^{dN^k}\), one has that
2.4 Geometric measure theory and sets of finite perimeter
Most of the facts collected in this section can be found in [22] and [23]; however, some notions as the slicing of sets of finite perimeter are presented there only in a formal way. For a better understanding of such topics we refer the reader to [24].
Let \(A \subset \mathbb {R}^N\) be a Borel set. The Gauss-Green measure \(\mu _A\) of A is the derivative of the characteristic function of A in the sense of distributions, i.e., \(\mu _A := \mathrm{D}(\mathbbm {1}_A)\). We say that A is a set of locally finite perimeter if and only if \(|\mu _A|\) is a vector-valued Radon measure in \(\mathbb {R}^N\). We write \(A \in \mathrm {BV}_\mathrm{{loc }}(\mathbb {R}^N)\) to express that A is a set of locally finite perimeter in \(\mathbb {R}^N\).
Let \(\omega \subset \subset \mathbb {R}^N\) be a Borel set. The perimeter in \(\omega \) of a set A with locally finite perimeter is defined as
The Radon–Nikodým differentiation theorem states that the set of points
has full \(|\mu _A|\)-measure in \(\mathbb {R}^N\); this set is commonly known as the reduced boundary of A. We will also use the notation
the measure theoretic normal of A.
In general, for \(s \ge 0\), we will denote by \(\mathscr {H}^s\) the s-dimensional Hausdorff measure in \(\mathbb {R}^N\). The following well-known theorem captures the structure of sets with finite perimeter in terms of the measure \(\mathscr {H}^{N-1}\):
Theorem 2.13
(De Giorgi’s Structure Theorem) Let A be a set of locally finite perimeter. Then
where
and each \(K_j\) is a compact subset of a \(\mathrm {C}^1\)-hypersurface \(S_j\) for every \(j \in \mathbb N\). Furthermore, \(\nu _A|_{S_j}\) is normal to \(S_j\) and
From De Giorgi’s Structure Theorem it is clear that \(\text {spt}~ \mu _A = \overline{\partial ^* A}\). Actually, up to modifying A on a set of zero measure, one has that \(\partial A = \overline{\partial ^* A}\) (see [22, Proposition 12.19]). From this point on, each time we deal with a set A of finite perimeter, we will assume without loss of generality that
For a set of locally finite perimeter A, the deviation from being a perimeter minimizer in \(\Omega \), at a given scale r, is quantified by the monotone function
The next result, due to Tamanini [25], states that a set of locally finite perimeter with small deviation \(\text {Dev}_\Omega \) at every scale is actually a \(\mathrm {C}^1\)-hypersurface up to a lower dimensional set.
Theorem 2.14
Let \(A \subset \mathbb {R}^N\) be a set of locally finite perimeter and let c(x) be a locally bounded function for which
Then the reduced boundary in \(\Omega \), \((\partial ^* A \cap \Omega )\), is an open \(\mathrm {C}^{1,\eta }\)-hypersurface and the singular set \(\Omega \cap (\partial A {\setminus } \partial ^* A)\) has at most Hausdorff dimension \((N-8)\).
2.4.1 Slicing sets of finite perimeter
Given a Borel set \(E \subset \mathbb {R}^N\) and a Lipschitz function \(g: \mathbb {R}^N \rightarrow \mathbb {R}\), we shall consider the level set slices
For a set \(A\subset \mathbb {R}^N\) of finite perimeter in \(\Omega \), the level set slice of the reduced boundary \((\partial ^* A)_t\) is \(\mathscr {H}^{N-2}\)-rectifiable for almost every \(t \in \mathbb {R}\). Furthermore, by the co-area formula, \(t \mapsto \mathscr {H}^{N-2}( (\partial ^*A)_t) \in \mathrm {L}^1_\mathrm{{loc}}(\mathbb {R})\).
If the set \(\{g = t\}\) is a \(\mathrm {C}^1\)-manifold and t is such that \(\mathscr {H}^{N-2}( (\partial ^*A)_t) < \infty \), we shall define the slice of A in \(g^{-1}\{t\}\) as
It turns out that, for \(g(x) = |x|\), the level set slice \(A_t\) is locally diffeomorphic to a set of finite perimeter in \(\mathbb {R}^{N-1}\). Even more,
Here, \(\partial ^*A_t\) is understood as the image, under local diffeomorphisms, of the reduced boundary of a set of finite perimeter. These properties can be inferred from the classical slicing by hyperplanes, see e.g., [22, Chapter 18.3].
We also define the cone extension of a set \(E \subset \mathbb {R}\) containing \(\{0\}\) by letting
For a.e. \(t > 0\) and \(g(x) = |x|\), the cone extension of \(A_t\) is a set of locally finite perimeter in \(\mathbb {R}^N\) with
In order to attend different variational problems involving the minimization of perimeter, a well-known technique is to modify a set A within balls \(B_t\) without modifying its Gauss-Green measure in \((B_t)^c\).
For almost every \(t > 0\), where \(\langle A, g, t\rangle \) is well-defined and (20), (21) hold, we construct a cone-like comparison set of A by setting
Exploiting the basic properties of reduced boundaries, it follows by (20) that
and, in particular,
On the other hand, again by the co-area formula,
Using the monotonicity of \(r \mapsto \mathrm {Per}(A;B_r)\) and the general version of the co-area formula (see [24, Theorem 3.2.22]) one can show that the derivative of \(r \mapsto \mathrm {Per}(A;B_r)\) exists at almost every \(t > 0\); even more, one has that
The previous estimate will play a crucial role in proving the Lower bound (LB).
3 Existence of solutions: proof of Theorem 1.1
We show an equivalence between the constrained problem (5) and the unconstrained problem (P) for which existence of solutions and regularity properties for minimizers are discussed in the present and subsequent sections. We fix \(\mathscr {A}: \mathrm {L}^2(\Omega \;\mathbb {R}^d) \rightarrow \mathrm {W}^{-k,2}(\Omega ;\mathbb {R}^{dN^k})\) an operator of gradient from as in Definition 2.1. We also fix \(A_0 \subset \mathbb {R}^N\), a set of locally finite perimeter.
Recall that, the minimization problem (5) under the state constraint (4) reads:
where \(w_A\) is the unique distributional solution to the state equation
On the other hand, the associated saddle point problem (P) readsFootnote 6:
where
Theorem 1.1 (Existence) There exists a solution (w, A) of problem (P). Furthermore, there is a one to one correspondence
between solutions of the problem (P) and solutions of the minimization problem (5) under the constraint (4).
Proof
We employ the direct method. We begin by proving existence of solutions to problem (P). To do so, we will first prove the following:
Claim
1. For any set \(A \subset \mathbb {R}^N\) as in the assumptions, there exists \(w_A \in \mathrm {W}^{\mathscr {A}}_0(\Omega )\) such that
The tensor \(\sigma _A\) is a positive definite tensor and therefore the mapping
is strictly concave. Observe that \(\sup _{u \in \mathrm {W}^{\mathscr {A}}_0(\Omega )} I_\Omega (u,A) \ge \text {Per }(A;\overline{\Omega })\); indeed, we may take \(u \equiv 0 \in \mathrm {W}^{\mathscr {A}}_0(\Omega )\). Hence,
Because of this, we may find a maximizing sequence \(\{w_h\}\) in \(\mathrm {W}^{\mathscr {A}}_0(\Omega )\), i.e.,
Even more, one has from (2) that
and consequently from (26) and (12) one infers that
A fast calculation shows that \(\Vert w_h\Vert _{\mathrm {L}^2(\Omega )} \le 2MC(\Omega )\Vert f\Vert _{\mathrm {L}^2(\Omega )}\); in return, (27) also implies that
Hence, using again the compactness property of \(\mathscr {A}\), we may pass to a subsequence (which we will not relabel) and find \(w_A \in \mathrm {W}^{\mathscr {A}}_0(\Omega )\) with
The concavity of \(-\sigma _A z \cdot z\) is a well-known sufficient condition for the upper semi-continuity of the functional \(\mathscr {A}u \mapsto -\int _\Omega \sigma _A \mathscr {A}u \cdot \mathscr {A}u\). Therefore,
This proves the claim.
Now, we use Claim 1 to find a minimizing sequence \(\{A_h\}\) for \(A \mapsto I_\Omega (w_{A},A)\). Since the uniform bound (27) does not depend on A, we may again assume (up to a subsequence) that there exists \(\tilde{w} \in \mathrm {W}^{\mathscr {A}}_0(\Omega )\) such that
Even more, since \(\{A_h\}\) is minimizing, it must be that \(\sup _h \{\mathrm {Per}(A_h;B_R)\} < \infty \), for some ball \(B_R\) properly containing \(\Omega \), and thus (for a further subsequence) there exists a set \(\tilde{A} \subset \mathbb {R}^N\) of locally finite perimeter with \(\tilde{A} \cap \Omega ^c \equiv A_0 \cap \Omega ^c\) and such that
Therefore
A consequence of Lemma 2.9 is thatFootnote 7
By taking the limit as h goes to infinity we get from (28) and the convergence above that
where the last equality is a consequence of the identity \(\tilde{w} = w_{\tilde{A}}\) which can be easily derived by using the equation and the strict concavity of \(I_\Omega \) in the first variable. Thus, the pair \((w_{\tilde{A}}, \tilde{A})\) is a solution to problem (P).
The equivalence of problem (P) and problem (5) under the state constraint (4) follows easily from (29), the strict concavity of \(I_\Omega (\cdot ,A)\), and a simple integration by parts argument. \(\square \)
4 The energy bound: proof of Theorem 1.2
Throughout this section and for the rest of the manuscript we fix \(\mathscr {A}: \mathrm {L}^2(\Omega ;\mathbb {R}^d)\rightarrow \mathrm {W}^{-k,2}(\Omega ;\mathbb {R}^{dN^k})\) in the class of operators of gradient form. Accordingly, the notations Z and \(\mathscr {B}\) shall denote the subspace of \(\mathbb {R}^{dN^k}\) and the homogeneous operator associated to \(\mathscr {A}\) (see Definition 2.1). We will also write (w, A) to denote a particular solution of problem (P).
Consider the energy \(J_\omega : \mathrm {L}^2(\Omega ;Z) \times \mathfrak B(\mathbb R^N) \rightarrow \mathbb {R}\) defined as
The goal of this section is to prove a local bound for the map \(x \mapsto J_{B_r(x)}(\mathscr {A}w,A)\). More precisely, we aim to prove that for every compactly contained subset K of \(\Omega \) there exists a positive number \(\Lambda _K\) such that
Our strategy will be the following. We first define a one-parameter family \(J^\varepsilon \) of perturbations of \(J_{B_1}\) in the perimeter term. In Theorem 4.2 we show that, as the perimeter term vanishes, these perturbations \(\Gamma \)-converge (with respect to the \(\mathrm {L}^2\)-weak topology) to the relaxation of the energy
for which we will assume certain regularity properties (cf. property (Reg)). Then, using a compensated compactness argument, we prove Theorem 1.2 (Upper bound) by transferring the regularity properties of the relaxed problem to our original problem.
Before moving forward, let us shortly discuss how the higher integrability property (Reg) stands next to the standard assumption that the materials \(\sigma _1\) and \(\sigma _2\) are well-ordered.
4.1 A digression on the regularization assumption
As commented beforehand in the introduction, a key assumption in the proof of the upper bound (30) is that generalized local minimizers of the energy
possess improved decay estimates. More precisely, we require that local minimizers \(\tilde{u}\) of the functional
possess a higher integrability estimate of the form
Only then, we will be able to transfer a decay estimate of order \(\rho ^{N-1}\) to solutions of our original problem.
Remark 4.1
(The case of gradients) In the case \(\mathscr {A}= \nabla \), condition (Reg) boils down to regularity above the critical \(\mathrm {C}^{0,1/2}\) local regularity. More specifically,
for all \(B_r(x) \subset B_{1/2}\).
By Poincaré’s inequality and Campanato’s Theorem one can easily deduce that \(w \in \mathrm {C}^{0,\frac{1}{2} + \varepsilon }_\mathrm{{{loc }}}(B_{1/2})\) (cf. [9]).
Let us give a short account of some cases where one may find (Reg) to be a natural assumption.
4.1.1 The well-ordered case
The notion of well-ordering in Materials Science is not only justified as the comparability of two materials, one being at least better than the other. It has also been a consistent assumption when dealing with optimization problems because it allows explicit calculations. See for example [1, 26, 27], where the authors discuss how the well-ordering assumption plays a role in proving the optimal lower bounds of an effective tensor made-up by two materials. If \(\sigma _1\) and \(\sigma _2\) are well-ordered, say \(\sigma _2 \ge \sigma _1\) as quadratic forms, then \(W(P) = \sigma _2 P \cdot P\). Hence, by Lemma 2.6, the desired higher integrability (Reg) holds with \(\delta = 0\).
4.1.2 The non-ordered case
Applications for this setting are mostly reserved for gradients of scalar valued functions. In this particular case one can ensure that \(Q_{\mathscr {B}}W = W^{**}\), where \(W^{**}\) is the convex envelope of W. For example, one may consider an optimal design problem involving the linear conductivity equations for two dielectric materials which happen to be incomparable as quadratic forms. In this setting, it is not hard to see that indeed \(QW = W^{**}\) and even that \(W^{**} \in \mathrm {C}^{1,1}(\mathbb {R}^{dN^k},\mathbb {R})\). In dimensions \(N =2,3\), one can employ a Moser-iteration technique for the dual problem as the one developed in [17] to show better regularity of minimizers of (31).
Regarding the case of systems, if no well-ordering of the materials is assumed, it is not clear to us that (Reg) necessary holds (compare to [28, 29]).
4.2 Proof of Theorem 1.2
We define an \(\varepsilon \)-perturbation of \(v \mapsto \int _{B_1} \sigma _A v\cdot v\) as follows. Consider the functional
By a scaling argument one can easily check that
Furthermore,
We also consider the following one-parameter family of functionals:
The next result characterizes the \(\Gamma \)-limit of these functionals as \(\varepsilon \) tends to zero.
Theorem 4.2
The \(\Gamma \)-limit of the functionals \(G^\varepsilon \), as \(\varepsilon \) tends to zero, and with respect to the weak-\(\mathrm {L}^2\) topology is given by the functional
Proof
We divide the proof into three steps. First, we will prove the following auxiliary lemma.
Lemma 4.3
Let \(\omega \subset \mathbb {R}^N\) be an open and bounded domain. Let \(p > 1\) and let \(F:\mathbb {R}^{dN^k} \rightarrow [0,\infty )\) be a continuous integrand with p-growth, i.e.,
If \(v \in \mathrm {L}^p(\omega ;Z)\) and \(\mathscr {B}v = 0\), then there exists a p-equi-integrable recovery sequence \(\{v_h\} \subset \mathrm {L}^p(\omega ;Z)\) for v such that
Proof
Since \(v \mapsto \int _\omega Q_\mathscr {B}F(v)\) is the lower semi-continuous envelope of \(v \mapsto \int _{\omega } F(v)\) (see (17), (18)) with respect to the weak-\(\mathrm {L}^p\) topology, we may find a sequence \(\{v_h\}\) with the following properties:
and
Passing to a subsequence if necessary, we may assume that the sequence \(\{v_h\}\) generates a \(\mathscr {B}\)-free Young measure which we denote by \(\mu \). We then apply [16, Lemma 2.15] to find a p-equi-integrable sequence \(\{v'_h\}\) (with \(\mathscr {B}v_h = 0\)) generating the same Young measure \(\mu \). On the one hand, the Fundamental Theorem for Young measures (Theorem 2.10) and the fact that \(\{v_h\}\) generates \(\mu \) yield
Even more, due to the same theorem and the equi-integrability of the sequence \(\{|v'_h|^p\}\) one gets the convergence \(F(v'_h) \rightharpoonup \langle \mu _x,F\rangle \in \mathrm {L}^1\). In other words,
The three relations above yield
We summon the characterization for \(\mathscr {B}\)-free Young measures from Theorem 2.12 to observe that
This inequality and (37) imply
We conclude by recalling that \(F(v'_h) \rightharpoonup \langle \mu _x,F\rangle \) in \(\mathrm {L}^1(\omega )\). \(\square \)
The lower bound. Let \(v \in \mathrm {L}^2(B_1;Z)\) and let \(\{v_\varepsilon \}\) be a sequence in \(\mathrm {L}^2(B_1;Z)\) such that \(v_\varepsilon \rightharpoonup v\) in \(\mathrm {L}^2(B_1;Z)\). We want to prove that
Notice that, we may reduce the proof to the case where \(\mathscr {B}v_\varepsilon = 0\) for every \(\varepsilon \). From the inequality \(\sigma _A \ge W \ge Q_{\mathscr {B}}W\) (as quadratic forms), we infer that
Next, we recall that \(v \mapsto \int _{B_1} Q_{\mathscr {B}}W(v)\) is lower semi-continuous in \(\{v \in \mathrm {L}^2(\Omega ;Z) : \mathscr {B}v = 0 \}\) with respect to the weak-\(\mathrm {L}^2\) topology. Hence,
This proves the lower bound inequality.
The upper bound. We fix \(v \in \mathrm {L}^2(B_1;Z)\), we want to show that there exists a sequence \(\{v_\varepsilon \}\) in \(\mathrm {L}^2(B_1;Z)\) with \(v_\varepsilon \rightharpoonup v\) in \(\mathrm {L}^2(B_1;Z)\) and such that
We may assume that \(\mathscr {B}v = 0\), for otherwise the inequality occurs trivially. Lemma 4.3 guarantees the existence of a 2-equi-integrable sequence \(\{v_h\}_{h = 1}^\infty \) for which
Next, we define an h-parametrized sequence of subsets of \(B_1\) in the following way:
Using the fact that smooth sets are dense in the broader class of subsets with respect to measure convergence, we may take a smooth set \(A'_h \subset B_1\) such that the following estimates hold for some strictly monotone function \(L: \mathbb N \rightarrow \mathbb N\) (with \(\lim _{h \rightarrow \infty }L(h) = \infty \)):
Observe that, by the 2-equi-integrability of \(\{v_h \}\), one gets that
The next step relies, essentially, on stretching the sequence \(\{v_h \}\). Define the \(\varepsilon \)-sequence
where \(K: \mathbb {R}_+ \rightarrow \mathbb N\) is the piecewise constant decreasing function defined as
Claim
-
1.
\(L \circ K (\varepsilon ) \le \varepsilon ^{-1}\), if \(\varepsilon \in (0,L(1)^{-1}]\).
-
2.
\(K (\varepsilon ) = h\), where h is such that \(\varepsilon \in R_h\).
Proof
To prove 1, observe from the strict monotonicity of L that \(\cup _{h = 1}^\infty R_h = (0, L(1)^{-1}]\). A simple calculation gives
where \(h_0\) is such that \(\varepsilon \in R_{h_0}\). The proof of 2 is an easy consequence of the definition of K and the fact that \(\{R_{h}\}\) is a disjoint family of sets. Indeed, if \(\varepsilon \in R_h\) then \(K(\varepsilon ) = h \cdot \mathbbm {1}_{R_h}(\varepsilon ) = h\). \(\square \)
Since K is a piecewise decreasing function and \(K(\mathbb {R}_+) = \mathbb N \cup \{0\}\), it remains true that
We are now in position to calculate the \(\limsup \) inequality:
Hence, by (38)
This proves the upper bound inequality. \(\square \)
Corollary 4.4
Let \(\{w_\varepsilon \} \subset \mathrm {W}^\mathscr {A}(B_1)\) be a sequence of almost local minimizers of the sequence of functionals
Assume that \(\{\mathscr {A}w_\varepsilon \}\) is 2-equi-integrable in \(B_s\) for every \(s < 1\). Assume also that there exists \(w \in \mathrm {W}^\mathscr {A}(B_1)\) such that
Then,
Moreover, w is a local minimizer of \(u \mapsto G(\mathscr {A}u)\).
Proof
The first step is to check that
The sequence \(\mathscr {A}w_\varepsilon \) generates (up to taking a subsequence) a \(\mathscr {B}\)-free Young measure \(\mu : B_1 \rightarrow \mathscr {M}(Z)\) so that by Theorem 2.10, Theorem 2.12 and the local 2-equi-integrability assumption,
Fix \(s \in (0,1)\) and consider the rescaled functions
It is not hard to see that, because of the (almost) minimization properties of \(\{w_\varepsilon \}\), the rescaled sequence \(\{w_\varepsilon ^s\}\) is also a sequence of almost local minimizers of the sequence of functionals \(\{u \mapsto G(\mathscr {A}u)\}.\) Footnote 8 Moreover, \(\mathscr {A}w^s_\varepsilon \rightharpoonup \mathscr {A}w^s\) in \(\mathrm {L}^2(B_1;Z)\).
From the proof of the lower bound in Theorem 4.2, we may find a 2-equi-integrable recovery sequence \(\{v_\varepsilon '\}\) for v, i.e., such that \(v_\varepsilon ' \rightharpoonup \mathscr {A}w^s\) and
Recall that, by the exactness assumption of \(\mathscr {A}\) and \(\mathscr {B}\), there are functions \(w'_\varepsilon \in \mathrm {W}^\mathscr {A}(B_1)\) such that
A recovery sequence with the same boundary values. The next step is to show that one may assume, without loss of generality, that \({{\mathrm{spt}}}(w_\varepsilon ' - w_\varepsilon ^s) \subset \subset B_1\).
We may further assume (without loss of generality) that \(\{w^s_\varepsilon \}\) and \(\{w_\varepsilon '\}\) are \(\mathrm {W}^{k,2}\)-uniformly bounded, and that \(w^s_\varepsilon - w_\varepsilon ' \rightharpoonup 0\) in \(\mathrm {W}^{k,2}(B_1;\mathbb {R}^{d})\).
Define
where, for every \(h \in \mathbb N\), \(\varphi _h \in \mathrm {C}^\infty (B_1;[0,1])\) with \(\varphi _h \equiv 1\) in \(B_{1 - 1/h}\). Since \(\Vert g(h)\Vert _{\mathrm {L}^2(B_1)} \rightarrow 0\) as \(\varepsilon \rightarrow 0\), we infer that
We now let \(h \rightarrow \infty \) and use the 2-equi-integrability of \(\{\mathscr {A}w^s_\varepsilon \}\) and \(\{\mathscr {A}w_\varepsilon '\}\) to get
Thus, we may find a diagonal sequence \(\tilde{v}_\varepsilon = \tilde{v}_{h(\varepsilon ),\varepsilon } = \mathscr {A}\tilde{w}^s_\varepsilon \) which is 2-equi-integrable, \({{\mathrm{spt}}}(w_\varepsilon ^s - \tilde{w}_\varepsilon ) \subset \subset B_1\), and such that
In particular, the (almost) local minimizing property of \(\{\mathscr {A}w^s_\varepsilon \}\) gives
Rescaling back, the inequality above yields
which together with (43) proves (42).
Local minimizer of G. The second step is to show that w is a local minimizer of \(u \mapsto G(\mathscr {A}u)\). We argue by contradiction: assume that w is not a local minimizer of \(u \mapsto G(\mathscr {A}u)\), then we would find \(s \in (0,1)\) and \(\eta \in \mathrm {C}^\infty _c(B_s;\mathbb {R}^{dN^k})\) for which
Again, using a re-scaling argument, this would imply that
Similarly to the previous step, we can find a 2-equi-integrable recovery sequence \(\{\mathscr {A}(\phi ^s_\varepsilon + \eta ^s)\}\) of \((\mathscr {A}w^s + \mathscr {A}\eta ^s)\) with the property that \({{\mathrm{spt}}}(\phi ^s_\varepsilon - w^s_\varepsilon ) \subset \subset B_1\), for every \(\varepsilon > 0\). On the other hand, the (almost) minimizing property of \(\mathscr {A}w^s_\varepsilon \) and (42) yield
which is a contradiction. This shows that w is a local minimizer of \(u \mapsto G(\mathscr {A}u)\). \(\square \)
Let us recall, for the proof of the next proposition, that the higher integrability assumption (Reg) on local minimizers \(\tilde{u}\) of \(u \mapsto G(\mathscr {A}u)\) reads:
Proposition 4.5
Let (w, A) be a saddle-point of problem (P). Assume that the higher integrability condition (Reg) holds for local minimizers of \(u \mapsto G(\mathscr {A}u)\). Then, for every \(K \subset \subset \Omega \) there exists a positive constant \(C(K) > 1\) and a smallness constant \(\rho \in (0,1/2)\) such that at least one of the following properties
-
1.
\(J_{B_r(x)}(\mathscr {A}w,A) \le C(K)r^{N-1}, \)
-
2.
\(J_{B_{\rho r}(x)}(\mathscr {A}w,A) \le \rho ^{N - (1 + \delta )/2} J_{B_r(x)}(\mathscr {A}w,A),\)
holds for all \(x \in K\) and every \(r \in (0,\mathrm{dist}(K,\partial \Omega ))\). Here,
Proof
Let (w, A) be a saddle-point of (P) and fix \(\rho \in (0,1)\) (to be specified later in the proof). We argue by contradiction through a blow-up technique: Negation of the statement would allow us to find a sequence \(\{(x_h,r_h)\}\) of points \(x_h \in K\) and positive radii \(r_h \downarrow 0\) for which
An equivalent variational problem. It will be convenient to work with a similar variational problem: Consider the saddle-point problem
where
Here we recall the notation \(\tau _A :=\sigma _A\mathscr {A}w_A\), where \(w_A \in \mathrm {W}^\mathscr {A}_0(\Omega )\) is the unique maximizer of \(u \mapsto I_\Omega (u,A)\). It follows immediately from the identity
that saddle-points (w, A) of problem (P) are also saddle-points of (\(\tilde{\mathrm{P}}\)) and vice versa; hence, in the following we will make no distinction between saddle-points of (P) and (\(\tilde{\mathrm{P}}\)). A special property of \(\tilde{I}\) is that, its density is always positive on saddle-points (w, A) of (P). Indeed, in this case \(w = w_A\) and therefore
A re-scaling argument. We re-scale and translate \(B_r(x)\) into \(B_1\) by letting
A further normalization on the sequence takes place by setting
and defining
It is easy to check that the scaling rule (33), and the relations (45) and (46) imply
In particular, due to the coercivity of \(\sigma _1\) and \(\sigma _2\), the norms \(\Vert \mathscr {A}w_{\varepsilon (h)}\Vert ^2_{\mathrm {L}^2(B_1)}\) are h-uniformly bounded by M.
Local almost-minimizers of \(G^{\varepsilon (h)}\). The next step is to show that \(\{w_{\varepsilon (h)}\}\) is \(O(\varepsilon )\)-close in \(\mathrm {L}^2\) to a sequence \(\{\tilde{w}_\varepsilon \}\) of almost minimizers of \(\{u \mapsto G^{\varepsilon (h)}(\mathscr {A}u)\}\). Observe that \(w_{\varepsilon (h)}\) is the unique solution to the equation
Let \(\tilde{w}_{\varepsilon (h)}\) be the unique minimizer of \(u \mapsto J^{\varepsilon (h)}(\mathscr {A}u,A_{\varepsilon (h)})\) – see (32) – in the affine space \(W^{\mathscr {A}}_{w_{\varepsilon (h)}}(B_1)\). Thus, in particular, \(\tilde{w}_{\varepsilon (h)}\) is the unique solution of the equation
A simple integration by parts, considering that \(\tilde{w}_{\varepsilon (h)} - w_{\varepsilon (h)} \in \mathrm {W}^\mathscr {A}_0(B_1)\), gives the estimate
where \(C(B_1)\) is the Poincaré constant from (12); and therefore \(\Vert w_{\varepsilon (h)} - \tilde{w}_{\varepsilon (h)}\Vert _{\mathrm {W}^{k,2}_0(B_1)} = O(h^{-1})\).
Lastly, we use strongly the fact that (w, A) is a saddle-point of (P) to see that \(\{(w_{\varepsilon (h)},A_{\varepsilon (h)})\}\) is also a local saddle-point of the energy
Moreover, by (33), (46) and (50) one has that
An immediate consequence of the two facts above is that \(\{\tilde{w}_{\varepsilon (h)}\}\) is a sequence of local almost minimizers of the sequence of functionals \(\{u \mapsto G^{\varepsilon (h)}(\mathscr {A}u)\}\). The local (almost) minimizing properties of the sequence \(\{\tilde{w}_{\varepsilon (h)}\}\) – with respect to the functionals \(\{u \mapsto G^{\varepsilon (h)}(\mathscr {A}u)\}\) – are not affected by subtracting \(\mathscr {A}\)-free fields; hence, using the compactness assumption of \(\mathscr {A}\) once more, we may assume without loss of generality that \(\sup _h \Vert \tilde{w}_{\varepsilon (h)}\Vert _{\mathrm {W}^{k,2}(B_1)} < \infty \). Upon passing to a further subsequence, we may also assume that there exists \(\tilde{w} \in \mathrm {W}^{k,2}(B_1)\) such that
Equi-integrability of \(\{\mathscr {A}\tilde{w}_{\varepsilon (h)}\}\). The last but one step is to show that \(\{\mathscr {A}\tilde{w}_\varepsilon \}\) is a 2-equi-integrable sequence in \(B_s\), for every \(s < 1\).
Since \(\sigma _{A_\varepsilon }\) is uniformly bounded, there exists \(\tilde{\tau }\in \mathrm {L}^2(B_1;\mathbb {R}^{dN^k})\) such that (upon passing to a further subsequence)
Let \(\varphi \in \mathscr {D}(B_1)\) and fix \(\varepsilon > 0\), integration by parts yields
Since the term in the right hand side of the equality depends only on \(\nabla ^{k-1} \tilde{w}_{\varepsilon (h)}\), the strong convergence \(\tilde{w}_\varepsilon \rightarrow \tilde{w}\) in \(\mathrm {W}^{k-1,2}(B_1;\mathbb {R}^d)\) gives
Therefore,
The positivity of \(\sigma _{A_\varepsilon }\mathscr {A} \tilde{w}_\varepsilon \cdot \mathscr {A} \tilde{w}_\varepsilon \), the Dunford-Pettis Theorem and the convergence above imply that the sequence
In turn, due to the uniform coerciveness and boundedness of \(\{\sigma _{A_\varepsilon }\}\), both sequences \(\{\mathscr {A} \tilde{w}_\varepsilon \}\) and \(\{\tilde{\tau }_\varepsilon \}\) are 2-equi-integrable in \(B_s\); for every \(s < 1\).
The contradiction. We are in position to apply Proposition 4.4 to the sequence \(\{\tilde{w}_\varepsilon \}\), which in particular implies
and that w is a local minimizer of \(u \mapsto G(\mathscr {A}u)\). On the other hand, the higher integrability assumption (Reg) tells us that
We set the value of \(\rho \in (0,1/2)\) to be such that \(2 c M^2 \rho ^{(1 - \delta )/2} \le 1\). Taking the limit in (48) and (49), using Fatou’s Lemma, (50), (51), (53) and (54), we get
a contradiction. \(\square \)
Theorem 1.2 (Upper bound) Let (w, A) be a variational solution of problem (P). Assume that the higher integrability condition
holds for local minimizers of the energy \(u \mapsto \int _{B_1} Q_{\mathscr {B}} W(\mathscr {A}u)\), where \(u \in \mathrm {W}^\mathscr {A}(B_1)\). Then, for every compactly contained set \(K \subset \subset \Omega \), there exists a positive constant \( \Lambda _K\) such that
Proof
Let \(x \in K\), and set
where we recall that
Proposition 4.5 tells us that there exists a positive constant \(\rho \in (0,1/2)\) such that if \(B_r(x) \subset \Omega \), then
An application of the Iteration Lemma (stated below) to \(r \in (0, \min \{1,\text {dist}(K,\partial \Omega \})\), and \(\alpha _1 :=N - (1 + \delta )/2 > \alpha _2 :=N-1\) yields the existence of positive constants \(c = c(x)\), and \(r = r(K)\) such that
Notice that the constants c and r depend continuously on \(x \in \Omega \). Hence, for any \(K \subset \subset \Omega \) we may find \(\Lambda _K > 0\) for which
\(\square \)
Lemma 4.6
(Iteration Lemma [30, Lemma 2.1, Chapter III]) Assume that \(\varphi (\rho )\) is a non-negative, real-valued, non-decreasing function defined on the (0, 1) interval. Assume further that there exists a number \(\tau \in (0, 1)\) such that for all \(r < 1\) we have
for some non-negative constant C, and positive exponents \(\alpha _1 > \alpha _2\). Then there exists a positive constant \(c=c(\tau ,\alpha _1,\alpha _2)\) such that for all \(0\le \rho \le r\le R\) we have
Corollary 4.7
(Compactness of blow-up sequences) Let (w, A) be a variational solution of problem (P). Under the assumptions of the Upper bound Theorem 1.2, there exists a positive constant \(C_K\) such that
Proof
The assertion follows directly from the Upper bound Theorem and the coercivity of \(\sigma _1\) and \(\sigma _2\). \(\square \)
5 The Lower bound: proof of estimate (LB)
During this section we will write (w, A) to denote a solution of problem (P) under the assumptions of Theorem 1.2. In light of the results obtained in the previous section we will assume, throughout the rest of the paper, that for every compact set \(K \subset \subset \Omega \) there exist positive constants \(C_K\), and \(\Lambda _K\) such that
for all \(x \in K\) and every \(r \in (0,\text {dist}(K,\partial \Omega ))\). Here, \(w^{x,r} :=w(x + ry)/r^{k - \frac{1}{2}}\).
The main result of this section is a lower bound on the density of the perimeter in \(\partial ^* A\). In other words, there exists a positive constant \(\lambda _K = \lambda _K(N,M)\) such that
There are two major consequences from estimate (LB). The first one (cf. Corollary 5.8) is that the difference between the topological boundary of A and the reduced boundary of A is at most a set of zero \(\mathscr {H}^{N-1}\)-measure. In other words, \((\partial A {\setminus } \partial ^* A) = \Sigma \) where \(\mathscr {H}^{N-1}(\Sigma ) = 0\) (cf. [7, Theorem 2.2]). The second is that (LB) is a necessary assumption for the Height bound Lemma and the Lipschitz approximation Lemma, which are essential tools to prove the flatness excess improvement in the next section.
Throughout this section and the rest of the manuscript we will constantly use the following notations:
The scaled Dirichlet energy
and the excess for \(\gamma \)-weighted energy
Granted that the spatial-, radius-, or (w, A)- dependence is clear, we will shorten the notations to the only relevant variables, e.g., D(r) and \(E_{\gamma }(r)\). Recall that, up to translation and re-scaling, we may assume
Bear also in mind that all the constants in this section are universal up to their dependence on \(\Lambda _K\) and \(C_K\).
We will proceed as follows. First we prove in Lemma 5.1 that if the density of the perimeter is sufficiently small, one may regard the regularity properties of solutions as those ones for an elliptic equation with constant coefficients. Then, in Lemma 5.2, we prove a lower bound on the decay of the density of the perimeter in terms of D. Combining these results, we are able to show a discrete monotonicity formula on the decay of \(E_{\gamma }\).
The proof of the Lower density bound (LB) follows easily from this discrete monotonicity formula, De Giorgi’s Structure Theorem, and the Upper bound Theorem of the previous section. Finally, we prove that the difference between \(\partial A\) and \(\partial ^* A\) is \(\mathscr {H}^{N-1}\)-negligible (Theorem 5.8) as a corollary of the estimate (LB).
Lemma 5.1
(Approximative solutions of the constant coefficient problem) For every \(\theta _1 \in (0,1/2)\), there exist positive constantsFootnote 9 \(c_1(\theta _1,N,M)\) and \(\varepsilon _1(\theta _1,N,M)\) such that either
or
where \(c = c(N,M)\) is the constant from Lemma 2.6; whenever
Proof
Since \(c \ge 2^N\), the result holds if we assume \(\rho \ge 1/2\), therefore we focus only on the case where \(\rho \in (\theta _1,1/2]\). Fix \(\theta _1 \in (0,1/2)\). We argue by contradiction: We would find a sequence of pairs \((w_h,A_h)\) (locally solving (P) in \(B_1\) for a source function \(f_h\)) and constants \(\rho _h \in [\theta _1,1/2]\), such that
and simultaneously
The estimate above yields \(\delta _h^{-1} f_h \rightarrow 0\) in \(\mathrm {L}^2(B_1;\mathbb {R}^d)\). Also, since \(\mathrm {Per}(A_h;B_1) \rightarrow 0\), the isoperimetric inequality yields that either \(\sigma _{A_h} \rightarrow \sigma _1\) or \(\sigma _{A_h} \rightarrow \sigma _2\) in \(\mathrm {L}^2\) as h tends to infinity. Let us assume that the former convergence \(\sigma _{A_h} \rightarrow \sigma _1\) holds.
Let \(u_h :=\delta _h^{-1} w_h\), and observe that
We use that \(w_h\) is a (local) solution to (P) for \(A_h\) as indicator set and \(f_h\) as source term, to see that
Up to modifying the sequence by \(\mathscr {A}\)-free fields and passing to a further subsequence, we may assume that \(u_h \rightharpoonup u\) in \(\mathrm {W}^{k,2}(B_1;\mathbb {R}^{dN^k})\). We may then apply the compensated compactness result from Lemma 2.9 to obtain that
and
Hence, by (57) and Fatou’s Lemma one gets
This is a contradiction to Lemma 2.6 because u is a solution for the problem with constant coefficients \(\sigma _1\). The case when \(\sigma _{A_h} \rightarrow \sigma _2\) can be solved by similar arguments. \(\square \)
The next lemma is the principal ingredient in proving the (LB) estimate. It relies on a cone-like comparison to show that the decay of the perimeter density is controlled by D(r) / r: The perimeter density cannot blow-up at smaller scales, while for a fixed scale, the perimeter density is small.
Lemma 5.2
(Universal comparison decay) There exists a positive constantFootnote 10 \(c_2 = c_2(N,M)\) such that
Proof
For a.e. \(r \in (0,1)\) the slice \(\langle A,g ,r \rangle \), where \(g(x) = | x |\), is well defined (see Sect. 2.4). Fix one such r and let \(\tilde{A}\) be the cone-like comparison set to A as in (23). By minimality of (w, A) and a duality argument, we get
for \(\tau _A = \sigma _A\mathscr {A}w\). Hence,
To reach the inequality in the last row we have used that the cone extension \(\tilde{A}\) is precisely built (cf. (24)) so that the Green-Gauss measures \(\mu _{\tilde{A}}\) and \( \mu _{A}\) agree in \((B_r)^c\); where, by (22),
We know from (25) that \(\frac{\,\mathrm {d}}{\,\mathrm {d}\rho }\big |_r\mathrm {Per}(A;B_\rho ) \ge \langle A,g,r \rangle (\mathbb R^N)\) for a.e. \(r > 0\). Since (58) and the previous inequality are valid almost everywhere in (0, 1), a combination of these arguments yields
The result follows for \(c_2 := M^3(N-1)\). \(\square \)
The following result is a discrete monotonicity for the weighted excess energy \(E_\gamma \). We remark that, in general, a monotonicity formula may not be expected in the case of systems.
Theorem 5.3
(Discrete monotonicity) There exist positive constants \(\gamma = \gamma (N,M)\), \(\varepsilon _2 = \varepsilon _2(\gamma ,N) \le \mathrm {vol}(B_1') \cdot \gamma /2\), and \(\theta _2 = \theta _2(N,M) \in (0,1/2)\) such that
Proof
We fix \(\gamma \) and \(\theta _1\) such that
Set \(\theta _2 := \theta _1\). Recall that \(c_2\) is the constant from Lemma 5.2, and c is the constant of Lemma 2.6.
Let also \(\varepsilon _2 = \varepsilon _2(\gamma ,\varepsilon _1)\) be a positive constant with \(\varepsilon _2 \le \min \{\gamma \varepsilon _1(\theta _2), \gamma \cdot \text {vol }(B_1') /2\}\). This implies
which in turn gives, for \(c_1 = c_1(\theta _2)\),
Now, we apply Lemmas 5.1 and 5.2 to \(s \in (\theta _2,1)\) to get
This proves the desired result. \(\square \)
Lemma 5.4
For every \(\varepsilon > 0\), there exist positive constants \(\theta _0(N,M,K,\varepsilon ) \in (0,1/2)\) and \(\kappa (N,M,K,\varepsilon ) > 0\) such that
whenever
Proof
The result follows by taking \(\theta _0\) such that \(2c\theta _0 C_K \le \varepsilon /2\) (recall that, \(D(s) \le C_K\) for every \(s \in (0,1)\)) and \(\kappa \le \min \bigg \{\frac{\varepsilon \theta ^{N-1}_0}{2\gamma },\varepsilon _1(\theta _0)\bigg \}\) and then simply applying Lemma 5.1. \(\square \)
Lemma 5.5
Let (w, A) be a saddle-point of (P) and let \(x \in K \subset \subset \Omega \). Then, for every \(\varepsilon > 0\) there exists a positive radius \(r_0 = r_0(N,M,K,\Vert f\Vert _{\mathrm {L}^\infty (B_1)},\varepsilon )\) for which
whenever \(r \le r_0\) and \(\mathrm {Per}\big (A;B_{\theta _0^{-1}r}\big ) \le \kappa (\varepsilon ) \cdot \big (\frac{r}{\theta }\big )^{N-1}\).
Proof
Let \(r_0\) be a positive constant such that \(c_1 r_0^{2k+1}\Vert f\Vert _{\mathrm {L}^\infty (B_1)}^2 \le \theta _0^{2k +1}\varepsilon \) and let us set \(s :=\theta _0^{-1}r\). Since
it follows from the previous lemma and a rescaling argument that
\(\square \)
Theorem 5.6
(Lower bound) Let (w, A) be a solution of problem (P) in \(\Omega \). Let \(K \subset \subset \Omega \) be a compact subset. Then, there exist positive constants \(\lambda _K\) and \(r_K\) depending only on K, the dimension N, the constant M in the assumption (2), and f such that
for every \(r \in (0,r_K)\) and every \(x \in \partial ^*A \cap K\).
Proof
Let \(p(\theta _2) :=\sum _{h = 0}^\infty \theta _2^{(2k + 1)h} \in \mathbb {R}\) and let \(r_1 \in (0,1)\) be a positive constant for which
We argue by contradiction. If the assertion does not hold, we would be able to find a point \(x \in \partial ^*A\) and a radius \(r \le \min \{r_0,r_1\}\) for which
After translation, we may assume that \(x =0\). The fact that \(r \le r_0\) and Lemma 5.5 yield the estimate
in return, Lemma 5.3 and a rescaling argument give (recall that \(f^r(y) = r^{k + \frac{1}{2}}f(ry)\))
A recursion of the same argument gives the estimate
Taking the limit as \(j \rightarrow \infty \) we get
This a contradiction to the fact that \(x = 0 \in \partial ^*A\) (cf. Sect. 2.4). \(\square \)
Corollary 5.7
Let (w, A) be a solution for problem (P) in \(\Omega \). Let \(K \subset \subset \Omega \) be a compact subset. Then, there exist positive constants \(\lambda _K\) and \(r_K\) depending only on K, the dimension N, and f such that
for every \(r \in (0,r_K)\) and for every \(x \in \partial A \cap K\).
Proof
The property (LB) from the Lower bound theorem is a topologically closed property, i.e., it extends to \(\overline{\partial ^*A} = {{\mathrm{spt}}}\mu _A = \partial A\) (cf. (19)). \(\square \)
Corollary 5.8
Under the same assumptions of Theorem 5.6, the following characterization for the topological boundary of A holds:
Proof
An immediate consequence of the previous corollary is that \(\mathscr {H}^{N-1}\llcorner \partial A \ll |\mu _A|\) as measures in \(\Omega \). The assertion follows by De Giorgi’s Structure Theorem. \(\square \)
6 Proof of Theorem 1.5
As we have established in the past section, we will assume that for every \(K\subset \subset \Omega \) there exist positive constants \(\lambda _K,C_K\) such that \(D(w;x,r) \le C_K\) and
Half-space regularity. Throughout this section we shall work with the additional assumption for solutions of the half-space problem: let \(H :=\{\;x \in \mathbb {R}^N \; : \; x_N > 0\;\}\) and let \(\sigma _H\) be the two-point valued tensor defined in (3) for \(\Omega = B_1\) (so that \(\sigma _ H = \sigma _1\) in \(H \cap B_1\)), then the operator
is hypoelliptic in \(B_1 {\setminus } \partial H\) in the sense that, if \(w \in \mathrm {L}^2(B_1;\mathbb {R}^d)\), thenFootnote 11
Furthermore, there exists a positive constant \(c^* = c^*(N,M,\mathscr {A})\) such that
Remark 6.1
(Half-space regularity in applications) For 1-st order operators of gradient form it is relatively simple to show that such estimates as in (61) hold. This case includes gradients and symmetrized gradients; while the linear plate equations may be also reduced to this case (cf. Remark 2.8).
A sketch of the proof is as follows: the first step is to observe that the tangential derivatives (\(i \ne N\)) \(\partial _i w\) of a solution w of \(P_H u = 0\) are also solutions of \(P_H u = 0\). The second step is to repeat recursively the previous step and use the Caccioppoli inequality from Lemma 2.5 to estimate
The third step consists in using the ellipticity of \(A_N = \mathbb A(\mathbf e_N)\) (cf. Remark 2.3) and the equation to express \(\partial _{NN} w\) in terms of the rest of derivativesFootnote 12: The tensor \((A^T_N \,\sigma \, A_N)\) is invertible, this can be seen from the inequality \(|\mathbb A(\mathbf e_N) z|^2 \ge \lambda (\mathscr {A})|z|^2\) for every \(z \in \mathbb {R}^d\) (cf. 2.3) and the fact that \(\sigma _H\) satisfies Gårding’s strong inequality (2) with \(M^{-1}\). Hence, using that \(P_H w = 0\), we may write
from which estimates for \(\partial _{NN} w\) of the form (62) in the upper half ball easily follow (similarly for the lower half ball). Further \(\partial _N\) differentiation of the equation in \(B^\pm _1\) and iteration of this procedure together with the Sobolev embedding yield bounds as in (61).
For arbitrary higher-order gradients and other general elliptic systems one cannot rely on the same method. However, the Schauder and \(\mathrm {L}^p\) boundary regularity of such systems has been systematically developed in [31, 32] through the so called complementing condition. In the case of strongly elliptic systems (cf. (2) and (11)) this complementing condition is fulfilled, see [32, pp 43-44]; see also [33] where a closely related natural notion of hypoellipticity of the half-space problem is assumed.
Flatness excess. Given a set \(A \subset \mathbb {R}^N\) of locally finite perimeter, the flatness excess of A at x for scale r and with respect to the direction \(\nu \in \mathbb S^{n-1}\), is defined as
Here, \(C(x,r,\nu )\) denotes for the cylinder centered at x with height 2, that is parallel to \(\nu \), of radius r.
Intuitively, the flatness excess expresses (for a set A) the deviation from being a hyperplane at a given scale r. Again, up to re-scaling, translating and rotating, it will be enough to work the case \( x = 0, \nu = \mathbf e_N\), and \(r = 1\). In this case, we will simply write \(e\mathrm {(A)}\). The hyper-plane energy excess is defined as
and as long as its dependencies are understood we will simply write \(H_{\text {ex}}(r) = e(r) + D(r)\).
The following result relies on the (LB) property, a proof can be found in [24, Section 5.3] or [22, Theorem 22.8].
Lemma 6.2
(Height bound) There exist positive constants \(c_1^* = c_1^*(N)\) and \(\varepsilon ^*_1 =\varepsilon ^*_1(N)\) with the following property. If \(A \subset \mathbb {R}^N\) is a set of locally finite perimeter with the (LB) property,
then
The next decay lemma is the half-space problem analog of Lemma 5.1. The proof is similar except that it relies on the half-space regularity assumptions (60), (61) (instead of the ones given by Lemma 2.6), and the Height bound Lemma stated above.
Lemma 6.3
(Approximative solutions of the half-space problem) Let (w, A) be a solution of problem (P) in \(B_1\). Then, for every \(\theta _1^* \in (0,1/2)\) there exist positive constants \(c^*_2(\theta _1^*,N,M)\) and \(\varepsilon _2^*(\theta ^*_1,N,M)\) such that either
or
where \(c^* = c^*(N,M)\) is the constant from the regularity condition (61); whenever
\(\square \)
Remark 6.4
Let \(\delta \in (0,1)\). Then there exists \(\kappa ^* = \kappa ^*(N,M,\delta )\) such that if \(e(1) \le \kappa ^*\), and if one further assumes that the excess function \(r \mapsto e(r)\) is monotone increasing, then the scaling \(w(ry)/r^{(k - \frac{\delta }{2})}\) and the Iteration Lemma 4.6 imply that
for some positive constant \(C_\delta = C_\delta (N,M)\). \(\square \)
The next crucial result can be found in [8, Section 5]. We have decided not to include a proof because the ideas remain the same. The ingredients for the proof are: the estimate (LB), the Height bound Lemma, the Lipschitz approximation Theorem, the estimates from Lemma 6.3 and the higher integrability for solutions to elliptic equations.Footnote 13
Lemma 6.5
(Flatness excess improvement) Let (w, A) be a saddle point of problem (P) in \(\Omega \). There exist positive constants \(\eta \in (0,1]\), \(c^*_3\), and \(\varepsilon _3\) depending only on K, the dimension N, the constant M in (2), and \(\Vert f\Vert _{\mathrm {L}^\infty }\) with the following properties: If (w, A) is a saddle point of problem (P) in \(B_9\), and
then, for every \(r \in (0,9)\), there exists a direction \(\nu (r) \in \mathbb S^{N-1}\) for which
\(\square \)
Theorem 1.5 (Partial regularity) Let (w, A) be a saddle point of problem (P) in \(\Omega \). Assume that the operator \(P_Hu = \mathscr {A}^*(\sigma \mathscr {A}u)\) is hypoelliptic and regularizing as in (60), (61), and that the higher integrability condition
holds for every local minimizer \(\tilde{u}\) of the energy \(u \mapsto \int _{B_1} Q_{\mathscr {B}} W(\mathscr {A}u)\), where \(u \in \mathrm {W}^{\mathscr {A}}(B_1)\). Then there exists a positive constant \(\eta \in (0,1]\) depending only on N such that
Moreover if \(\mathscr {A}\) is a first-order differential operator, then \(\mathscr {A}w \in \mathrm {C}^{0,\eta /8}_\mathrm{{loc}}(\Omega {\setminus } (\partial A {\setminus } \partial ^* A))\); and hence, the trace of \(\mathscr {A}w\) exists on either side of \(\partial ^* A\).
Proof
The reduced boundary is an open hypersurface. The first assertion \(\mathscr {H}^{N-1}((\partial A {\setminus } \partial ^* A) \cap \Omega ) = 0\) is a direct consequence of Corollary 5.8.
To see that \(\partial ^* A\) is relatively open in \(\partial A\) we argue as follows: De Giorgi’s Structure Theorem guarantees that for every \(x \in \partial ^*A\) there exist \(r > 0\) (sufficiently small) and \(\nu \in \mathbb S^{N-1}\) such that
The map \(y \mapsto \mu _A(B_r(y)) = 0\) is continuous at x, therefore we may find \(\delta (x) \in (0,1)\) such that
We may then apply Lemma 6.5 to get an estimate of the form
This and the first assertion of Lemma 6.5 imply that \(y \in \partial ^* A\) for every \(y \in B_\delta (x) \cap \partial A\). Therefore, the reduced boundary \(\partial ^* A\) is a relatively open subset of the topological boundary \(\partial A\).
We proceed to prove the regularity for \(\partial ^* A\). It follows from the last equation that
for some constant \(C = C(C_{B_\delta (x)},\Lambda _{B_\delta (x)},N,M)\).
Through a simple comparison, we observe from (64) and the property that (w, A) is a local saddle point of problem (P) in \(B_\delta (x)\), that
We conclude with an application of Tamanini’s Theorem 2.14:
The assertion follows by observing that the regularity of \(\partial ^* A\) is a local property.
Jump conditions for the hyper-space problem. Let \(\tau \in \mathrm {L}_\mathrm{{loc}}^2(B_1;Z) \cap (\mathrm {C}^\infty (\overline{B_\rho ^+};Z) \cup \mathrm {C}^\infty (\overline{B_\rho ^-};Z))\) for every \(\rho \in (0,1)\), assume furthermore that \(\tau \) is a solution of the equation
Let \(\eta \in \mathrm {C}^{\infty }_c(B_1';\mathbb {R}^d)\) be an arbitrary test function and choose a function \(\varphi \in \mathrm {C}^{\infty }_c(B_1;\mathbb {R}^d)\) with the following property:
Then, integration by parts and Green’s Theorem yield that
where \([\mathbb A(\mathbf e_N)^T \cdot \tau ] = \mathbb A(\mathbf e_N)^T \cdot (\tau ^+ - \tau ^-)\). Here, \(\tau ^+\) and \(\tau ^-\) are the traces of \(\tau \) in \(\partial H\) from \(B_1^+\) and \(B_1^-\) respectively. Since \(\eta \) is arbitrary, a density argument shows that
Regularity of \(\mathscr {A}w\). From this point and until the end of the proof we will assume that \(\mathscr {A}\) is a first-order differential operator of gradient form; we may as well assume that \(\partial ^* A\) is locally parametrized by \(\mathrm {C}^{1,\eta /2}\) functions.
Due to Campanato’s Theorem (\(\mathrm {C}^{0,\eta /8} \simeq \mathrm {L}^{2,\,N + (\eta /4)}\) on Lipschitz domains), our goal is to show local boundedness of the map
and a similar result for \(A^c\) instead of A. \(\square \)
Also, since Campanato estimates in the interior are a simple consequence of Lemma 2.6, we may restrict our analysis to show only local boundedness at points \(x \in \partial ^* A\). We first prove the following decay for solutions of the half-space:
Lemma 6.6
Let \(\tilde{w} \in \mathrm {W}^\mathscr {A}(B_1)\) be such that
Then \(\tilde{w}\) satisfies an estimate of the form
for all \(0 < \rho \le 1\), where we have defined
Proof
Since for \(\rho \ge 1/2\) one can use \(c :=2^{{(N+2)}}\), we only focus on proving the estimate for \(\rho \in (0,1/2)\). It is easy to verify that \(\mathscr {A}^* (\sigma _H \mathscr {A}(\partial _i \tilde{w} - \lambda )) = 0\) in \(\mathscr {D}'(B_1;\mathbb {R}^d)\) for all \(\lambda \in \mathbb {R}^d\), and every \(i = 1,\dots ,{N-1}\). In particular, by (61) we know that
for every \(\rho \in (0,1/2)\), and every \(i = 1,\dots ,N - 1\). Here, \(C = C(N)\) is the standard scaled Poincaré constant for balls. Summation over \(i \in \{1,\dots ,N-1\}\) yields an estimate of the form (68) for \(\nabla ' \tilde{w}\).
We are left to calculate the decay estimate for \(g_H(\tilde{w}) :=A_N^T (\sigma _H \mathscr {A}\tilde{w}) = \mathbb A(\mathbf e_N) \cdot (\sigma _H \mathscr {A}\tilde{w})\). By the hypoellipticity assumption (60) and the jump condition (65), we infer that \(g_H(\tilde{w}) \in \mathrm {W}_\mathrm{{loc}}^{1,2}(B_1;\mathbb {R}^d)\).
Even more, by the classical Poincaré’s inequality
for every \(\rho \in (0,1/2)\). On the other hand, it follows from the equation in \((B_1 {\setminus } \partial H)\) and (63) that one may write \(\nabla g(\tilde{w})\) in terms of \(\nabla (\nabla ' \tilde{w})\) for almost every \(x \in (B_r {\setminus } \partial H)\). We may then find a constant \(C' = C'(\sigma _1,\sigma _2,\mathscr {A})\) such that
Using the same calculation as in the derivation of (69), it follows from (70) that
for every \(\rho \in (0,1/2)\). The assertion follows by letting \(c(N,\sigma _1,\sigma _2) :=c^*\, C\max \{1,C'\}\). \(\square \)
The next corollary can be inferred from (68) by following the strategy of Lin in [8, pp 166–167]:
Corollary 6.7
Let \(\tilde{w} \in \mathrm {W}^\mathscr {A}(B_2)\) solve the equation
where \(A :=\{\; x \in B'_2 \times \mathbb {R}\; : \; x_N > \varphi (x')\;\}\) for some function \(\varphi \in \mathrm {C}^{1,\eta /2}(B_2')\) with \(\varphi (0) = |\nabla \varphi |(0) = 0\), and \(\Vert \varphi \Vert _{\mathrm {C}^{1,\eta /2}(B_2')} \le 1\). Then there exist positive constants \(\theta (N,\sigma _1,\sigma _2)\in (0,1/2)\), and \(C(N,\sigma _1,\sigma _2)\) such that either
or
\(\square \)
We are now in the position to prove (66). Let \(\delta \in (0,\eta /2)\) and let (w, A) be solution of problem (P). Since local regularity properties of the pair (w, A) are inherited to any (possibly rotated and translated) re-scaled pair \((w^{x,r},A^{x,r})\) – as defined in (47), where in particular the source \(f^{x,r}\) tends to zero – with \(r \le \text {dist}(x,\partial \Omega )\), we may do the following assumptions without any loss of generality: \(B_4 \subset \Omega \) and \(x = 0 \in \partial ^* A\), \(\partial A^*\) is parametrized in \(B_{2}\) by a function \(\varphi \in \mathrm {C}^{1,\eta /2}(B_{2}')\) such that \(\varphi (0) = |\nabla \varphi (0)| = 0\), and \(\Vert \varphi \Vert _{\mathrm {C}^{1,\eta /2}(B_{2}')}, \Vert f\Vert _{\mathrm {L}^\infty (B_2;\mathbb {R}^d)} \le \min \{1,\kappa ^*\}\) where \(\kappa ^* = \kappa ^*(\delta ,N,M)\) is the constant of Remark 6.4. Additionally, since (w, A) is a solution of problem (P), we know that
and
where \(C_\delta (N,M)\) is the constant from Remark 6.4.
Notice that the rescaled functionsFootnote 14 \(w^r(y) :=(w(ry) - v_r(ry))/r^{1 - (\delta /2)}\) and \(\varphi ^r(y) :=\varphi (ry)/r\) still solve (74) for \(f^r(y) :=r^{1 + (\delta /2)} f(ry)\) and \(A^r :=A/r\) with \(\Vert \varphi ^r\Vert _{\mathrm {C}^{1,\eta /2}(B_{2}')}, \Vert f^r\Vert _{\mathrm {L}^\infty (B_2;\mathbb {R}^d)} \le \min \{1,\kappa ^*\}\). In particular, by (75) and Poincaré’s inequality
Recall that \(\Vert \varphi ^r\Vert _{\mathrm {C}^{1,\eta /2}(B_1')}\) scales as \(r^{\eta /2}\Vert \varphi \Vert _{\mathrm {C}^{1,\eta /2}(B_r')}\) and, in view of its definition, \(\Vert f^r\Vert ^2_{\mathrm {L}^\infty (B_1)}\) scales as \(r^{2 + \delta }\). In view of these properties, we are in position to apply Corollary 6.7 to \(w^r/\max \{1,\overline{C}^{1/2}\}\): We infer that either
or
where \(\theta = \theta (N,\sigma _1,\sigma _2) \in (0,1/2)\) is the constant from Corollary 6.7.
It is not difficult to verify, with the aid of the Iteration Lemma 4.6, that re-scaling in (76) and (77) conveys a decay of the form
and some constant \(c' = c'(\delta ,N, \sigma _1,\sigma _2,\Vert \mathscr {A}w\Vert _{\mathrm {L}^2(B_2)}\)).
The last step of the proof consists in showing that \(R_A (w - \nu _r)\) dominates \(\nabla (w - \nu _r)\). By the definition of \(R_A\), it is clear that \(|\nabla ' (w - \nu _r)(x) - (\nabla ' (w - \nu _r))_{B_r \cap A}|^2\le |R_A (w - \nu _r)(x) - (R_A (w - \nu _r))_{B_r \cap A}|^2\) for all \(x \in B_1\) and every \(r \in (0,1)\). We show a similar estimate for \(\partial _N (w - \nu _r)\):
The pointwise Gårding inequality (2) and (11) imply, in particular, that the tensor \((\mathbb A(\mathbf e_N)^T \sigma _1 \; \mathbb A(\mathbf e_N)) = (A_N^T \sigma _1 A_N) \in \text {Lin }(\mathbb {R}^d;\mathbb {R}^d)\) is invertible (use, e.g., Lax-Milgram in \(\mathbb {R}^d\)). Hence,
from where we deduce that
for some constant \(c'' = c''(\sigma _1,\mathscr {A}) \ge 1\) bounding the right hand side of (79) in terms of \(\nabla '(w - v_r)\) and \(g(w - v_r)\).
By (78) and the estimate above we obtain
for every \(r \in (0,1)\). The assertion follows by taking \(\delta = \eta /4\).
Notice that the dependence on \(\Vert \mathscr {A}w \Vert _{\mathrm {L}^2(B_2)}\) is local since we assumed \(B_4 \subset \Omega \); this means that in general we may not expect a uniform boundedness of the decay. Similar bounds for A replaced by \(A^c\) can be derived by the same method. \(\square \)
Remark 6.8
(Regularity I) In general, for a k’-th order operator \(\mathscr {A}\) of gradient form, the only feature required to prove the regularity of \(\nabla ^k w\) up to the boundary \(\partial ^*A\) by the same methods as for first-order operators of gradient form is to obtain an analog of Lemma 6.6 (and its Corollary 6.7) for higher-order operators.
More specifically, if \(\tilde{w} \in \mathrm {W}^\mathscr {A}(B_1)\) is a solution of the equation
then \(\tilde{w}\) satisfies an estimate of the form
for all \(0 < \rho \le 1\),
where
Unfortunately, for 2k’-th order systems of elliptic equations (with \(k> 1\)) it is not clear to us whether one can prove such decay estimates by standard methods. While a decay estimate for \(\nabla ^{k-1}(\nabla ' u)\) can be shown by the very same method as the one in the proof of Theorem 1.5, the main problem centers in proving a decay estimate for the term \(\mathbb A(\mathbf e_N)^T (\sigma \mathscr {A}u) \in \mathrm {W}^{1,2}(B_1)\) – cf. (65). Technically, the issue is that one cannot use the equation on half-balls to describe \(\partial ^{(0,\dots ,0,k)} u\) in terms of \(\nabla ^{k-1}(\nabla ' u)\).
Remark 6.9
(Regularity II: linear plate theory) In the particular case of models in linear plate theory (\(\mathscr {A}= \nabla ^2, N= 2\), and \(d = 1\)) it is possible to show a decay estimate as in (80) for solutions \(w \in \mathrm {W}^{2,2}_0(B_2)\) of the equation
By Remark 2.8, there exists a field \(w \in W^{1,2}(B_2;\mathbb {R}^2)\) which turns out to be a solution of the equation
where \(\mathbf S\) is a positive fourth-order symmetric tensor such that \(\sigma _H(x) = \mathbf R_\perp \mathbf S_H^{-1}(x) \mathbf R_\perp \); furthermore, \(\mathbf R_\perp \mathscr {E} w =\mathbf \sigma _H \nabla ^2 u\). Since \(\mathscr {A}= \nabla ^2\), it is easy to verify that \(A_\alpha = A_{(i,j)} = \mathbf e_i \otimes \mathbf e_j\) for \(i,j \in \{1,2\}\), a simple calculation shows that
and thus, since \(\mathscr {E}\) is an operator of gradient form of order one, it follows form the proof of Theorem 1.5 that an estimate of the form (80) indeed holds for \(g_H(u)\).
Notes
Here, \(\mathrm {Per}(A;\overline{\Omega }) = |\mu _A|(\overline{\Omega })\), where \(\mu _A\) is the Gauss–Green measure of A; see Sect. 2.4.
Due to the nature of the problem, we cannot replace \(\mathrm {Per}(A;\overline{\Omega })\) with \(\mathrm {Per}(A;\Omega )\) in E(A) because it possible that minimizing sequences tend to accumulate perimeter in \(\partial \Omega \).
Here, \(\mathrm {W}^\mathscr {A}(\Omega ) = \big \{u \in \mathrm {L}^2(\Omega ;\mathbb {R}^d) : \mathscr {A}u \in \mathrm {L}^2(\Omega ;\mathbb {R}^{dN^k}) \big \}\) is the \(\mathscr {A}\)-Sobolev space of \(\Omega \).
Here, \(\mathscr {B}\) is a second order operator expressing the Saint-Venant compatibility conditions.
As stated in Sect. 2.4, we write \(A \in \mathrm {BV}_\mathrm{{loc }}(\mathbb {R}^N)\) to express that A is a Borel set of locally finite perimeter in \(\mathbb {R}^N\).
The convergence of the total energy is not covered by Lemma 2.9; however, this can be deduced using integration by parts and the fact that \(w_h\) has zero boundary values for every \(h \in \mathbb N\).
This scaling has the property that \(s^{N-1}J_{B_1}(\mathscr {A}w^s,A^s) = J_{B_s}(\mathscr {A}w,A)\).
As it can be seen from the proof of Lemma 5.1, the constant \(c_1\) does not depend on K.
The constant \(c_2\) is independent of the compact set K; indeed, this is the result of universal comparison estimates in \(\Omega \).
The notation \(B_r^\pm \) stands for the upper and lower half ball of radius r: \(B_r \cap H\) and \(B_r \cap - H\) respectively.
Recall that, for a 1-st order operator as in (7), the coefficients \(A_\alpha \) can be simply denoted by \(A_i\) with \(i = 1,\dots ,N\).
\(\mathrm {L}^{2^*}(\Omega )\)-integrability of \(\mathscr {A}w\), for some exponent \(2^* > 2\), can be established by standard methods through the use of the Caccioppoli inequality in Lemma 2.5.
Here, \(\nu _r\) is the \(\mathscr {A}\)-free corrector function for w in \(B_r\), see Definition 2.1.
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Acknowledgements
I wish to extend many thanks to Prof. Stefan Müller for his advice and fruitful discussions in this beautiful subject. I would also like to thank the reviewer and the editor for their patience and care which derived in the correct formulation of Theorem 1.5. The support of the University of Bonn and the Hausdorff Institute for Mathematics is gratefully acknowledged. The research conducted in this paper forms part of the author’s Ph.D. thesis at the University of Bonn.
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Communicated by L. Ambrosio.
Glossary of constants
Glossary of constants
-
N spatial dimension
-
M coercivity and bounding constant for the tensors \(\sigma _1\) and \(\sigma _2\) (as quadratic forms)
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K an arbitrary compact set in \(\Omega \)
-
\(\lambda _K\) local upper bound constant
Other constants Groups of constants are numbered in non-increasing order, e.g., \(c_1^* \ge c_2^* \ge c_3^*\). The following constants play an important role in our calculations:
Constant | Dependence | Description |
|---|---|---|
\(\theta _1\) | arbitrary in (0, 1 / 2) | Ratio constant |
\(c_1\) | \(\theta _1,N,M\) | Universal constant |
\(\varepsilon _1\) | \(\theta _1, N, M\) | Smallness of perimeter density |
\(c_2\) | N, M | Universal constant |
\(\gamma \) | N, M | Universal constant |
\(\theta _2\) | N, M | Universal constant |
\(\varepsilon _2\) | N, M | Smallness of excess energy |
\(\theta _0(\varepsilon )\) | N, M, K | Smallness of perimeter density |
\(c^*_1\) | \(\lambda _K, N\) | Constant in the Height bound Lemma |
\(\theta _1^*\) | arbitrary in (0, 1 / 2) | Ratio constant |
\(c_2^*\) | \(\theta _1^*, N,M\) | Universal constant |
\(\epsilon _2^*\) | \(\theta ^*_1,N, M\) | Smallness of flatness excess |
\(c_3^*\) | K, N, M, f | Flatness excess improvement scaling constant |
\(\varepsilon _3^*\) | K, N, M, f | Smallness of flatness excess |
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Arroyo-Rabasa, A. Regularity for free interface variational problems in a general class of gradients. Calc. Var. 55, 154 (2016). https://doi.org/10.1007/s00526-016-1085-5
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DOI: https://doi.org/10.1007/s00526-016-1085-5