Abstract
We prove existence and uniqueness of minimizers for a family of energy functionals that arises in Elasticity and involves polyconvex integrands over a certain subset of displacement maps. This work extends previous results by Awi and Gangbo to a larger class of integrands. We are interested in Lagrangians of the form \(L(A,u)=f(A)+H(\det A)-F\cdot u \). Here the strict convexity condition on \(f \) and \(H \) have been relaxed to a convexity condition. Meanwhile, we have allowed the map \(F \) to be non-degenerate. First, we study these variational problems over displacements for which the determinant is positive. Second, we consider a limit case in which the functionals are degenerate. In that case, the set of admissible displacements reduces to that of incompressible displacements which are measure preserving maps. Finally, we establish that the minimizer over the set of incompressible maps may be obtained as a limit of minimizers corresponding to a sequence of minimization problems over general displacements provided we have enough regularity on the dual problems. We point out that these results do not rely on the direct methods of the calculus of variations.
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1 Introduction
We are interested in Euler-Lagrange equations, existence and uniqueness of minimizers for some problems in the vectorial calculus of variations emanating from elasticity theory. These variational problems are related to an open problem in Partial Differential Equations that we describe as follows: let \(T>0\) and let \(\varOmega \) and \(\varLambda \) be two open subsets of \(\mathbb{R}^{d}\); suppose that \(\mathbf{u}_{0}\) is a diffeomorphism between \(\varOmega \) and \(\varLambda \); we seek \(\mathbf{u} : \varOmega \times (0,T)\longrightarrow \mathbb{R}^{d} \) such that \(\mathbf{u}(\cdot ,t) (\varOmega )=\varLambda \) for each \(t\) and
in the sense of distributions. In (1.1), we assume that the map \(\mathbb{R} ^{d\times d}\ni \xi \mapsto L(\xi )\) is quasiconvex. We refer the reader to [2], [7], [5], [11], and [12] for further details on these gradient flows. Understanding variational problems associated to the time-discretization of (1.1) is arguably an important step toward the construction of a solution. In that regard, several partial results are available in the literature (see for instance [7] and [5]).
In [2], the authors have focused on a class of Lagrangians that arises in elastic materials. More precisely, they have considered polyconvex Lagrangians of the form \(\xi \mapsto L(\xi )=f(\xi )+H( \det \xi )\). Here \(f\) is a \(C^{1}(\mathbb{R}^{d} )\) strictly convex function with \(p\)-th order growth, and the map \(H\) is a \(C^{1}(0, \infty )\) convex function that satisfies
As a result, a variational problem emerges from the time discretization and has a relaxation that takes the general form:
where \(F\in L^{1}(\varOmega ,\mathbb{R} ^{d}) \) and
Although the existence of minimizers in (1.3) follows from the direct methods in the calculus of variations, the uniqueness is a rather challenging problem. Indeed, because of (1.2) and the non-convexity of the integrand, standard techniques in calculus of variations do not apply.
To bypass these difficulties, the authors of [2] have introduced a pseudo-projected gradient operator \(\mathscr{U}_{\mathcal{S}}\ni u\mapsto \nabla _{\mathcal{S}}u\) defined as follows: for a given \(u\in \mathscr{U}_{\mathcal{S}}\), the map \(\nabla _{ \mathcal{S}}u\) is the unique minimizer of
over
Here, \(\mathcal{S}\) is a finite-dimensional subspace of \(W_{0}^{1,q}( \varOmega ,\mathbb{R} ^{d\times d})\), \(q\) is the conjugate of \(p\), \(\mathscr{U}_{\mathcal{S}}\) is the set of all \(u:\varOmega \to \bar{ \varLambda }\) measurable such that there exists a \(c=c(u,\varOmega ,\varLambda ) >0\) satisfying:
We point out that the pseudo-projected gradient operator depends also on \(f\), though the dependence is not exhibited in its notation. As a first step to approaching (1.3), they have considered the following perturbed problem:
The choice of problem (1.6) is justified by the construction of a family of finite dimensional subspaces \(\lbrace \mathcal{S} _{n} \rbrace _{n} \) dense in \(W_{0}^{1,q}(\varOmega ,\mathbb{R} ^{d\times d}) \) such that for \(u\in W^{1,p}(\varOmega ,\mathbb{R} ^{d})\), one has
We note that a \(L^{p}(\varOmega ,\; \mathbb{R}^{d})\)-bounded subset of \(\mathscr{U}_{\mathcal{S}}\) whose image by the operator \(\nabla _{\mathcal{S}}\) is bounded in \(L^{p}(\varOmega ,\; \mathbb{R}^{d \times d})\) is not in general strongly pre-compact with respect to the \(L^{p}(\varOmega ,\; \mathbb{R}^{d})\) topology. As a result, compactness of level subsets of the functional in (1.6) cannot be guaranteed. Nevertheless, the authors of [2] have successfully shown existence and, more importantly, uniqueness in (1.6) under the assumption that \(F\) is non-degenerate (see definition below). This condition of non-degeneracy for uniqueness is crucial in a similar problem, the so-called Brenier polar factorization, and more generally, in optimal transport problems. Confer [1], [3], [9], [8], [10] and [15].
In this paper, we investigate the respective roles played by the strict convexity of \(f\), the convexity and smoothness of \(H\), and the non-degeneracy of \(F\) in problem (1.6). More precisely, we impose less stringent conditions so that either the map \(F\) is allowed to be degenerate or \(f\) is allowed to be merely convex or \(H\) is neither smooth nor strictly-convex. These considerations are not just technicalities. Indeed we note that a prominent case of mere convexity, \(f(\xi )=|\xi | \), is typical for the study of minimal surfaces as well as for the study of functionals involving the total variation (see for instance [4]). Furthermore, we observe that cases where \(H\) is taken to be the characteristic function of a singleton of ℝ arise in the study of incompressible deformations in Elasticity theory (see for instance [12] and [15]). Finally, the non-degeneracy condition tests the extent to which one can hope for uniqueness in the variational problem we considered. To deal with these weaker assumptions, we introduce a family of operators \(\lbrace V_{\mathcal{S}}^{f} : \mathcal{S}\subset W_{0}^{1,q}(\varOmega ,\mathbb{R} ^{d}), \;f \text{ convex} \rbrace \) defined by
We note that the operator \(V_{\mathcal{S}}^{f}\) is actually well defined on the set of measurable functions \(u\) defined from \(\varOmega \) to \(\bar{\varLambda }\) when the set \(\mathcal{S}\) is a finite dimensional nonempty set and the function \(f\) satisfies appropriate growth conditions. As a family, these operators extend the pseudo-projected gradient operators and the distributional gradient. Indeed, \(V_{\mathcal{S}}^{f}[u]=\int _{\varOmega}f(\nabla _{\mathcal{S}}u)\;dx\) if \(\mathcal{S}\) is a finite dimensional subspace of \(W_{0}^{1,q}(\varOmega ,\mathbb{R} ^{d\times d})\) and \(u\in \mathscr{U}_{\mathcal{S}}\) and furthermore \(V_{\mathcal{S}}^{f}[u]=\int _{\varOmega }f(\nabla u)\;dx\) if \(\mathcal{S}=W_{0}^{1,q}(\varOmega ,\mathbb{R} ^{d\times d})\) and \(u\in W^{1,p}(\varOmega ,\mathbb{R} ^{d})\). These extensions are only valid under appropriate conditions on \(f\). It is worth pointing out that if \(f(\xi )=|\xi |\) and \(\mathcal{S}=W_{0}^{1,q}(\varOmega,\mathbb{R} ^{d \times d})\) then \(V_{\mathcal{S}}^{f}(u)\) is nothing but the total variation of \(u\) on the set \(\varOmega \). We show that for a collection of sets \(\lbrace \mathcal{S}_{n} \rbrace _{n=1}^{\infty } \) of \(W_{0}^{1,q}(\varOmega ,\mathbb{R} ^{d})\) satisfying Hypothesis (H1) or Hypothesis (H2) (see Sect. 2), we have a convergence result in the same spirit as (1.7):
for any \(u\in W^{1,p}(\varOmega ,\mathbb{R} ^{d\times d})\) and appropriate conditions on \(f\). We thus proceed to study a more general problem:
where \(\mathcal{S}\) is an element of a collection of sets satisfying Hypothesis (H1) or Hypothesis (H2), and
Sublevel sets of the integrand in (1.10) are not compact. Nor is \(f\) necessarily strictly convex. However, we show existence and uniqueness in problem (1.10). In fact, this result holds for \(F\) non-degenerate as well as for a class of degenerate \(F\) provided that the set \(\mathcal{S}\) is chosen accordingly (see Corollaries 3.6 and 3.7). Unlike optimal transport theory, this analysis suggests that the non-degeneracy condition is not essential for a uniqueness result in (1.3).
Existence and uniqueness results for problem (1.10) are established thanks to the discovery of suitable dual problems. Indeed, call \(\mathcal{C}\) the set of all functions \((k,l)\) with \(k,\;l:\mathbb{R}^{d} \rightarrow \mathbb{R} \cup \{\infty \}\) Borel measurable, finite at least at one point, and satisfying the relation \(l\equiv \infty \) on \(\mathbb{R}^{d} \setminus \bar{\varLambda }\) and such that
Let \(\mathscr{A}\) be the set of \((k,l,\varphi )\) such that \((k,l) \in \mathcal{C}\) and \(\varphi \in \mathcal{S}\). Define the following functional over the set \(\mathscr{A}\):
Next, assume that the map \(F \) and the set \(\mathcal{S}\) are such that for all \(\varphi \in \mathcal{S}\),
Then \(-J\) admits a maximizer \((k_{0},l_{0},\varphi _{0}) \) with \(k_{0} \) convex and \(\operatorname{diam}(\varLambda )\)-Lipschitz. As a consequence, problem (1.10) admits a unique minimizer \((u_{0},\beta _{0}) \) and \(u_{0} \) satisfies
Here, we have denoted by \(\varPhi _{\mathcal{S}}(u_{0})\), the non-empty set of maximizers of problem (1.8) (see Proposition 2.8). In order to obtain condition (1.12), we consider two distinct situations.
First, we assume that \(F \) has a countable range, thus degenerate. If \(\mathcal{S}\) is an element of a collection of sets satisfying hypothesis (H2) then it holds that \(F+\operatorname{div} \varphi \) is non-degenerate.
Second, we assume \(F \) non-degenerate and \(\mathcal{S}\) is a finite dimensional vector space, as in [2]. It holds again that \(F+\operatorname{div} \varphi \) is non-degenerate. However, unlike the hypotheses in [2], we have allowed the map \(f\) to be as singular as the map \(\mathbb{R} ^{d\times d}\ni \xi \mapsto |\xi |\).
We have also studied (1.10) when \(H\) is replaced by \(H_{0}:(0,\infty )\to \mathbb{R} \cup \{\infty \}\) defined by \(H_{0}(1)=0\) and \(H_{0}(t)=\infty \) if \(t\neq 1\). This case corresponds to the case of measure preserving maps. Note that \(H_{0}\) is not even continuous. However, it may be obtained as a limit of functions \(H_{n}\) which are \(C^{1}(0,\infty )\) convex functions and satisfy (1.2). We show that for such singular \(H_{0}\), the corresponding problem
with
admits a unique minimizer. (See Theorem 4.3.)
To obtain existence and uniqueness results in problem (1.14), we exploit a dual formulation and maximize \(-J\) over the set that consists of \((k,l,\varphi )\) such that \(\varphi \in \mathcal{S}\) and \(k,\;l:\mathbb{R}^{d} \rightarrow \mathbb{R} \cup \{\infty \}\) are Borel measurable, finite at least at one point, and satisfy the relations \(l\equiv \infty \) on \(\mathbb{R} ^{d} \setminus \bar{\varLambda }\) and
One shows that \(-J\) admits a maximizer \((k_{0},l_{0},\varphi _{0}) \) with \(k_{0} \) convex and Lipschitz and the unique minimizer of problem (1.14) is \(u_{0} \) given by
Finally, we show convergence of a sequence of problems of the form (1.10) to (1.14). More precisely, we show that the minimizer of problem (1.14) may be obtained as limit of minimizers of problems of the form (1.10) provided that the dual problems admit regular enough maximizers. In fact, suppose the map \(F \) and the set \(\mathcal{S}\) are such that for all \(\varphi \in \mathcal{S}\), the map \(F+\operatorname{div} \varphi \) is non-degenerate. For \((u,\beta ) \in \mathscr{U}_{\mathcal{S}}\), define
and set
Thanks to Theorem 3.5, the problem
admits a unique minimizer that we denote \((u_{n},\beta _{n})\) with \(u_{n}=\nabla k_{n}(F+\operatorname{div}\varphi _{n})\) for some \(k_{n}:\mathbb{R} ^{d}\to \mathbb{R} \) convex and \(\varphi _{n}\in \mathcal{S}\). Denote \(u_{0}\) the unique minimizer of (1.14). If for all \(n\in \mathbb{N} ^{*}\) the map \(k_{n}\) is differentiable then the sequence \(\{ u_{n} \}_{n\in \mathbb{N} ^{*}}\) converges almost everywhere to \(u_{0}\) and in addition, the minima \(\{ I_{n}(u_{n},\beta _{n})\}_{n\in \mathbb{N} ^{*}}\) converge to \(I_{0}(u_{0})\) (cf. Theorem 4.7).
2 Preliminaries
2.1 Notation and Definitions
-
Throughout this manuscript, \(\varOmega \) and \(\varLambda \subset \mathbb{R}^{d}\) are two bounded open convex sets; \(r^{*}>1\) is such that \(B(0,1/r^{*})\subset \varLambda \subset B(0,r^{*}/2)\); \(p\in (1,\infty )\) and \(q\) is its conjugate, that is, \(p^{-1}+q^{-1}=1\).
-
Given \(A\subset \mathbb{R} ^{d}\), the indicator function of \(A\) is defined as
$$ \chi _{A}(x)= \textstyle\begin{cases} 0&\text{if } x\in A,\\ \infty&\text{otherwise.} \end{cases} $$ -
For any subset \(\mathcal{S}\) of \(W_{0}^{1, q}( {\varOmega ,\mathbb{R} ^{d\times d}})\), we denote by \(\operatorname{span}(\mathcal{S})\) the linear subspace of \(W_{0}^{1, q}( {\varOmega , \mathbb{R} ^{d\times d}})\) generated by \(\mathcal{S}\).
-
We denote by \(f^{*}\) the Legendre transform of a map \(f:\mathbb{R} ^{d\times d}\longrightarrow \mathbb{R} \) so that
$$ f^{*}\bigl(\xi ^{*}\bigr)=\sup_{\xi \in \mathbb{R} ^{d\times d}} \bigl\{ \xi \cdot \xi ^{*}-f(\xi ) \bigr\} . $$ -
If \(h : \mathbb{R} ^{d}\longrightarrow \mathbb{R} \cup \{\infty \}\) is convex then the subdifferential \(\partial h (x)\) of \(h\) at \(x\in \operatorname{Dom} (h)\) is closed and convex. If \(\partial h (x)\) is non-empty we denote by \(\operatorname{grad}[h](x)\) the element of \(\partial h (x)\) with minimum norm:
$$ \bigl|\operatorname{grad}[h](x)\bigr|= \min \bigl\lbrace |y|: y\in \partial h (x) \bigr\rbrace ; \quad x\in \operatorname{Dom}(h). $$ -
Let \(\mathcal{S}\subset W_{0}^{1, q}( {\varOmega , \mathbb{R} ^{d\times d}})\). We denote by \(\mathscr{S}_{f}\) the set
$$ \mathscr{S}_{f} := \biggl\{ \varphi \in \mathcal{S}: \int _{\varOmega }f ^{*}(\varphi ) \text{ is finite} \biggr\} . $$(2.1) -
Let \(F :\mathbb{R}^{d} \longrightarrow \mathbb{R} ^{d} \) be measurable. We say that \(F\) is non-degenerate if for any \(N\subset \mathbb{R}^{d} \) such that \(\mathcal{L}^{d}(N) =0 \) we have \(\mathcal{L}^{d}(F^{-1}(N)) =0\).
2.2 Assumptions
- (A0) :
-
We additionally assume that there exists a strictly convex function that is \(C^{1}(\bar{\varOmega }) \) and vanishes on the boundary of \(\varOmega \).
- (A1) :
-
The set \(\mathcal{S}\) is a subset of \(W_{0}^{1, q}( {\varOmega ,\mathbb{R} ^{d\times d}})\). In addition, the map \(f:\mathbb{R} ^{d\times d}\to \mathbb{R} \) is convex and satisfies the following three properties:
-
(i)
There exist \(a,b,c >0\) such that for all \(\xi \in \mathbb{R} ^{d\times d}\),
$$ c\frac{|\xi |^{p}}{p}+b\geq f(\xi )\geq a|\xi |-b $$(2.2)and for all \(\xi ^{*}\in \partial f(\xi ) \),
$$ \bigl|\xi ^{*}\bigr|^{q}\leq c|\xi |^{p} + b. $$(2.3) -
(ii)
The set \(\mathscr{S}_{f}\) is non-empty.
-
(iii)
One of the following two conditions holds:
-
(a)
The map \(f\) is such that \(\partial f^{*}(x^{*})\) is non-empty and \(\operatorname{grad}[f^{*}](x^{*})=0\) for each \(x^{*}\in \operatorname{Dom} f^{*}\).
-
(b)
The map \(f\) is strictly convex and there exist \(\bar{a}, \bar{b}>0\) such that for all \(\xi ^{*}\in \mathbb{R}^{d\times d}\), one has
$$ f^{*}\bigl(\xi ^{*}\bigr)\leq \bar{a}+\bar{b}\bigl|\xi ^{*}\bigr|^{q} \quad \mbox{and} \quad \bigl| \nabla f^{*}\bigl(\xi ^{*}\bigr)\bigr|\leq \bar{a} +\bar{b}\bigl| \xi ^{*}\bigr|^{q-1}. $$(2.4)
-
(a)
-
(i)
- (A2) :
-
The map \(H\) is \(C^{1}(0,\infty )\), strictly convex, and such that
$$ \lim_{t\rightarrow 0^{+}} H(t)= \lim_{t\rightarrow \infty } \frac{H(t)}{t}=+\infty . $$ - (A3) :
-
The function \(F\) is measurable and belongs to \(L^{1}(\varOmega )\).
Let \(\mathcal{S}\) be a subset of \(W_{0}^{1,q }({\varOmega ,\mathbb{R} ^{d\times d}})\). We say that \(F\) satisfies the condition \(\textbf{(ND)}_{\mathcal{S}}\) if
$$ \operatorname{div}(\varphi ) + F \text{ is non-degenerate} $$for all \(\varphi \in \mathcal{S}\).
Remark 2.1
-
(i)
As\(f\)satisfies (2.2), we have
$$ -b+c^{p}\frac{|\xi ^{*}|^{q}}{q}\leq f^{*}\bigl(\xi ^{*}\bigr)\leq \chi _{\bar{B}(0,a)} \bigl( \xi ^{*}\bigr)+b $$(2.5)for all\(\xi ^{*}\in \mathbb{R} ^{d\times d}\).
-
(ii)
If\(f\)satisfies case (b) in (iii) of Assumption\(\mathbf{(A1)}\), then\(f^{*}\)is continuously differentiable. In that case, \(\operatorname{grad}[f^{*}]=\nabla f^{*}\).
-
(iii)
If\(f\)satisfies case (a) of Assumption\(\mathbf{(A1)}\)(iii) then\(0\in \partial f^{*}(x^{*}) \)for every element\(x^{*} \in \operatorname{Dom}(f^{*} )\). Consequently, the map\(f^{*} \)is constant on\(\operatorname{Dom}(f^{*} )\)and the following equalities are satisfied for all\(x^{*} \)and\(y^{*} \)in\(\operatorname{Dom}(f^{*} ) \):
$$ f^{*}\bigl(x^{*}\bigr)-f^{*} \bigl(y^{*}\bigr)=\operatorname{grad}\bigl[f^{*}\bigr] \bigl(x^{*}\bigr)= \operatorname{grad}\bigl[f^{*}\bigr] \bigl(y^{*}\bigr)=0. $$(2.6) -
(iv)
Assumption\(\mathbf{(A0)}\)is satisfied by\(\varOmega =B(0,1) \subset \mathbb{R}^{d}\)with the strictly convex function being the map\(\mathbb{R}^{d}\ni x\mapsto |x|^{2}-1 \).
-
(v)
The map\(f:\mathbb{R} ^{d\times d}\to \mathbb{R} \)defined by\(f(\xi )=|\xi |\)satisfies case (a) in (iii) of Assumption\(\mathbf{(A1)}\). The map\(f:\mathbb{R} ^{d\times d}\to \mathbb{R} \)defined by\(f(\xi )=|\xi |^{p}\)satisfies case (b) in (iii) of Assumption\(\mathbf{(A1)}\).
The following lemma summarizes some elementary properties of \(H \). We refer the reader to Remark 2.1 in [2].
Lemma 2.2
Assume\(\mathbf{(A2)}\)holds. Then,
-
(i)
The map\(H':(0,\infty )\to \mathbb{R} \)is a strictly increasing bijection.
-
(ii)
The Legendre transform\(H^{*}\)of\(H\)is a strictly increasing bijection from ℝ to ℝ.
-
(iii)
Let\(g:\mathbb{R} \to \bar{\mathbb{R} } \)be defined by\(g(s)=\alpha s-\beta H^{*}(s)\), with\(\alpha ,\beta >0\). Then
$$ \lim_{s\to -\infty }g(s)=\lim_{s\to \infty }g(s)=-\infty . $$
Define \(H_{0}\) by
and, for \(n\geq 1\),
The following lemma is straightforward.
Lemma 2.3
Assume\(\mathbf{(A2)}\)holds. Then,
-
(i)
There exists\(\bar{H}\in \mathbb{R} \)such that
$$ \bar{H}= \min_{t\in [0,\infty )} H(t). $$ -
(ii)
The collection\(\lbrace H_{n} \rbrace _{n=1} ^{\infty } \)is a non-decreasing sequence of functions that converges pointwise to\(H_{0}\). In addition, for all\(n\in \mathbb{N} ^{*}\), the map\(H_{n}\)is a\(C^{1}(0,\infty )\)strictly convex function that satisfies
$$ \lim_{t\rightarrow 0^{+}} H_{n}(t)= \lim_{t\rightarrow \infty } \frac{H _{n}(t)}{t}=+\infty . $$ -
(iii)
Let\(t>0\). If\(\lbrace H_{n}(t) \rbrace _{n=1} ^{\infty } \)is uniformly bounded above by a constant\(c_{0}\)then
$$ n(t-1)^{2} \leq c_{0} +H(1)-\bar{H} $$and\(t=1\).
2.3 Hypothesis on the Underlying Sets of Pseudo-Gradients
We recall that in [2], the construction of \(\nabla_{\mathcal{S}^{\tau }} u\) has relied on hypothesis on the underlying sets \(\mathcal{S}^{\tau } \) that we summarize in Hypothesis (H1) below.
Hypothesis (H1).
A collection \(\{ \mathfrak{A}_{n }\}^{\infty }_{ n=1} \) of subsets of \(W_{0}^{1,q }({\varOmega ,\mathbb{R}^{d\times d}}) \) satisfies Hypothesis (H1) if
-
(i)
\(\mathfrak{A}_{n }\) of a finite dimensional subspace of \(W_{0}^{1,q }({\varOmega ,\mathbb{R}^{d\times d}}) \) for each \(n\in \mathbb{N^{*}}\).
-
(ii)
The map \(\nabla \varphi \) has a countable range whenever \(\varphi \in \mathfrak{A}_{n} \), for any \(n\in \mathbb{N} ^{*} \).
-
(iii)
The set \(\bigcup _{n\in \mathbb{N} ^{*}}\mathfrak{A} _{n} \) is dense in \(W_{0}^{1,q }({\varOmega ,\mathbb{R} ^{d\times d}} )\).
-
(iv)
For \(i\leq j\), we have the inclusion \(\mathfrak{A}_{i } \subset \mathfrak{A}_{j}\).
An explicit construction of sets satisfying Hypothesis (H1) is provided in [2]. Here, we build on the conditions of Hypothesis (H1) and we relax conditions on the underlying sets:
Hypothesis (H2).
A collection \(\{ \mathfrak{Q}_{n }\}^{\infty }_{ n=1} \) of subsets of \(W_{0}^{1,q }({\varOmega ,\mathbb{R}^{d\times d}}) \) satisfies Hypothesis (H2) if
-
(i)
\(\text{Span}(\mathfrak{Q}_{n})\) is of finite dimension and \(\mathfrak{Q}_{n }\) is a non-empty closed and convex subset of \(W_{0}^{1,q }({\varOmega ,\mathbb{R}^{d\times d}})\).
-
(ii)
The map \(\operatorname{div} \varphi \) is non-degenerate whenever \(\varphi \in \mathfrak{Q}_{n} \), for any \(n\in \mathbb{N} ^{*} \).
-
(iii)
The set \(\bigcup _{n\in \mathbb{N} ^{*}}\mathfrak{Q} _{n} \) is dense in \(W_{0}^{1,q }({\varOmega ,\mathbb{R} ^{d\times d}} )\).
-
(iv)
For \(i\leq j\), the inclusion \(\mathfrak{Q}_{i }\subset \mathfrak{Q}_{j} \) holds.
The next lemma asserts that a collection of sets can be constructed to satisfy Hypothesis (H2).
Lemma 2.4
Assume\(\mathbf{(A0)}\)holds. Then, there exists a collection of sets\(\{\mathfrak{Q}_{n }\}^{\infty } _{n=1}\)satisfying the requirements of Hypothesis (H2).
Remark 2.5
The condition\(\mathbf{(A0)}\)in Lemma 2.4 is only needed for requirement (ii) of Hypothesis (H2).
Proof
Suppose that \(\psi \) is a strictly convex function that is \(C^{1}(\bar{\varOmega }) \) and vanishes on the boundary of \(\varOmega \) as given by Assumption (A0). Let \(\varphi _{0}:\varOmega \to \mathbb{R} ^{d\times d} \) be defined by
As \(\psi \) is \(C^{1}(\bar{\varOmega }) \), we have \(\varphi _{0}\in W_{0} ^{1,q }({\varOmega ,\mathbb{R}^{d\times d}}) \) and it follows that \(\operatorname{div}\varphi _{0}=\nabla \psi \). Thus, for almost every \(x \) in \(\varOmega \), we have
Thanks to Lemma 5.5.3 in [1], the map \(\operatorname{div}\varphi _{0} \) is non-degenerate. Let \(\{ \mathfrak{A}_{n }\}^{\infty }_{ n=1} \) be a collection of sets satisfying Hypothesis (H1). One readily checks that the family of sets defined by
for \(n\in \mathbb{N} ^{*}\), satisfies hypothesis (H2). □
2.4 Special Displacements
To \(\mathcal{S}\subset W_{0}^{1,q}(\varOmega ,\mathbb{R} ^{d\times d})\) we associate \(\mathscr{U}_{\mathcal{S}}\), the set of all \(u:\varOmega \to \bar{\varLambda }\) measurable such that there exists \(\bar{c}= \bar{c}(u,\varOmega ,\varLambda ) >0 \) satisfying:
Remark that if \(u \in \mathscr{U}_{\mathcal{S}}\), then \(u \) belongs to \(L^{\infty }(\varOmega ,\mathbb{R}^{d}) \) since \(u\) has values in \(\bar{\varLambda }\) which is bounded. If \(\operatorname{span}(\mathcal{S})\) is of finite dimension then \(\mathscr{U}_{\mathcal{S}}\) is the set of all measurable maps \(u:\varOmega \to \bar{\varLambda }\). In fact, the linear map \(\operatorname{span}(\mathcal{S})\ni \varphi \mapsto \int _{\varOmega }u \operatorname{div}\varphi \) is continuous with respect to the \(L^{q} \)-norm as in finite dimension, all norms are equivalent. Therefore, we may find \(c\) for which inequality (2.9) holds for all \(\varphi \in \operatorname{span}(\mathcal{S})\) and in particular for all \(\varphi \in \mathcal{S}\).
At any rate, \(\mathscr{U}_{\mathcal{S}}\) contains \(W^{1,p}(\varOmega , \mathbb{R} ^{d})\). Indeed, notice that for a fixed \(u\in W^{1,p}( \varOmega ,\mathbb{R} ^{d})\), we have, for all \(\varphi \in \mathcal{S}\):
We introduce the following set
and
Notice that \(\mathscr{U}_{\mathcal{S}}^{1}=\{u\in \mathscr{U}_{ \mathcal{S}}: (u,1)\in \mathscr{U}_{\mathcal{S}}^{*}\}\). This corresponds to measure preserving displacements.
2.5 Extended Pseudo-Projected Gradient
Let \(\mathcal{S}\subset W_{0}^{1,q}(\varOmega ,\mathbb{R} ^{d\times d})\) and \(u \in \mathscr{U}_{\mathcal{S}}\). Define
Consider the operator
We denote by \(\varPhi _{S}( u )\) the set of maximizers of problem (2.10).
Lemma 2.6
Let\(\mathcal{S}\subset W_{0}^{1,q}(\varOmega ,\mathbb{R} ^{d\times d})\)and\(u \in \mathscr{U}_{\mathcal{S}}\).
-
1.
We have
$$ \mathcal{G}_{\mathcal{S}}( u )= \biggl\{ G\in L^{p}\bigl( \varOmega , \mathbb{R} ^{d\times d}\bigr) : \int _{\varOmega }u \operatorname{div}\varphi \,dx=- \int _{\varOmega } \langle {G},{\varphi } \rangle\, dx;\ \forall \varphi \in \operatorname{span}(\mathcal{S}) \biggr\} . $$ -
2.
If\(\operatorname{span}(\mathcal{S})\)is finite dimensional, then\(\mathcal{G}_{\mathcal{S}}( u )\)is nonempty.
Proof
Set
As \(\mathcal{S}\subset \operatorname{span}(\mathcal{S})\), we have \(\bar{\mathcal{G}}_{\mathcal{S}}(u) \subset \mathcal{ G}_{\mathcal{S}}(u) \). Next, let \(G\in \mathcal{ G}_{\mathcal{S}}(u)\). Assume that \(\varphi \in \operatorname{span}(\mathcal{S})\). We may find \(n\in \mathbb{N}\), \(\lambda _{1},\dots ,\lambda _{n}\in \mathbb{R} \) and \(\varphi _{1},\dots ,\varphi _{n}\in \mathcal{S}\) such that \(\varphi = \sum_{i=1}^{n}\lambda _{i}\varphi _{i}\). Then
and
Thus \(G\in \bar{\mathcal{G}}_{\mathcal{S}}(u) \). We deduce that \(\mathcal{G}_{\mathcal{S}}(u) \subset \bar{\mathcal{G}}_{\mathcal{S}}(u) \). It follows that part (1.) holds. To obtain part (2.), we use part (1.) and the Riesz Representation Theorem. □
The following results are essentially found in Proposition 3.1 in [2].
Proposition 2.7
Suppose that the set\(\mathcal{S}\)is a finite dimensional subspace of\(W_{0}^{1,q }({\varOmega ,\mathbb{R}^{d\times d}})\)and\(f\)is\(C^{1}\)and strictly convex. Suppose, in addition that there exist constants\(c_{1},c_{2},c_{3}>0\)such that
for all\(\xi \in \mathbb{R} ^{d\times d}\). Then, there exists a unique map denoted\(\nabla _{\mathcal{S}}u \)that minimizes
Moreover, \(\nabla _{\mathcal{S}}u \)is the unique map\(G\in \mathcal{G}_{\mathcal{S}}( u )\)that satisfies\(Df(G)\in \mathcal{S}\).
In the next proposition, we establish similar results as in Proposition 2.7 but under weaker assumptions on \(\mathcal{S}\) and \(f\) (except in part 4).
Proposition 2.8
Assume\(\mathbf{(A1)}\)holds. Assume\(\mathcal{S}\)is a finite dimensional non-empty closed and convex subset of\(W_{0}^{1,q }( {\varOmega ,\mathbb{R}^{d\times d}})\)and let\(u \in \mathscr{U}_{ \mathcal{S}}\).
-
1.
For all\(G\in \mathcal{G}_{\mathcal{S}}( u )\), \(\varphi \in \mathcal{S}\), we have
$$ \int _{\varOmega }f(G) \;dx\geq \int _{\varOmega } \bigl(- u \operatorname{div}\varphi -f^{*}(\varphi ) \bigr)\;dx. $$ -
2.
The supremum in problem (2.10) is attained.
-
3.
A map\(\bar{\varphi }\)belongs to\(\varPhi _{\mathcal{S}}( u ) \)if and only if\(\bar{\varphi }\)belongs to\(\mathscr{S}_{f} \)and
$$ \int _{\varOmega } \bigl( \operatorname{grad}\bigl[f^{*} \bigr](\bar{\varphi }) \cdot (\varphi -\bar{\varphi } ) + u \cdot ( \operatorname{div} \varphi - \operatorname{div}\bar{\varphi } ) \bigr)\;dx\geq 0 $$for all\(\varphi \in \mathscr{S}_{f} \).
-
4.
Suppose that the hypotheses of Proposition 2.7are satisfied. Then we have
$$ \int _{\varOmega }f(\nabla _{\mathcal{S}}u )\;dx=V_{\mathcal{S}}^{f}( u ) $$and\(\varPhi _{\mathcal{S}}( u )=\{Df(\nabla _{\mathcal{S}}u )\}\).
Proof
(1.) Let \(\varphi \in \mathcal{S}\) and \(G\in \mathcal{G}_{ \mathcal{S}}( u)\). By using the Legendre transformation,
(2.) Let \(\varphi \in \mathcal{S}\). We use (2.9) and (2.5) to get
In light of (2.11), \(q>1\) implies that the map
is \(L^{q}\)-coercive. Moreover, the convexity of \(f^{*}\) guarantees that \(T\) is lower semi-continuous. The direct methods of the calculus of variations thus yield the existence of a maximizer in problem (2.10).
(3.) Let \(\bar{\varphi }\in \varPhi _{\mathcal{S}}( u ) \) so that \(\bar{\varphi }\in \mathscr{S}_{f}\). Let \(\varphi \in \mathscr{S}_{f}\) and \(\epsilon \in (0,1)\). The convexity of \(f^{*}\) ensures that \(\bar{\varphi }+\epsilon (\varphi -\bar{\varphi })\in \mathscr{S}_{f}\) and the maximality property of \(\bar{\varphi }\) implies that
We rewrite (2.12), in turn, as
Note that \(\operatorname{grad}[f^{*}](\bar{\varphi }+\epsilon ( \varphi -\bar{\varphi })) \) belongs to the set \(\partial f^{*}((\bar{ \varphi }+\epsilon (\varphi -\bar{\varphi }))) \) whenever \((\bar{ \varphi }+\epsilon (\varphi -\bar{\varphi })) \) is in the domain of \(f^{*} \). It follows that
that is,
We combine (2.13) and (2.14) to get
First, we assume that \(\mathbf{(A1)}\)(iii)(a) holds. In light of (2.6), we have \(\operatorname{grad}[f^{*}](\bar{\varphi }+\epsilon (\varphi -\bar{\varphi }))=\operatorname{grad}[f^{*}](\bar{ \varphi })\). Equation (2.15) becomes
Second, we assume that \(\mathbf{(A1)}\)(iii)(b) holds. In light of Remark 2.1(ii), we use the growth condition on \(\nabla f^{*} \) in (2.4), the Lebesgue dominated convergence theorem and let \(\epsilon \) go to 0 in (2.15) to obtain that:
We next show the converse implication. Let \(\varphi \in \mathscr{S} _{f} \) such that
for all \(\varphi \in \mathscr{S}_{f}\). We notice that, as \(f^{*}\) is convex, the range of the map \(\operatorname{grad}[f^{*}](\bar{\varphi })\) lies in the sub-differential of \(f^{*}\) so that \(f^{*}(\varphi )-f ^{*}(\bar{\varphi })\geq \operatorname{grad}[f^{*}](\bar{\varphi })(\varphi -\bar{\varphi } )\) for all \(\varphi \in \mathscr{S}_{f}\). Then, the inequality (2.16) implies that
for all \(\varphi \in \mathscr{S}_{f}\), that is,
for all \(\varphi \in \mathscr{S}_{f}\). We conclude that \(\bar{\varphi }\in \varPhi _{\mathcal{S}}( u ) \).
(4.) Thanks to Proposition 2.7, \(D f( \nabla _{\mathcal{S}}u)\in \mathcal{S}\). Next, we set \(\varphi _{0}:= D f(\nabla _{\mathcal{S}}u)\). By definition of \(f^{*}\),
for all \(\varphi \in \mathcal{S}\). As \(f\) is convex and \(\varphi _{0} = D f(\nabla _{\mathcal{S}}u)\), we have
Thus,
and
We deduce that \(\varphi _{0}\in \varPhi _{\mathcal{S}}( u )\). Since \(f^{*}\) is strictly convex, we conclude that \(\varPhi _{\mathcal{S}}( u )= \{D f(\nabla _{\mathcal{S}}u)\}\) and moreover, \(\int _{\varOmega }f( \nabla _{\mathcal{S}}u )=V^{f}_{\mathcal{S}}( u )\), see (2.10). □
In the next proposition, we establish a convergence result in the spirit of (1.7). We also connect the operator \(V_{ \mathcal{S}}^{f}\) with the usual notions of gradient and total variation.
Proposition 2.9
Assume\(\mathbf{(A1)}\)holds. Assume that\(\mathcal{S}_{n}\)is a finite dimensional non-empty closed and convex subset of\(W_{0}^{1,q }( {\varOmega ,\mathbb{R}^{d\times d}})\)for each\(n\geq 1\). The following holds.
-
1.
If\(\lbrace \mathcal{S}_{n} \rbrace _{n=1}^{\infty } \)is a monotonically increasing family of subsets of some set\(\mathcal{S}_{0} \)and\(\bigcup _{n\in \mathbb{N} ^{*}}\mathcal{S}_{n} \)is dense in\(\mathcal{S}_{0} \)with respect to the\(W_{0}^{1,q}(\varOmega ,\mathbb{R} ^{d\times d}) \)norm then
$$ \lim_{n\rightarrow \infty } V_{\mathcal{S}_{n}}^{f}[u]=V_{\mathcal{S} _{0}}^{f}[u] $$for any\(u\in \mathscr{U}_{\mathcal{S}_{0}}\).
-
2.
If\(\mathcal{S}=W_{0}^{1,q}(\varOmega ,\mathbb{R} ^{d\times d} )\)and\(u\in W^{1,p}(\varOmega ,\mathbb{R} ^{d} )\)then\(V_{\mathcal{S}}^{f}[u]= \int _{\varOmega }f(\nabla u)\,dx \).
-
3.
Assume\(u\in BV(\varOmega ,\mathbb{R} ^{d\times d}) \)and\(f(\xi )=|\xi | \)for all\(\xi \in \mathbb{R} ^{d\times d}\). If\(\mathcal{S}=W_{0}^{1,q}(\varOmega ,\mathbb{R} ^{d\times d}) \)then\(V_{\mathcal{S}}^{f}[u] \)is the total variation of\(u \).
Remark 2.10
A consequence of Proposition 2.9is the following: If the sequence of sets\(\{\mathcal{S}_{n} \}_{n\in \mathbb{N} ^{*}}\)is monotonically increasing to\(W_{0}^{1,q}(\varOmega ,\mathbb{R} ^{d\times d})\)and\(u\in W^{1,p}(\varOmega ,\mathbb{R} ^{d} )\)we have
Proof
(1.) Recall that
As \(\lbrace \mathcal{S}_{n} \rbrace _{n=1}^{\infty } \) is a monotonically increasing, \(\lim_{n\rightarrow \infty } V_{S_{n}} ^{f}[u]\) exists. Moreover, since \(\mathcal{S}_{n}\subset \mathcal{S} _{0} \) for all \(n\geq 1\),
Let \(\epsilon >0 \) and choose \(\varphi ^{\epsilon }\in \mathcal{S} _{0}\) such that
Let \(\{\varphi ^{\epsilon }_{n}\} _{n\in \mathbb{N} ^{*}} \) be a sequence converging to \(\varphi ^{\epsilon }\) in \(W_{0}^{1,q}(\varOmega ,\mathbb{R} ^{d\times d}) \) and such that \(\varphi _{n}^{\epsilon } \in \mathcal{S}_{n} \) for all \(n\in \mathbb{N} ^{*} \). Then, we use the growth conditions on \(f^{*} \) in (2.4) and (2.5), the continuity of \(f^{*}\) on its domain and the Lebesgue dominated convergence theorem to obtain that
It follows that
As \(\epsilon \) is arbitrary, we have
From (2.17) and (2.18), we conclude that \(\lim_{n\rightarrow \infty } V_{\mathcal{S}_{n}}^{f}[u]=V_{\mathcal{S}_{0}}^{f}[u]\).
(2.) One has
The inequality above is obtained by using the definition of the Legendre transform \(f^{*}\) of \(f\). Let \(\bar{\varphi }\in \partial f (\nabla u) \). Then \(f^{*}(\bar{\varphi })+f(\nabla u)=\nabla u\cdot \bar{ \varphi }\). Thanks to the growth conditions (2.2) and (2.3) on \(f \), it holds that \(\bar{\varphi }\in L ^{q}(\varOmega ,\mathbb{R} ^{d\times d}) \). Since \(W_{0}^{1,q}(\varOmega , \mathbb{R} ^{d\times d})\) is dense in \(L^{q}(\varOmega ,\mathbb{R} ^{d \times d}) \) for the \(L^{q}(\varOmega ,\mathbb{R} ^{d\times d}) \) norm, we get
We conclude that \(V^{f}_{S}[u]= \int _{\varOmega }f(\nabla u)\;dx\).
(3.) The total variation of \(u\in BV(\varOmega ,\mathbb{R} ^{d\times d}) \) is
while, using the Legendre transform of \(f(\xi )= |\xi |\), we obtain for every \(q>1\)
It follows directly from (2.19) and (2.20) that \(\|Du\|(\varOmega )\leq V^{f}_{\mathcal{S}}[u]\). The converse inequality \(\|Du\|(\varOmega )\geq V^{f}_{\mathcal{S}}[u]\) follows from the density of \(C_{c}^{1}({\varOmega ,\mathbb{R} ^{d\times d}})\) in \(W_{0}^{1,q}(\varOmega ,\mathbb{R} ^{d\times d})\) and an argument similar to the one made in the proof of (2) in the proposition. □
3 Minimization with General Displacements
We consider the following:
This problem will be studied via a dual problem that we will formulate next. We assume in this section that Assumption (A2) holds.
3.1 An Auxiliary Problem
For \(l, k:\mathbb{R}^{d} \rightarrow (-\infty ,\infty ]\), define for \(u,v\in \mathbb{R} ^{d}\)
and
Under Assumption (A2), it is known that \(((l^{\#})_{\#})^{\#}=l ^{\#}\) and \(((k_{\#})^{\#})_{\#}=k_{\#}\) (see for instance Lemma A1 of [11]). Call \(\mathcal{C}\) the set of all functions \((k,l)\) with \(k,\;l:\mathbb{R}^{d} \rightarrow \mathbb{R} \cup \{ \infty \}\) Borel measurable, finite at least at one point, and satisfying \(l\equiv \infty \) on \(\mathbb{R}^{d} \setminus \bar{\varLambda }\) and such that
Call \(\mathcal{C}'\) the set of all functions \((k,l)\in \mathcal{C}\) such that \(l=k_{\#} \) and \(k=l ^{\#} \). The set \(\mathcal{C}'\) is nonempty. Indeed, \((\chi _{\bar{\varLambda }}^{\#},(\chi _{\bar{\varLambda }}^{\#})_{\#}) \in \mathcal{C}'\) as \(((\chi _{\bar{\varLambda }}^{\#})_{\#})^{\#}= \chi _{\bar{\varLambda }}^{\#}\).
Let \(\mathscr{A}\) be the set of \((k,l,\varphi )\) such that \((k,l) \in \mathcal{C}\) and \(\varphi \in \mathcal{S}\). Consider the following functional defined on \(\mathscr{A}\):
The following problem will play an important role in this section:
The value of the expression (3.5) is the opposite of the value of the following expression:
Let \(\mathscr{A}'\) denote the subset of \(\mathscr{A}\) consisting of all \((k,l,\varphi )\in \mathscr{A}\) that satisfy \((k,l)\in \mathcal{C}' \). It holds that
Indeed, the key observation to this end is that for \((k,l,\varphi ) \in \mathscr{A}\), one has \(l\geq k_{\#}\) and \(k\geq (k_{\#})^{\#}\) so that
For \(R>0\), we set
Lemma 3.1
Assume that\(\mathbf{(A1)}\), \(\mathbf{(A2)}\)and\(\mathbf{(A3)}\)hold. Let\((k,l,\varphi )\in \mathscr{A}_{R}\). Set\(s_{l}:=- \inf_{u\in \bar{\varLambda}}l(u)\). Then,
Moreover, there exists\(M:= M(R,F,f,\varOmega ,\varLambda )>0 \)such that
Proof
As \(\varLambda \) is bounded and \(l\) is convex, we choose \(u_{l}\in \overline{\varLambda }\) such that \(-l(u_{l})=s_{l}\). Since \(k:=l ^{\#}\), in view of (3.2), we have
Using the last inequality in (3.9), one gets
We have used the fact that \(u_{l}\) is a constant vector and \(\varphi \in W_{0}^{1,q}(\varOmega ,\mathbb{R} ^{d\times d} )\) to obtain the equality in (3.11). Hence,
Thus,
Thanks to Lemma 2.2(iii), \(s_{l} \) is bounded uniformly in \(l \). □
Lemma 3.2
Assume that\(\mathbf{(A1)}\), \(\mathbf{(A2)}\)and\(\mathbf{(A3)}\)hold.
-
1.
There exists\(M>0 \)such that for all\((k,l,\varphi )\in \mathscr{A}_{R}\)one has
$$ \int _{\varLambda }\bigl|l(y)\bigr|\,dy \leq M. $$(3.12) -
2.
There exist\(a_{0},b_{0},c_{0}>0 \)such that for all\((k,l,\varphi )\in \mathscr{A}_{R}\), the map\(k \)is\(r^{*} \)-Lipschitz, and one has for all\(v\in \mathbb{R}^{d} \)
$$ -c_{0}+a_{0}|v|\leq k(v)\leq b_{0}+r^{*}|v|. $$(3.13)
Proof
(1.) Recall that for \((k,l,\varphi )\in \mathcal{A}_{R}\), one has
By Lemma 3.1, for all \((k,l,\varphi )\in \mathcal{A}_{R}\), if we define \(s_{l}:=-\inf_{u\in \bar{\varLambda }}l(u)\), we get
Rearranging the terms, we get:
By definition of \(s_{l}\) we also have \(-s_{l}\mathcal{L}^{d}{(\varOmega )} \leq \int _{\varLambda } l(y) \,dy\) and thus
We consider the negative part of \(l\) defined by \(l^{-}:=\max \{-l,0\}\) and note that
Observe that, by the definition of \(s_{l}\), we have \(l^{-}\leq |s_{l}|\). This, combined with (3.14), (3.15) and (3.8) yields (3.12).
(2.) Let \((k,l,\varphi )\in \mathscr{A}_{R}\). Since \(k=l^{\#}\), by Eq. (3.2), \(k\) is a \(r^{*}\)-Lipschitz as \(\varLambda \) has diameter less or equal to \(r^{*}\). Next, we have
As \(s_{l}\) is uniformly bounded, the growth condition on \(H\) ensures that \(|k(0)|\) is uniformly bounded say by some \(b_{0}>0\). We get then the inequality \(k(v)\leq b_{0}+r^{*}|v|\) for all \(v\in \mathbb{R} ^{d}\).
Because of the hypothesis on the domain \(\varLambda \), we take \(a_{0}>0\) such that \(B(0,a_{0})\subset \varLambda \). As \((k,l,\varphi ) \in \mathcal{A}_{R}\), we use relation (3.4) to obtain for \(v\neq 0\)
Thanks to inequality (3.12), \(\int _{\varLambda }|l|\,dy\) is uniformly bounded in \(l \). We use in addition the fact that \(l \) is bounded to deduce that \(\sup_{y\in \bar{B}(0,a_{0})}|l|(y) \) is bounded by a constant independent of \(l\) (see for instance Theorem 1, p. 236 in [6]). Thus Eq. (3.16) implies that there exists \(c_{0}>0\) such that \(k(v)\geq a_{0}|v|-c_{0}\) for all \(v\in \mathbb{R} ^{d}\). □
Proposition 3.3
Assume that\(\mathbf{(A1)}\), \(\mathbf{(A2)}\), and\(\mathbf{(A3)}\)hold. Assume\(\mathcal{S}\)is a finite dimensional non-empty closed and convex subset of\(W_{0}^{1,q }({\varOmega ,\mathbb{R}^{d\times d}})\). Then, the functional\(J \)admits a minimizer\(( k_{0}, l_{0},\varphi _{0})\)in\(\mathscr{A}' \).
Proof
Let \((\bar{k},\bar{l},\bar{\varphi })\in \mathscr{A} \). Set \(R=J(\bar{k},\bar{l},\bar{\varphi })\). Take a minimizing sequence \(\{(k_{n},l_{n},\varphi _{n})\}_{n\in \mathbb{N} ^{*}} \) of problem (3.5) that is in \(\mathscr{A}_{R} \). By Lemma 3.1 and the growth condition on \(f^{*}\) we may assume without loss of generality that \(\{ \varphi _{ n}\}^{\infty } _{ n=1} \) converges to some \(\varphi _{0}\in \mathcal{S}\) weakly in \(L^{q}(\varOmega ,\mathbb{R} ^{d\times d}) \). Since \(\text{Span}( \mathcal{S})\) is finite dimensional, \(\{ \varphi _{ n}\}^{\infty } _{ n=1} \) converges to some \(\varphi _{0}\in \mathcal{S}\) strongly in the \(L^{q}(\varOmega ,\mathbb{R} ^{d\times d}) \) norm. We deduce
From Lemma 3.2, as \(l_{n}\) is convex, we use Ascoli-Arzelà Theorem together with Theorem 1, p. 236 in [6] to deduce that up to a subsequence, we may assume that \((k_{n},l_{n}) \) converges locally uniformly \(\mathbb{R} ^{d} \times \varLambda \) to \((k_{0},l_{0}) \in \mathcal{C}' \). The Lebesgue dominated convergence together with inequality (3.13) yield
Since \(\{ l_{n}\}^{\infty } _{n=1} \) is uniformly bounded below (thanks to Lemma 3.1), by Fatou’s Lemma we get
By inequalities (3.17), (3.18) and (3.19), we get
and \((k_{0},l_{0},\varphi _{0})\) is a minimizer of \(J\) over \(\mathscr{A}'\). □
3.2 A Uniqueness Result
Here, we prove the main result of this section. We will need the following lemma which is in the spirit of Lemma 4.3 and Lemma 4.4 in [2]. A proof of Lemma 3.4 is given in Sect. A.1.
Lemma 3.4
Assume that(A2)holds. Consider a lower semicontinuous function\(l_{0}:\mathbb{R}^{d} \to \bar{\mathbb{R}} \)such that\(\inf_{\bar{\varLambda }}l_{0}>-\infty \); \(l_{0}\)is finite on\(\varLambda \)and\(l_{0}\equiv +\infty \)on\(\mathbb{R}^{d} \setminus \bar{ \varLambda }\). Set\(k_{0}= ({l_{0}})^{\#}\). Let\(v\in \mathbb{R}^{d} \)be such that\(k_{0}\)is differentiable at\(v\).
-
1.
There exist unique\(u_{0}\in \bar{\varLambda }\)and\(t_{0}>0\)such that\(k_{0}( v ) =-t_{0}l_{0}(u_{0})-H(t_{0})-u_{0}\cdot v\). In addition, \(u_{0} \)and\(t_{0} \)are characterized by\(u_{0}=\nabla k_{0}( v ) \)and\(H'(t_{0})+l(u_{0})=0\).
-
2.
Let\(\hat{l}\in C_{b}(\mathbb{R}^{d} )\)and let\(1\geq \epsilon >0\). Define\(l_{\epsilon }=l_{0}+\epsilon \hat{l}\)and\(k_{\epsilon }= { (l_{\epsilon } )}^{\#}\).
-
(a)
There exists a constant\(M\)independent of\(v\)and\(\epsilon \)such that
$$ \biggl\vert \frac{k_{\epsilon }(v)-k_{0}( v ) }{\epsilon } \biggr\vert \leq M. $$ -
(b)
We have
$$ \lim_{\epsilon \rightarrow 0 } \frac{k_{\epsilon }(v)-k_{0}( v ) }{ \epsilon }=- t_{0} \hat{l}(u_{0}). $$
-
(a)
Next, we give the main result of this section.
Theorem 3.5
Assume that\(\mathbf{(A1)}\), \(\mathbf{(A2)}\), and\(\mathbf{(A3)}\)hold. Assume\(\mathcal{S}\)is a finite dimensional non-empty closed and convex subset of\(W_{0}^{1,q }({\varOmega ,\mathbb{R}^{d\times d}})\). Assume\(F\)satisfies the condition\(\textbf{(ND)}_{\mathcal{S}}\). Then, problems (3.1) and (3.6) are dual. Problem (3.6) admits a maximizer\((k_{0},l_{0}, \varphi _{0}) \)with\(k_{0}=l_{0}^{\#} \)and\(l_{0}=(k_{0})_{\#} \). Problem (3.1) admits a unique minimizer\((u_{0},\beta _{0}) \). Moreover\(u_{0} \)satisfies
Proof
Step 1. For \((u,\beta )\in \mathscr{U}^{*}_{ \mathcal{S}}\) and \((k,l,\varphi )\in \mathscr{A}\), one has
Thus \(I(u,\beta )\geq -J(k,l,\varphi )\) with equality if and only if \(\varphi \in \varPhi _{\mathcal{S}}(u)\) and
Note that if \(k\) is convex, the map \(\nabla k (F+\operatorname{div} \varphi )\) is well defined as the map \(F+\operatorname{div} \varphi \) is non-degenerate. Using Lemma 3.4(i), it follows that if \(k\) is convex, then \(I(u,\beta )=-J(k,l,\varphi )\) if and only if
Step 2. Thanks to Eq. (3.7), we may find a maximizer \((k_{0},l_{0},\varphi _{0}) \) of problem (3.5) satisfying \(k_{0}=l_{0}^{\#} \) and \(l_{0}=(k_{0})_{ \#} \). The function \(u_{0}=\nabla k_{0}(F+\operatorname{div} \varphi _{0})\) is well defined as \(k_{0}\) is convex and we set \(\beta _{0} =(H')^{-1}(-l(u _{0}))\). We have to show that \((u_{0},\beta _{0})\ \in \mathscr{U}^{*} _{\mathcal{S}}\) and \(\varphi _{0} \in \varPhi _{\mathcal{S}}(u_{0})\).
Step 3. Let \(\bar{l}\in C_{c}(\mathbb{R}^{d} ) \). For \(\epsilon \in (0,1) \), define \(l_{\epsilon }=l_{0}+\epsilon \bar{l} \) and \(k_{\epsilon }=(l_{\epsilon })^{\#} \). Using Lemma 3.4, one has
Since \(J(k_{0},l_{0},\varphi _{0})\leq J(k_{\epsilon },l_{\epsilon }, \varphi _{0}) \), we deduce that \(-\int _{\varLambda }\bar{l}\;dy + \int _{\varOmega }\beta _{0}\bar{l}(u_{0})\;dx\leq 0\). As we can replace \(\bar{l} \) by \(-\bar{l} \), one deduces that \(\int _{\varLambda } \bar{l} \;dy=\int _{\varOmega }\beta _{0}\bar{l}(u_{0})\;dx\). Therefore \((u_{0},\beta _{0})\in \mathscr{U}_{S}^{*} \).
Step 4. Let \(\varphi \in \mathcal{S}\). For \(\epsilon \in (0,1) \), set \(\varphi _{\epsilon }=\epsilon \varphi +(1-\epsilon ) \varphi _{0} \). By the convexity of \(\mathcal{S}\), the map \(\varphi _{\epsilon }\) belongs to \(\mathcal{S}\). As \(J(k_{0},l_{0}, \varphi _{0})\leq J(k_{0},l_{0},\varphi _{\epsilon }) \), we have
Thanks to Lemma 3.4, Inequality (3.22) implies
It follows from Proposition 2.8 that \(\varphi _{0}\in \varPhi _{\mathcal{S}}(u_{0}) \).
Step 5. Since \((u_{0},\beta _{0}) \in \mathscr{U}^{*}_{ \mathcal{S}}\), \(\varphi _{0} \in \varPhi _{\mathcal{S}}(u_{0})\), \(u_{0}=\nabla k_{0}(F+\operatorname{div} \varphi _{0})\), and \(\beta _{0} = (H')^{-1}(-l(u_{0}))\), we deduce that \(I(u_{0},\beta _{0})=J(k _{0},l_{0},\varphi _{0})\) and \(u_{0}\) is a minimizer of problem (3.1) thanks to relation (3.20). Suppose \((u_{1},\beta _{1})\ \in \mathscr{U}^{*}_{S}\) is another minimizer of problem (3.1). Then we have \(I(u_{1},\beta _{1})=J(k_{0},l _{0},\varphi _{0})\) and by relation (3.20), we get \(u_{1}= \nabla k_{0}(F+\operatorname{div} \varphi _{0})\) which implies \(u_{1}=u_{0}\). Next the strict convexity of \(H\) yields that \(\beta _{0}=\beta _{1}\). We conclude that \((u_{0},\beta _{0})\) is the unique minimizer of problem (3.1) and \(u_{0}\) is characterized by
□
Corollary 3.6
Assume that\(\mathbf{(A0)}\), \(\mathbf{(A1)}\), \(\mathbf{(A2)}\), and\(\mathbf{(A3)}\)hold. Assume\(\mathcal{S}\)is a finite dimensional non-empty closed and convex subset of\(W_{0}^{1,q }({\varOmega , \mathbb{R}^{d\times d}})\)and\(\nabla \varphi \)is non-degenerate whenever\(\varphi \in \mathcal{S}\). Suppose\(F \)has a countable range (thus degenerate). Then, \(F\)satisfies the condition\(\textbf{(ND)} _{\mathcal{S}}\)and problem (3.1) admits a unique solution.
Corollary 3.7
Assume that\(\mathbf{(A1)}\), \(\mathbf{(A2)}\), and\(\mathbf{(A3)}\)hold. Assume\(\mathcal{S}\)is a finite dimensional subspace of\(W_{0}^{1,q }({\varOmega ,\mathbb{R}^{d\times d}})\)and\(\nabla \varphi \)has a countable range whenever\(\varphi \in \mathcal{S}\). Suppose\(F \)is non-degenerate. Then, \(F\)satisfies the condition\(\textbf{(ND)}_{ \mathcal{S}}\)and problem (3.1) admits a unique solution.
4 The Incompressible Case
Throughout this section, we assume that \(\mathcal{S}\) is a subset of \(W_{0}^{1,q}(\varOmega ,\mathbb{R} ^{d\times d})\). We consider the following problem:
and we recall that the set \(\mathcal{U}_{\mathcal{S}}^{1}\) is defined as
We assume \(\mathcal{L}^{d}(\varOmega )=\mathcal{L}^{d}(\varLambda )\) so that \(\mathscr{U}_{\mathcal{S}}^{1}\) is non-empty.
4.1 Existence and Uniqueness via Duality
We study problem (4.1) via duality. Let \(u\in \mathscr{U}^{1}_{\mathcal{S}}\), \(\varphi \in \mathcal{S}\), \(l\in C( \varLambda )\) and \(k:\mathbb{R}^{d} \to \mathbb{R} \) satisfy \(k(v)+l(u)\geq u\cdot v\) for all \(u\in \varLambda \) and all \(v\in \mathbb{R}^{d} \). One has
This suggests that we consider the dual problem
with \(A_{0} \) being the set of all \((k,l,\varphi ) \) such that \(\varphi \in \mathcal{S}\), \(l\in C( \varLambda )\), \(\inf_{\varLambda }l=0 \) and \(k:\mathbb{R}^{d} \to \mathbb{R} \) satisfies \(k(v)+l(u)\geq u \cdot v\) for all \(u\in \bar{\varLambda }\), and all \(v\in \mathbb{R}^{d} \). Remark that we have
4.1.1 Existence and Regularity of Minimizers of Problem (4.5)
Denote by \(\mathcal{C}\) the set of all \((k,l) \) such that \(k:\mathbb{R} ^{d}\to \mathbb{R} \) and \(l:\mathbb{R} ^{d}\to \mathbb{R} \cup \{\infty \} \) satisfy
and \(l\equiv \infty \) on \(\mathbb{R} ^{d}\setminus \bar{\varLambda }\). Consider the subset \(\mathcal{C}_{0}\) of \(\mathcal{C}\) consisting of \((k,l)\in \mathcal{C}\) such that \(l\in C( \varLambda )\) and \(\inf_{\varLambda }l=0\). The following lemma is standard:
Lemma 4.1
Let\((k,l)\in \mathcal{C}\). It holds that\((l^{*}, l^{**})\in \mathcal{C}\), \(l^{*}\leq k\), \(0\leq l^{**}\leq l\)and\(l^{***}=l^{*}\). If\((k,l)\in \mathcal{C}_{0}\)then\(l^{*}(0)=0\).
Let us denote by \(C_{0}' \) the set of all \((k,l)\in C_{0}\) such that \(l^{*}=k\), \(k^{*}=l\), \(k(0)=0\), and \(l\geq 0\), and by \(A'_{0}\) the set of all \((k,l,\varphi )\) with \((k,l)\in C_{0}'\) and \(\varphi \in \mathcal{S}\). Remark that an element in \(C_{0}' \) is the couple \((\chi _{\bar{\varLambda }},(\chi _{\bar{\varLambda }})^{*})\). Hence \(A'_{0}\) is nonempty when \(\mathcal{S}\) is nonempty. One readily checks that, in light of Lemma 4.1, problem (4.5) has the same infimum value as
We recall that \(r^{*}\) is such that \(B(0,1/r^{*})\subset \varLambda \subset B(0,r^{*}/2)\).
Lemma 4.2
Assume that\(\mathbf{(A1)}\)and\(\mathbf{(A3)}\)hold. Assume that the set\(\mathcal{S}\)is a finite dimensional non-empty closed and convex subset of\(W_{0}^{1,q }({\varOmega ,\mathbb{R}^{d\times d}})\). Then, problem (4.8) admits a minimizer\((k_{0},l_{0}, \varphi _{0})\in A_{0}' \)with\(k_{0} \)convex and\(r^{*} \)-Lipschitz and\(k_{0}(0)=0 \).
Proof
Consider a minimizing sequence \(\lbrace (k_{n},l_{n}, \varphi _{n}) \rbrace _{n=1}^{\infty }\) of problem (4.8). Since \(k_{n}=l_{n}^{*}\) and \(l_{n}=(k_{n})^{*}\), \(k_{n}\) is \(r^{*}\)-Lipschitz. As \(k_{n}(0)=0\), we use Ascoli-Arzelà theorem to deduce that a subsequence of \(\lbrace k_{n} \rbrace _{n=1}^{\infty }\) converges locally uniformly to some \(k_{0}\). Next, using the growth condition (2.5) on \(f^{*}\) as well as the facts that \(k_{n}\) is \(r^{*}\)-Lipschitz, \(k_{n}(0)=0\), we establish the following estimate:
As the left hand side of (4.9) is bounded, \(l_{n}\geq 0\) and \(\mathcal{S}\) is finite dimensional, we deduce from (4.9) that a subsequence of \(\lbrace \varphi _{n} \rbrace _{n=1}^{\infty }\) converges strongly to some \(\varphi _{0}\) in \(W_{0}^{1,q}(\varOmega ,\mathbb{R} ^{d\times d})\). Invoking (4.9) again, we show that \(\lbrace \int _{\varLambda }l_{n}(y)\,dy \rbrace _{n=1}^{\infty } \) is bounded. This, combined with the fact that \(l_{n}\) is non-negative and convex, yields the existence of a subsequence of \(\lbrace l_{n} \rbrace _{n=1}^{\infty }\) that converges locally uniformly to some \(l_{0}\) (see for instance Theorem 1, p. 236 in [6]). One readily checks that \((k_{0}, l_{0}, \varphi _{0})\in A_{0}'\). We next exploit lower semi-continuity properties of the functional \(J\) to conclude that \((k_{0}, l_{0}, \varphi _{0})\) is a minimizer of \(J\) over \(A_{0}'\). □
4.1.2 A Duality Result
We have the following theorem.
Theorem 4.3
Assume that\(\mathbf{(A1)}\)and\(\mathbf{(A3)}\)hold. Assume\(\mathcal{S}\)is a finite dimensional non-empty closed and convex subset of\(W_{0}^{1,q }({\varOmega ,\mathbb{R}^{d\times d}})\). Suppose that the map\(F \)satisfies the condition\(\textbf{(ND)}_{\mathcal{S}}\). Then problems (4.1) and (4.6) are dual. Problem (4.6) admits a maximizer\((k_{0},l_{0}, \varphi _{0}) \)with\(k_{0}=l_{0}^{*} \)and\(l_{0}=(k_{0})^{*} \). Problem (4.1) admits a unique minimizer\(u_{0} \). Moreover\(u_{0} \)satisfies
Proof
Suppose \(u\in \mathscr{U}_{S}^{1} \) and \(( k,l,\varphi ) \in A_{0} \). Using (4.3) and (4.4), we see that \(I_{0}(u)\geq -J(k,l,\varphi )\) with equality if and only if \(\varphi \in \varPhi _{S}(u) \) and \(l(u)+k(F+\operatorname{div}\varphi )=u\cdot (F+\operatorname{div} \varphi ) \) for almost every \(x\in \varOmega \). The latter condition reduces to \(u(x)=\nabla k(F(x)+\operatorname{div} \varphi (x)) \) if \(k\) is convex, under the assumption \(F+\operatorname{div} \varphi \) is non-degenerate. Now, let \((k_{0},l_{0},\varphi _{0}) \in A_{0}'\) be a minimizer of \(J \) over \(A_{0} \). Since \(F+\operatorname{div} \varphi _{0} \) is non-degenerate and \(k_{0} \) is convex, the map \(u_{0}=\nabla k_{0}(F+\operatorname{div}\varphi _{0}) \) is well defined.
Variation around\(l_{0}\). Let \(\bar{l}\in C_{c}( \mathbb{R}^{d} ) \). For \(\epsilon \in (0,1) \), set \(l_{\epsilon }=l _{0}+\epsilon \bar{l} \) and \(k_{\epsilon }=(l_{\epsilon })^{*}\). Let \(v\in \mathbb{R} ^{d} \) be a point where \(k_{0} \) is differentiable. Using the measurable selection theorem, one deduces that there exists \(T_{\epsilon }:\mathbb{R} ^{d}\to \mathbb{R}^{d} \) measurable such that for all \(\epsilon \in [0,1) \)
Then, for \(\epsilon \in (0,1)\), we have
and
Moreover,
We refer the reader to Lemma A.3 for (4.10)–(4.12). Hence, as
using again (4.12), one has
Since \(J(k_{0},l_{0},\varphi _{0})\leq J(k_{\epsilon },l_{\epsilon }, \varphi _{0}) \), we deduce from (4.13) that \(-\int _{\varLambda }\bar{l} +\int _{\varOmega }\bar{l}(u_{0})\leq 0\). By replacing \(l \) by \(-l \) in the above argument, one deduces that \(\int _{\varLambda }\bar{l} =\int _{\varOmega }\bar{l}(u_{0})\). As a result, \(u_{0}\in \mathscr{U}_{S}^{1} \).
Variation around\(\varphi _{0}\). Let \(\varphi \in \mathcal{S}\). For \(\epsilon \in (0,1) \), by convexity of \(\mathcal{S}\), we have \(\varphi _{\epsilon }:=\epsilon \varphi +(1- \epsilon ) \varphi _{0}\in \mathcal{S}\). Then \(J(k_{0},l_{0},\varphi _{0})\leq J(k_{0},l_{0},\varphi _{\epsilon }) \). This implies that
As \(\epsilon \) tends to \(0^{+}\), the above equation yields
It follows from Proposition 2.8 that \(\varphi _{0}\in \varPhi _{\mathcal{S}}(u_{0}) \). □
Corollary 4.4
Assume that\(\mathbf{(A0)}\), \(\mathbf{(A1)}\), and\(\mathbf{(A3)}\)hold. Assume that\(\mathcal{S}\)is a finite dimensional non-empty closed and convex subset of\(W_{0}^{1,q }({\varOmega ,\mathbb{R}^{d\times d}})\)and\(\nabla \varphi \)is non-degenerate whenever\(\varphi \in \mathcal{S}\). Suppose\(F \)has a countable range (thus degenerate). Then, \(F \)satisfies the condition\(\textbf{(ND)}_{\mathcal{S}}\)and problem (4.1) admits a unique solution.
Corollary 4.5
Assume that\(\mathbf{(A1)}\)and\(\mathbf{(A3)}\)hold. Assume that\(\mathcal{S}\)is a finite dimensional subspace of\(W_{0}^{1,q }({\varOmega , \mathbb{R}^{d\times d}})\)and\(\nabla \varphi \)has a countable range whenever\(\varphi \in \mathcal{S}\). Suppose\(F \)is non-degenerate. Then, \(F \)satisfies the condition\(\textbf{(ND)}_{\mathcal{S}}\)and problem (4.1) admits a unique solution.
4.2 A Link Between Problem (3.1) and Problem (4.1)
Here, we explore the relationships between problem (3.1) and problem (4.1). For this purpose, we make a further assumption of the domains \(\varOmega \) and \(\varLambda \) by requiring that \(\varOmega =\varLambda \). Assume \(\mathbf{(A1)}\) holds and recall \(\lbrace H_{n} \rbrace _{n=0}^{\infty }\) as defined in (2.7) and (2.8). Then, Lemma 2.3 ensures that \(\mathbf{(A2)}\) holds for \(H_{n}\) for all \(n\geq 1\). Define
and
Recall that \(C_{0}\) is the set of all \((k,l)\) such that \(l\in C(\bar{ \varLambda })\), \(\inf l=0\) and \(k:\mathbb{R} ^{d}\to \mathbb{R} \) satisfies for all \(u\in \varLambda \) and all \(v\in \mathbb{R} ^{d}\):
Let \(C_{n}\) be the set of all \((k,l)\) such that \(l\in C(\bar{\varLambda})\) and \(k:\mathbb{R} ^{d}\to \mathbb{R} \) satisfy:
We denote by \(\mathcal{A}_{0}\) the set of all \((k,l,\varphi )\) satisfying \((k,l)\in C_{0}\) and \(\varphi \in S\). Similarly \(\mathcal{A}_{n}\) denotes the set of all \((k,l,\varphi )\) satisfying \((k,l)\in C_{n}\) and \(\varphi \in S\). If \((k,l,\varphi )\in \mathcal{A}_{0}\cup \mathcal{A}_{n}\), we still set
Lemma 4.6
Assume that\(\mathbf{(A1)}\), \(\mathbf{(A2)}\), and\(\mathbf{(A3)}\)hold. Assume that\(\mathcal{S}\)is a finite dimensional non-empty closed and convex subset of\(W_{0}^{1,q }({\varOmega ,\mathbb{R}^{d\times d}})\). For each\(n\in \mathbb{N} \), let\((u_{n},\beta _{n})\)be the unique minimizer of\(I_{n}\)over\(\mathscr{U}_{\mathcal{S}}^{*}\)as given by Theorem 3.5and let\((k_{n},l_{n},\varphi _{n})\)be a minimizer of\(J\)over\(\mathcal{A}_{n}\)with\(k_{n}\)convex and\(r^{*}\)-Lipschitz as ensured by Proposition 3.3and Lemma 4.2. Then,
-
1.
The sequence\(\{I_{n}(u_{n},\beta _{n})\}_{n\in \mathbb{N} ^{*}}\)is bounded.
-
2.
The sequence\(\{ \beta _{n} \}_{n\in \mathbb{N} ^{*}}\)converges to 1 in\(L^{2}(\varOmega ) \).
-
3.
The sequence\(\{ \varphi _{n} \}_{n\in \mathbb{N} ^{*}}\)admits a subsequence that converges to some\(\bar{\varphi }\)in\(S\)with respect to the\(W_{0}^{1,q} (\varOmega ,\mathbb{R}^{d\times d} )\)-norm.
Proof
Step 1. Let \(\bar{u}\in \mathscr{U}_{\mathcal{S}}^{1}\). We have \((\bar{u},1)\in \mathscr{U}_{\mathcal{S}}^{*}\) and thus \(I_{n}(u_{n},\beta _{n})\leq I_{n}(\bar{u},1)\) for all \(n\geq 1\). As \(H_{n}(1)=0\), it holds that \(I_{n}(\bar{u},1)=V_{\mathcal{S}}^{f}( \bar{u})-\int _{\varOmega }\bar{u}\cdot F\;dx\) which is finite. Hence
On the other hand, we use growth condition (2.5) to get
Finally, we use (4.16) and (4.17) to prove (1).
Step 2. Let \(\varphi _{0} \in \mathcal{S}\). As \(u_{n}\) has values in \(\varLambda \), it holds that
and
We combine (4.16), (4.17), (4.18), (4.19) to get
Setting \(c_{0} \mathcal{L}^{d}(\varOmega ):= R_{0}- R_{2}+r^{*}\|F\|_{ L ^{1}(\varOmega ,\mathbb{R} ^{d}) }\), we use Lemma 2.3 and (4.20) to obtain
This establishes (2).
Step 3. As \(\lbrace H_{n} \rbrace _{n=1}^{ \infty } \) is a non-decreasing sequence that converges to \(H_{0}\), it holds that \(C_{n+1}\subset C_{n}\subset C_{0}\) for all \(n\in \mathbb{N} \). Thus, as \((k_{n},l_{n})\in C_{n}\), we have \((k_{n},l _{n})\in C_{0}\) so that
Since \(-J(k_{n},l_{n},\varphi _{n})=I_{n}(u_{n},\beta _{n})\), we have \(J(k_{n},l_{n},\varphi _{n})\leq R_{1}\) for all \(n\in \mathbb{N} ^{*}\). This, combined with \(\varOmega =\varLambda \), and (4.21) yields
In view of the growth condition (2.5) and boundedness of \(\varOmega \), (4.22) implies
As the space \(\mathcal{S}\) is of finite dimension and the \(\operatorname{div}\) operator is continuous on \(\mathcal{S}\), we conclude that \(\lbrace \varphi _{n} \rbrace _{n=1}^{\infty } \) is convergent up to a subsequence in \(W_{0}^{1,q} (\varOmega ,\mathbb{R}^{d\times d} ) \) which allows us to conclude (3). □
Theorem 4.7
Assume that\(\mathbf{(A1)}\), \(\mathbf{(A2)}\), and\(\mathbf{(A3)}\)hold. Assume that\(\mathcal{S}\)is a finite dimensional non-empty closed and convex subset of\(W_{0}^{1,q }({\varOmega ,\mathbb{R}^{d\times d}})\). Assume\(F\)satisfies the condition\(\mathbf{(ND)_{\mathcal{S}}}\). For each\(n\in \mathbb{N} \), let\((u_{n},\beta _{n})\)be the unique minimizer of\(I_{n}\)over\(\mathscr{U}_{\mathcal{S}}^{*}\)as given by Theorem 3.5and let\((k_{n},l_{n},\varphi _{n})\)be a minimizer of\(J\)over\(\mathcal{A}_{n}\)with\(k_{n}\)convex and\(r^{*}\)-Lipschitz as ensured by Proposition 3.3and Lemma 4.2. Suppose that\(k_{n}\)is differentiable for all\(n\in \mathbb{N} ^{*}\). Then, the sequence\(\{ u_{n} \}_{n\in \mathbb{N} ^{*}}\)converges almost everywhere to the unique minimizer\(u_{0}\)of\(I_{0}\)over \(\mathscr{U}_{S}^{1}\). In addition, the minima\(\lbrace I_{n}(u_{n},\beta _{n}) \rbrace _{n=1}^{\infty } \)converge to\(I_{0}(u_{0})\).
Proof
Step 1. For \(n\in \mathbb{N} ^{*}\), set \(\bar{k}_{n}=k _{n}-k_{n}(0)\). Note that we have \(\bar{k}_{n}(0)=0\). Since the functions \(k_{n}\) are \(r^{*}\)-Lipschitz, so are the functions \(\bar{k}_{n}\) and we obtain that, up to a subsequence, the sequence \(\lbrace \bar{k}_{n} \rbrace _{n=1}^{\infty }\) converges locally uniformly to a certain function \(\bar{k}\). Since \(F+ \operatorname{div}\varphi _{n}\) is non-degenerate, we have that \(\nabla \bar{k}_{n}(F+\operatorname{div}\varphi _{n})\) is well-defined. Furthermore, Lemma 4.6 ensures that \(\lbrace \varphi _{n} \rbrace _{n=1}^{\infty } \) converges up to a subsequence to some \(\bar{\varphi }\in S\) with respect to the \(W^{1,q} (\varOmega , \mathbb{R}^{d} )\)-norm. As a result, \(\lbrace \operatorname{div}\varphi _{n} \rbrace _{n=1}^{ \infty }\) converges to \(\operatorname{div}\bar{\varphi }\) in \(L^{q}(\varOmega ,\mathbb{R}^{d})\). Since \(\mathcal{S}\) is of finite dimension, the \(L^{q}\) convergence of \(\lbrace \operatorname{div} \varphi _{n} \rbrace _{n=1}^{\infty }\) reduces to a pointwise convergence. Next, using the convexity of the \(\bar{k}_{n}\) and the pointwise convergence of \(\lbrace \operatorname{div}\varphi _{n} \rbrace _{n=1}^{\infty }\) to \(\operatorname{div}\varphi \), we deduce that up to a subsequence \(\lbrace \nabla \bar{k}_{n}(F+ \operatorname{div}\varphi _{n}) \rbrace _{n=1}^{\infty } \) converges a.e to \(\nabla \bar{k}(F+\operatorname{div}\bar{\varphi })\) (cf. [13] Theorem 25.7).
As a duality result, Theorem 3.5 ensures that \(\nabla \bar{k}_{n}(F+\operatorname{div}\varphi _{n})=u_{n}\). If we denote \(\bar{u}:= \nabla \bar{k}(F+\operatorname{div}\bar{\varphi })\), then, up to a subsequence, the sequence \(\{u_{n}\}_{n\in \mathbb{N} }\) converges a.e to \(\bar{u}\).
Step 2. Let \(l\in C_{b}(\mathbb{R}^{d} )\). The strong convergence in \(L^{2}(\varOmega )\) of \(\lbrace \beta _{n} \rbrace _{n=1}^{\infty } \) to 1 established in Lemma 4.6 and the almost everywhere convergence of \(\{u_{n}\}_{n\in \mathbb{N} }\) to \(\bar{u}\) obtained in Step 1 ensure that \(\lim_{n\rightarrow \infty }\int _{\varOmega }\beta _{n} l(u_{n})\,dx= \int _{\varOmega }l(\bar{u} (x))\,dx\). As \((u_{n},\beta _{n})\in \mathscr{U} _{S}^{*}\), \(\int _{\varOmega }\beta _{n}(x) l(u_{n})\,dx=\int _{\varOmega }l(y)\,dy\) for all \(l\in C_{b}(\mathbb{R}^{d} )\). It follows that in the limit \(\int _{\varOmega }l(\bar{u})\,dx=\int _{\varOmega }l(y)\,dy\) for all \(l\in C_{b}( \mathbb{R}^{d} )\) and thus \(\bar{u}\in \mathscr{U}_{S}^{1}\).
Step 3. We recall that
Since \(u\mapsto V_{S}^{f}(u)\) is lower-semicontinuous as a supremum of affine functions, by applying the Fatou’s Lemma, we have
Let \(u_{0}\) be the unique minimizer of \(I_{0}\) over \(\mathscr{U}_{S} ^{1}\) as given by Theorem 4.3. Then,
Meanwhile, as \(C_{n}\subset C_{0}\) and \((k_{0},l_{0},\varphi _{0})\) is a minimizer of \(J\) over \(C_{0}\), we have
This, along with the duality established in Theorem 3.5 imply that
We combine (4.25) and (4.26) to obtain \(I_{0}(\bar{u})=I_{0}(u_{0})\). As \(u_{0}\) is the unique minimizer of \(I_{0}\) over \(\mathscr{U}_{S}^{1}\) we have \(u_{0}=\bar{u}\). We note that the limit \(\bar{u}\) does not depend on the subsequence of \(\{u_{n}\} _{n}\) chosen. Thus, the whole sequence \(\{u_{n}\}_{n}\) converges a.e. to \(u_{0}\). In addition, \(\{I_{n}(u_{n},\beta _{n})\}_{n}\) converges to \(I_{0}(u_{0})\). □
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Most of the work presented in this paper was carried out while R.A. was a postdoctoral fellow at the Institute for Mathematics and its Applications during the IMA’s annual program on Control Theory and its Applications.
M.S. gratefully acknowledges the support of the King Abdullah University of Science and Technology.
Appendix
Appendix
1.1 A.1 Proof of Lemma 3.4
We will prove Lemma 3.4 through two lemmas. The results of the first lemma can be found in Lemma 4.3 of [2]. We give here a sketch of the proof for the convenience of the reader.
Lemma A.1
Assume that\(\mathbf{(A2)}\)holds. Consider a lower semicontinuous function\(l:\mathbb{R}^{d} \to \bar{\mathbb{R}} \)such that\(\inf_{\bar{\varLambda }}l>-\infty \); \(l\)is finite on\(\varLambda \)and\(l \equiv +\infty \)on\(\mathbb{R}^{d} \setminus \bar{\varLambda }\). Set\(k= l^{\#}\)and let\(w\in \mathbb{R}^{d} \). Then:
-
1.
There exist\(\bar{u}\in \bar{\varLambda }\)and\(\bar{t}>0\)such that
$$ k( w ) =-\bar{t}l(\bar{u})-H(\bar{t})-\bar{u}\cdot w. $$(A.1)Moreover, \(\bar{u} \)and\(\bar{t} \)satisfy\(\bar{u}\in \partial k( w) \)and\(H'(\bar{t})+l(\bar{u})=0\).
-
2.
If\(k\)is differentiable at\(w\)then\(\bar{u}\)and\(\bar{t}\)are uniquely determined by\(\bar{u}=\nabla k(w)\)and\(\bar{t}=(H')^{-1}(-l( \bar{u}))\).
Proof
(1.) We have
Consider a maximizing sequence \(\{(u_{n},t_{n})\}_{n=1}^{\infty } \) in (A.2). As \(0\in \varLambda \), we may assume without loss of generality that
for \(n\geq 1\). It follows that
for \(n\geq 1\). In light of the growth condition on \(H \) in (A2) there exists a positive real number \(\alpha \) such that \(\{t_{n}\}_{n=1}^{\infty }\subset [\alpha ,\alpha ^{-1}] \). As \(\varLambda \) is bounded, we may assume without loss of generality that the sequence \(\{(u_{n},t_{n})\}_{n=1}^{\infty }\) converges to some \((\bar{u},\bar{t})\in \bar{\varLambda }\times [\alpha ,\alpha ^{-1}] \). We next use the lower semicontinuity of \(H \) and \(l \) to deduce that
Note that \(k(w)\geq \bar{u}\cdot w-l(\bar{u})t-H( t) \text{ for all }t>0\). In view of (A.3), it follows that \(g: (0, \infty )\to \mathbb{R}\) defined by \(g(t)=\bar{u}\cdot w-l(\bar{u})t-H( t) \) admits a maximum at \(\bar{t} \). As \(g\) is differentiable at \(\bar{t}\), we have \(g'(\bar{t})=0 \), that is, \(l(\bar{u})+H'( \bar{t})=0 \). Next, observe that \(k(z)\geq \bar{u}\cdot z-l(\bar{u}) \bar{t}-H( \bar{t}) \text{ for all }z\in \mathbb{R} ^{d}\). In light of the convexity of \(k\) we have that \(\bar{u}\in \partial k(w) \).
(2.) Assume that \(k\) is differentiable at \(w\). Then, \(\bar{u}\) is uniquely determined as \(\bar{u}=\nabla k (w) \). As \(H'(\bar{t})=-l_{0}(\bar{u}) \) and \(H' \) is a bijection, we obtain that \(\bar{t} \) is also uniquely determined as \(\bar{t}=(H')^{-1}(-l(\bar{u}))\). □
The second lemma which is inspired by Lemma 4.4 in [2] is the following:
Lemma A.2
Assume that\(\mathbf{(A2)}\)holds. Consider a lower semicontinuous function\(l_{0}:\mathbb{R}^{d} \to \bar{\mathbb{R}} \)such that\(\inf_{\bar{\varLambda }}l_{0}>-\infty \); \(l_{0}\)is finite on\(\varLambda \)and\(l_{0}\equiv +\infty \)on\(\mathbb{R}^{d} \setminus \bar{ \varLambda }\). Set\(k_{0}= ({l_{0}})^{\#}\). Let\(\hat{l}\in C_{b}( \mathbb{R}^{d} )\)and let\(1\geq \epsilon >0\). Define\(l_{\epsilon }=l _{0}+\epsilon \hat{l}\)and\(k_{\epsilon }={ (l_{\epsilon } )} ^{\#}\). Let\(v\in \mathbb{R}^{d} \)be such that\(k_{0}\)is differentiable at\(v\).
-
1.
There exists a constant\(M\)independent of\(v\)and\(\epsilon \)such that
$$ \biggl\vert \frac{k_{\epsilon }(v)-k_{0}( v ) }{\epsilon } \biggr\vert \leq M. $$(A.4) -
2.
We have
$$ \lim_{\epsilon \rightarrow 0 } \frac{k_{\epsilon }(v)-k_{0}( v ) }{ \epsilon }=- t_{0}\hat{l}(u_{0}). $$(A.5)
Proof
Note that the map \(l_{\epsilon }=l_{0}+\epsilon \hat{l} \) is bounded below by \(m-|\hat{l}|_{\infty }\). As \(k_{\epsilon }={ (l _{\epsilon } )}^{\#}\) and \(k_{0}={ (l_{0} )}^{\#}\), Lemma A.1 ensures that there exist \(t_{0}, t_{\epsilon }>0 \) and \(u_{0}, u_{\epsilon }\in \bar{\varLambda }\) such that
and
We then have
and
We combine (A.6) and (A.7) to get
Using again Lemma A.1 we have
As \(l_{\delta }\) is bounded below by \(m-|l|_{\infty }\), we use the fact that \(H' \) is a continuous and strictly increasing bijection from \((0,\infty )\) to ℝ to deduce that \(t_{\delta }\) is bounded above by \(M_{1}>0 \) given by \(M_{1}:=(H')^{-1}(-m+|\hat{l}|_{\infty })\). This bound on \(t_{\delta }\) combined with (A.8) yields a constant \(M:=|\hat{l}|_{\infty }(H')^{-1}(-m+| \hat{l}|_{\infty })\) such that (A.4) holds. As a result \(\lim_{\epsilon \to 0^{+}}k_{\epsilon }(v)=k_{0}(v)\). Next, let \(\{e_{n}\}_{n=1}^{\infty }\subset (0,1] \) converging to 0 such that \(\limsup_{\epsilon \to 0} \hat{l}(u_{\epsilon })t_{\epsilon }= \lim_{n\to \infty }\hat{l}(u _{e_{n}} )t_{e_{n}}\). Without loss of generality, we may assume that \(\{u _{e_{n}}\}_{n=1}^{\infty } \) converges to some \(\bar{u}\in \bar{\varLambda }\) and \(\{ t _{e_{n}}\} _{n=1}^{\infty } \) converges to \(\bar{t} \in [0,M_{1}]\). Exploiting the lower semicontinuity of \(l_{0} \), \(\hat{l} \) and \(H \), we get:
It follows that \(k_{0}(v)=\bar{u} v-l_{0}(\bar{u})\bar{t}-H(\bar{t}) \). As \(k_{0}\) is differentiable at \(v\), we have \(t_{0}=\bar{t} \) and \(u_{0}=\bar{u} \). We use (A.8), the definition of \(\{e_{n}\}_{n=1}^{\infty }\), the convergence of \(\{u _{e_{n}}\}_{n=1} ^{\infty } \) and \(\{ t _{e_{n}}\}_{n=1}^{\infty } \) to obtain
As a result, \(\lim_{\epsilon \to 0}-t_{\epsilon }\hat{l}(u_{\epsilon })=-t_{0}\hat{l}(u_{0}) \). We invoke one more time Eq. (A.8) to obtain (A.5). □
1.2 A.2 Some Properties of the Legendre Transform
We have the following lemma which is similar to Lemma 3.4 but uses the Legendre transform instead of the \((\cdot )^{\#}\) operator.
Lemma A.3
Consider a lower semicontinuous function\(l_{0}:\mathbb{R}^{d} \to \bar{ \mathbb{R}} \)such that\(\inf_{\bar{\varLambda }}l_{0}>-\infty \); \(l_{0}\)is finite on\(\varLambda \)and\(l_{0}\equiv +\infty \)on\(\mathbb{R}^{d} \setminus \bar{\varLambda }\). Set\(k_{0}= ({l_{0}})^{*}\).
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1.
There exists a measurable map\(T_{0}:\mathbb{R}^{d} \to \mathbb{R} ^{d}\)such that\(k_{0}(v)=v\cdot T_{0}(v)-l_{0}(T_{0}(v)) \)for all\(v\in \mathbb{R}^{d} \)and\(T_{0}(v)=\nabla k_{0}(v) \)whenever\(k_{0} \)is differentiable at\(v\in \mathbb{R}^{d} \).
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2.
Let\(\hat{l}\in C_{b}(\mathbb{R}^{d} )\)and let\(1\geq \epsilon >0\). Define\(l_{\epsilon }=l_{0}+\epsilon \hat{l}\)and\(k_{\epsilon }= { (l_{\epsilon } )^{*}}\).
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(a)
For all\(v\in \mathbb{R}^{d} \)we have:
$$ \biggl\vert \frac{k_{\epsilon }(v)-k_{0}( v ) }{\epsilon } \biggr\vert \leq | \hat{l}|_{\infty }. $$ -
(b)
For\(\epsilon \in (0,1)\), there exists a map\(T_{\epsilon }: \mathbb{R}^{d} \to \mathbb{R}^{d} \)satisfying for all\(v\in \mathbb{R}^{d} \): \(k_{\epsilon }(v)=vT_{\epsilon }(v)-l_{\epsilon }(T _{\epsilon }(v))\). When\(k_{0} \)is differentiable at\(v\in \mathbb{R}^{d} \), we have\(\lim_{\epsilon \to 0}T_{\epsilon }(v)= \nabla k_{0}(v)\)and
$$ \lim_{\epsilon \rightarrow 0 } \frac{k_{\epsilon }(v)-k_{0}( v ) }{ \epsilon }=- t_{0} \hat{l}\bigl(\nabla k_{0}(v)\bigr). $$
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(a)
Proof
(1.) Let \(v\in \mathbb{R}^{d} \). We have
We use the lower semicontinuity of \(l_{0} \) and the compactness of \(\bar{\varLambda }\) to deduce that there exists \(\bar{u}\in \varLambda \) such that \(k_{0}(v)=\bar{u} v-l_{0}(\bar{u})\). We have \(k_{0}(w)-( \bar{u}w-l_{0}(\bar{u}))\geq 0 \) for all \(w\in \mathbb{R}^{d} \) while \(k_{0}(v)-(\bar{u}v-l_{0}(\bar{u}))= 0 \). Since \(k_{0} \) is convex, we deduce that \(\bar{u} \in \partial k_{0}(v) \).
Next, for \(v\in \mathbb{R}^{d} \), define
Assume \(\{u_{n}\}_{n\in \mathbb{N}}\subset \mathbb{R}^{d} \) converges to \(u \); \(\{v_{n}\}_{n\in \mathbb{N}}\subset \mathbb{R}^{d} \) converges to \(v \) and for all \(n\in \mathbb{N} \), one has \(u_{n}\in \varGamma (v_{n}) \). Then \(u\in \varGamma (v) \). Indeed, one has
Therefore, \(uv-l_{0}(u)=k_{0}(v) \) and \(u\in \varGamma (v) \). As a result, the multifunction \(\varGamma :\mathbb{R}^{d} \rightrightarrows \mathbb{R}^{d} \) is closed and nonempty valued. By the Measurable Selection Theorem [14, Corollary 14.6], there exists a measurable map \(T_{0}:\mathbb{R}^{d}\to \mathbb{R}^{d} \) such that for all \(v\in \mathbb{R}^{d} \), one has \(T_{0}(v)\in \varGamma (v) \). That is \(k_{0}(v)=vT_{0}(v)-l_{0}(T_{0}(v)) \). As \(T(v)\in \varGamma (v)\subset \partial k_{0}(v) \), we also have \(T_{0}=\nabla k_{0} \) almost everywhere.
(2.) For \(\epsilon >0 \), \(l_{\epsilon }\) is bounded below and satisfies the hypothesis on \(l_{0} \). Let \(k_{\epsilon }=l_{\epsilon }^{*} \) and consider a map \(T_{\epsilon }\) satisfying for all \(v\in \mathbb{R} ^{d} \): \(k_{\epsilon }(v)=vT_{\epsilon }(v)-l_{\epsilon }(T_{\epsilon }(v))\) as given by part 1.). We have for \(v\in \mathbb{R}^{d} \):
Similarly, for \(v\in \mathbb{R}^{d} \) we have
We combine (A.10) and (A.11) to get
which leads to
Consider a sequence \(\{\epsilon _{n}\}_{n} \) converging to 0. The sequence \(\{T_{\epsilon _{n}}(v)\}_{n} \) is bounded so we may find a subsequence \(\{\epsilon '_{n}\}_{n} \) of \(\{\epsilon _{n}\}_{n} \) such that the sequence \(\{T_{\epsilon '_{n}}(v)\}_{n} \) converges to \(u\in \bar{\varLambda }\). We then have:
We use (A.14) to obtain \(k_{0}(v)=vu-l_{0}(u)\) and thus \(u=\nabla k_{0}(v) \) as \(k_{0} \) is differentiable at \(v \). It follows that \(\lim_{\epsilon \to 0}T_{\epsilon }(v)= \nabla k_{0}(v)\). We use Eq. (A.12) and the continuity of \(\hat{l} \) to obtain
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Awi, R., Sedjro, M. On the Uniqueness of Minimizers for a Class of Variational Problems with Polyconvex Integrand. Acta Appl Math 168, 137–167 (2020). https://doi.org/10.1007/s10440-019-00282-0
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DOI: https://doi.org/10.1007/s10440-019-00282-0