%% This document created by Scientific Word (R) Version 3.5 % \usepackage{graphicx} % \setcounter{MaxMatrixCols}{30} %NEW %ENDNEW % \newtheorem{acknowledgements}[theorem]{Acknowledgements} % \newtheorem{algorithm}[theorem]{Algorithm} % \newtheorem{axiom}[theorem]{Axiom} % \newtheorem{case}[theorem]{Case} % \newtheorem{claim}[theorem]{Claim} % \newtheorem{conclusion}[theorem]{Conclusion} % \newtheorem{condition}[theorem]{Condition} % \newtheorem{conjecture}[theorem]{Conjecture} % \newtheorem{criterion}[theorem]{Criterion} % \newtheorem{definition}[theorem]{Definition} % \newtheorem{example}[theorem]{Example} % \newtheorem{exercise}[theorem]{Exercise} % \newtheorem{notation}[theorem]{Notation} % \newtheorem{problem}[theorem]{Problem} % \newtheorem{summary}[theorem]{Summary} \documentclass{article}% \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{graphicx}% \setcounter{MaxMatrixCols}{30} \date{May 8, 2003} %TCIDATA{OutputFilter=latex2.dll} %TCIDATA{Version=4.10.0.2345} %TCIDATA{CSTFile=LaTeX article.cst} %TCIDATA{Created=Thursday, October 17, 2002 15:36:53} %TCIDATA{LastRevised=Wednesday, May 07, 2003 13:44:16} %TCIDATA{} %TCIDATA{} %TCIDATA{Language=American English} %TCIDATA{ComputeDefs= %$F\left( A\right) :=g\left( \frac{M\left( A\right) }{{}}\right) \qquad %A\in M^{n\times n}$ %} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em} \medskip\smallskip } \numberwithin{equation}{section} \begin{document} \title{Higher Order Quasiconvexity Reduces to Quasiconvexity} \author{Gianni Dal Maso \and Irene Fonseca \and Giovanni Leoni \and Massimiliano Morini} \maketitle \begin{abstract} In this paper it is shown that higher order quasiconvex functions suitable in the variational treatment of problems involving second derivatives may be extended to the space of all matrices as classical quasiconvex functions. Precisely, it is proved that a smooth strictly $2$-quasiconvex function with $p$-growth at infinity, $p>1$, is the restriction to symmetric matrices of a $1$-quasiconvex function with the same growth. As a consequence, lower semicontinuity results for second-order variational problems are deduced as corollaries of well-known first order theorems. \end{abstract} \section{Introduction} In recent years there has been a renewed interest in higher order variational problems motivated by various mathematical models in engineering and materials science: in connection with the so-called gradient theories of phase transitions within elasticity regimes (see \cite{CFL}, \cite{KM}, \cite{Mue}); in the study of equilibria of micromagnetic materials where mastery of second order energies (here accounting for the exchange energy) is required (see \cite{CKO}, \cite{DS}, \cite{Mue}, \cite{RS}); in the theory of second order structured deformations (SOSD) (see \cite{OP}), in the Blake-Zisserman model for image segmentation in computer vision (see \cite{CLT}); etc.. In the study of lower semicontinuity, relaxation and $\Gamma$% -convergence\ problems for second order functional the natural notion of convexity, $2$-quasiconvexity, was introduced by Meyers in \cite{M} (see also \cite{BCO}, \cite{F}). We recall that a real valued function $f$, defined on the space $\mathbb{M}_{\operatorname*{sym}}^{n\times n}$ of ${n\times n}$ symmetric matrices, is \textit{$2$-quasiconvex} if \[ \int_{Q}\left[ f\left( A+\nabla^{2}\phi\right) -f\left( A\right) \right] \,dx\geq0 \] for every $A\in\mathbb{M}_{\operatorname*{sym}}^{n\times n}$ and every $\phi\in C_{c}^{2}\left( Q\right) $, where $Q:=\left( 0,1\right) ^{n}$ is the unit cube. While lower semicontinuity properties of functionals depending \textit{only} on second order derivatives can be proved easily, when lower order terms are present, the question is significantly more difficult, since sufficient tools to handle localization and truncation of gradients are still missing. To bypass these difficulties one would be tempted to transform higher order into first order problems, where one uses the standard notion of quasiconvexity, called $1$-quasiconvexity in this paper. We recall that a real valued function $f$, defined on the space $\mathbb{M}^{n\times n}$ of ${n\times n}$ matrices, is \textit{$1$-quasiconvex} if \[ \int_{Q}\left[ f\left( A+\nabla\varphi\right) -f\left( A\right) \right] \,dx\geq0 \] for every $A\in\mathbb{M}^{n\times n}$ and every $\varphi\in C_{c}^{1}\left( Q;\mathbb{R}^{n}\right) $. Thus we are led to the following question. \begin{enumerate} \item[\textbf{(Q)} ] \textit{Is every }$2$\textit{-quasiconvex function the restriction of a }$1$\textit{-quasiconvex function to the space of symmetric matrices?} \end{enumerate} A good indication of the plausibility of an affirmative answer is that it holds for polyconvex functions as noticed by Dacorogna and Fonseca. Indeed if \[ f\left( A\right) =g\left( M\left( A\right) \right) \qquad A\in \mathbb{M}_{\operatorname*{sym}}^{n\times n}, \] where $g$ is a convex function and $M\left( A\right) $ stands for the vector whose components are all the minors of $A$, then the function \[ F\left( A\right) :=g\left( \frac{M\left( A\right) +M\left( A\right) ^{t}}{2}\right) \qquad A\in\mathbb{M}^{n\times n}% \] is a polyconvex extension of $f$ to the whole space $\mathbb{M}^{n\times n}$ of ${n\times n}$ matrices. It is known that $2$-gradient Young measures, i.e. Young measures generated by second order gradients, may be characterized by duality via Jensen's inequality with respect to $2$-quasiconvex functions (see \cite{FM}), just as gradient Young measures are characterized by duality with $1$-quasiconvex functions (see \cite{KP}). Therefore, the understanding of the structure of $2$-gradient Young measures helps the study of $2$-quasiconvex functions, and, accordingly, the following result by \v{S}ver\'{a}k in \cite[Lemma 1]{S} provides further evidence that $1$-quasiconvexity and $2$-quasiconvexity are somehow strictly linked: If a Young measure $\nu$ on $\mathbb{M}^{n\times n}$ is generated by a sequence $\left\{ \nabla u_{k}\right\} $ of gradients, with $\left\{ u_{k}\right\} $ bounded in $W^{1,p}\left( \Omega ;\mathbb{R}^{n}\right) $ for some $p>1$, and $\operatorname*{supp}\nu _{x}\subset\mathbb{M}_{\operatorname*{sym}}^{n\times n}$ for $\mathcal{L}^{n}$ a.e. $x\in\Omega$, then $\nu$ is generated also by a sequence $\left\{ \nabla^{2}w_{k}\right\} $, with $\left\{ w_{k}\right\} $ bounded in $W^{2,p}\left( \Omega\right) $. A partial answer to \textbf{(Q)} was given by M\"{u}ller and \v{S}ver\'{a}k (see \cite{MS}). Indeed, as an auxiliary result to construct a counter-example to regularity for elliptic systems, they proved that any smooth, strictly $2$-quasiconvex function $f:\mathbb{M}_{\operatorname*{sym}}^{2\times 2}\rightarrow\mathbb{R}$, with bounded second derivatives, is the restriction of a $1$-quasiconvex function. The main purpose of this paper is to extend their result to any space dimension and to a larger class of strictly $2$-quasiconvex functions with $p$-growth at infinity, with $p>1$. \begin{theorem} \label{theorem 2}Let $f\in C^{1}\left( \mathbb{M}_{\operatorname*{sym}% }^{n\times n}\right) $ satisfy the following conditions for suitable constants $p>1$, $\mu\geq0$, $L\geq\nu>0$: \begin{enumerate} \item[(a) ] (strict $2$-quasiconvexity)% \[ \int_{Q}\left[ f\left( A+\nabla^{2}\phi\right) -f\left( A\right) \right] \,dx\geq\nu\int_{Q}\left( \mu^{2}+\left\vert A\right\vert ^{2}+\left\vert \nabla^{2}\phi\right\vert ^{2}\right) ^{\!\!\frac{p-2}{2}}\left\vert \nabla^{2}\phi\right\vert ^{2}\,dx \] for every $A\in\mathbb{M}_{\operatorname*{sym}}^{n\times n}$ and every $\phi\in C_{c}^{2}\left( Q\right) $; \item[(b)] (Lipschitz condition for gradients)% \begin{equation} \left\vert \nabla f\left( A+B\right) -\nabla f\left( A\right) \right\vert \leq L\left( \mu^{2}+\left\vert A\right\vert ^{2}+\left\vert B\right\vert ^{2}\right) ^{\frac{p-2}{2}}\left\vert B\right\vert \label{3}% \end{equation} for every $A$, $B\in\mathbb{M}_{\operatorname*{sym}}^{n\times n}$. \end{enumerate} \noindent Then there exists a $1$-quasiconvex function $F:\mathbb{M}^{n\times n}\rightarrow\mathbb{R}$ such that \begin{align} & F\left( A\right) =f\left( A\right) \qquad\forall A\in\mathbb{M}% _{\operatorname*{sym}}^{n\times n},\label{5}\\ & \left\vert F\left( A\right) \right\vert \leq c_{f}\left( 1+\left\vert A\right\vert ^{p}\right) \qquad\forall A\in\mathbb{M}^{n\times n}, \label{6}% \end{align} for a suitable constant $c_{f}>0$ depending on $f$. \end{theorem} We remark that a $1$-quasiconvex function $F$ satisfying (\ref{5}) and (\ref{6}) is constructed explicitly if $p\geq2$ (see (\ref{900})), while in the case $1
1$, $\mu\geq0$, $\nu>0$, $M>0$: \begin{enumerate} \item[(a) ] (strict $2$-quasiconvexity)% \[ \int_{Q}\left[ f\left( A+\nabla^{2}\phi\right) -f\left( A\right) \right] \,dx\geq\nu\int_{Q}\left( \mu^{2}+\left\vert A\right\vert ^{2}+\left\vert \nabla^{2}\phi\right\vert ^{2}\right) ^{\!\!\frac{p-2}{2}}\left\vert \nabla^{2}\phi\right\vert ^{2}\,dx \] for every $A\in\mathbb{M}_{\operatorname*{sym}}^{n\times n}$ and every $\phi\in C_{c}^{2}\left( Q\right) $; \item[(b)] (growth condition)% \begin{equation} \left\vert f\left( A\right) \right\vert \leq M\left( 1+\left\vert A\right\vert ^{p}\right) \label{8}% \end{equation} for every $A\in\mathbb{M}_{\operatorname*{sym}}^{n\times n}$. \end{enumerate} \noindent Then there exists an increasing sequence $\left\{ F_{k}\right\} $ of $1$-quasiconvex functions $F_{k}:\mathbb{M}^{n\times n}\rightarrow \mathbb{R}$ such that \begin{align} & \lim_{k\rightarrow\infty}F_{k}\left( A\right) =f\left( A\right) \qquad\forall A\in\mathbb{M}_{\operatorname*{sym}}^{n\times n},\label{1}\\ & \left\vert F_{k}\left( A\right) \right\vert \leq c_{k}\left( 1+\left\vert A\right\vert ^{p}\right) \qquad\forall A\in\mathbb{M}^{n\times n}, \label{7}% \end{align} for a suitable sequence of constants $\left\{ c_{k}\right\} $ depending only on $k$ and on the structural constants $p$, $\mu$, $\nu$, $M$, but not on the specific function $f$. \end{theorem} Theorem \ref{theorem 1} allows us to reduce lower semicontinuity problems for $2$-quasi\allowbreak convex normal integrands of the form $f=f(x,u,\nabla u,\nabla^{2}u)$ to first order problems (see Section \ref{section lower} for more details). Indeed, as a consequence of Theorem \ref{theorem 1} we can prove the following result, which extends to the second order setting a lower semicontinuity property of $1$-quasiconvex functions in $SBV(\Omega ;{\mathbb{R}}^{d})$ due to Ambrosio \cite{AFP} and later generalized by Kristensen \cite{K}. For the definition and properties of the space $SBH(\Omega)$ we refer to \cite{CLT0} and~\cite{CLT}. \begin{theorem} \label{theorem4}Let $\Omega\subset{\mathbb{R}}^{n}$ be a bounded open set and let \[ f:\Omega\times{\mathbb{R}}\times{\mathbb{R}}^{n}\times\mathbb{M}% _{\operatorname*{sym}}^{n\times n}\rightarrow\lbrack0,+\infty) \] be an integrand which satisfies the following conditions: \begin{enumerate} \item[(a)] the function $f(x,\cdot,\cdot,\cdot)$ is lower semicontinuous on ${\mathbb{R}}\times{\mathbb{R}}^{n}\times\mathbb{M}_{\operatorname*{sym}% }^{n\times n}$ for $\mathcal{L}^{n}$ a.e. $x\in\Omega$; \item[(b)] the function $f(x,u,\xi,\cdot)$ is $2$-quasiconvex on $\mathbb{M}_{\operatorname*{sym}}^{n\times n}$ for $\mathcal{L}^{n}$ a.e. $x\in\Omega$ and every $\left( u,\xi\right) \in{\mathbb{R}}\times {\mathbb{R}}^{n}$; \item[(c)] there exist a locally bounded function $a:\Omega\times{\mathbb{R}% }\times{\mathbb{R}}^{n}\rightarrow\lbrack0,+\infty)$ and a constant $p>1$ such that% \[ 0\leq f(x,u,\xi,A)\ \leq a\left( x,u,\xi\right) (1+|A|^{p}) \] for $\mathcal{L}^{N}$ a.e.\ $x\in\Omega$ and every $(u,\xi,A)\in{\mathbb{R}% }^{n}\times{\mathbb{R}}^{n}\times\mathbb{M}_{\operatorname*{sym}}^{n\times n}$. \end{enumerate} \noindent Then \[ \int_{\Omega}f(x,u,\nabla u,\nabla^{2}u)\,dx\leq\liminf_{j\rightarrow\infty }\int_{\Omega}f(x,u_{j},\nabla u_{j},\nabla^{2}u_{j})\,dx \] for every $u\in SBH(\Omega)$ and any sequence $\{u_{j}\}\subset SBH(\Omega)$ converging to $u$ in $W^{1,1}(\Omega)$ and such that \begin{equation} \sup_{j}\left( \left\Vert \nabla^{2}u_{j}\right\Vert _{L^{p}}+\int_{S(\nabla u_{j})}\theta(\left\vert \left[ \nabla u_{j}\right] \right\vert )\,d\mathcal{H}^{n-1}\right) <\infty, \label{700}% \end{equation} where $\theta:\left[ 0,\infty\right) \rightarrow\left[ 0,\infty\right) $ is a concave, nondecreasing function such that% \[ \lim_{t\rightarrow0^{+}}\frac{\theta\left( t\right) }{t}=\infty, \] and $\left[ \nabla u_{j}\right] $ denotes the jump of $\nabla u_{j}$ on the jump set $S(\nabla u_{j})$. \end{theorem} An analogous result has been proved in \cite{FLP} in the space $BH(\Omega)$ in the case where (\ref{700}) is replaced by \[ \sup_{j}\left\Vert \nabla^{2}u_{j}\right\Vert _{L^{p}}<\infty\qquad\left\vert D_{s}^{2}u_{j}\right\vert (\Omega)\rightarrow0, \] where $D_{s}^{2}u_{j}$ is the singular part of the $\mathbb{M}% _{\operatorname*{sym}}^{n\times n}$-valued measure $D^{2}u_{j}$. Note that condition (\ref{700}) arises naturally in the context of free-discontinuity problems (see~\cite{AFP}). As a corollary of Theorem \ref{theorem4} we have the following result. \begin{corollary} \label{corollary lower}Let $\Omega$ and $f$ be as in Theorem \ref{theorem4}. Then \[ \int_{\Omega}f(x,u,\nabla u,\nabla^{2}u)\,dx\leq\liminf_{j\rightarrow\infty }\int_{\Omega}f(x,u_{j},\nabla u_{j},\nabla^{2}u_{j})\,dx \] for every $u\in W^{2,p}(\Omega)$ and any sequence $\{u_{j}\}\subset W^{2,p}(\Omega)$ weakly converging to $u$ in $W^{2,p}(\Omega)$. \end{corollary} In this generality Corollary \ref{corollary lower} was proved in \cite{FLP} and under stronger hypotheses in \cite{F}, \cite{GP}, and \cite{M}. \begin{remark} \emph{All of the above are still valid in a vectorial setting, i.e.,} $u:\Omega\rightarrow\mathbb{R}^{d}$, \emph{with} $\mathbb{M}% _{\operatorname*{sym}}^{n\times n}$ \emph{replaced now by} $\left( \mathbb{M}_{\operatorname*{sym}}^{n\times n}\right) ^{d}$. \emph{The proofs are entirely similar to those presented in this paper for the case} $d=1$, \emph{and we leave the obvious adaptations to the reader. } \end{remark} The paper is organized as follows. In Section \ref{section auxiliary} we present some auxiliary results including the Korn-type inequality mentioned above. In Section \ref{section proofs} we prove Theorems \ref{theorem 2} and \ref{theorem 1}, while Theorem \ref{theorem4} is addressed in the last section. \section{Auxiliary results\label{section auxiliary}} We begin with some results on the Helmholtz Decomposition and on Korn's type inequalities. A function $w:\mathbb{R}^{n}\rightarrow\mathbb{R}^{d}$ is said to be \textit{$Q$--periodic }if $w(x+e_{i})=w(x)$ for a.e. $x\in\mathbb{R}^{n}$ and every $i=1,\ldots,n$, where $(e_{1},\ldots,e_{n})$ is the canonical basis of $\mathbb{R}^{n}$. The spaces of $Q$-periodic functions of $W_{\operatorname*{loc}}^{1,p}\left( \mathbb{R}^{n};\mathbb{R}^{n}\right) $, $W_{\operatorname*{loc}}^{2,p}\left( \mathbb{R}^{n}\right) $, and $C^{\infty}(\mathbb{R}^{n};\mathbb{R}^{n})$ are denoted by $W_{\operatorname*{per}}^{1,p}\left( Q;\mathbb{R}^{n}\right) $, $W_{\operatorname*{per}}^{2,p}\left( Q\right) $, and $C_{\operatorname*{per}% }^{\infty}(Q;\mathbb{R}^{n})$, respectively. \begin{lemma} [Helmholtz decomposition]\label{lemma 6}For every $p>1$ and every $\varphi\in W_{\operatorname*{per}}^{1,p}\left( Q;\mathbb{R}^{n}\right) $ there exist two functions $\phi\in W_{\operatorname*{per}}^{2,p}\left( Q\right) $ and $\psi\in W_{\operatorname*{per}}^{1,p}\left( Q;\mathbb{R}^{n}\right) $ such that \[ \varphi=\nabla\phi+\psi,\quad\operatorname*{div}\psi=0. \] The function $\psi$ is uniquely determined, while $\phi$ is determined up to an additive constant. \end{lemma} \begin{proof} Since, by periodicity, $\operatorname*{div}\varphi$ has zero average on $Q$, there exists a $Q$-periodic\ solution $\phi$ of the equation $\Delta \phi=\operatorname*{div}\varphi$, which is unique up to an additive constant. It is clear now that $\psi:=\varphi-\nabla\phi$ is $Q$-periodic and $\operatorname*{div}\psi=0$. \end{proof} Throughout the paper, for every $A\in\mathbb{M}^{n\times n}$, we denote the symmetric and antisymmetric parts of $A$ by \[ A^{s}:=\frac{A+A^{t}}{2},\qquad A^{a}:=\frac{A-A^{t}}{2}, \] where $A^{t}$ is the transpose matrix of $A$. If $\psi:\Omega\subset\mathbb{R}^{n}\rightarrow\mathbb{R}^{d}$ is any function, then $\nabla\psi$\ is a $d\times n$ matrix in $\mathbb{M}^{d\times n}$, with $\left( \nabla\psi\right) _{ij}:=\frac{\partial\psi_{i}}{\partial x_{j}}.$\ Also, differential operators applied to matrix-valued fields are understood on a row-by-row basis, e.g., if $\psi:\Omega\subset\mathbb{R}% ^{n}\rightarrow\mathbb{R}^{n}$, then \[ \operatorname*{div}\nabla\psi:=\left( \begin{array} [c]{c}% \operatorname*{div}\nabla\psi_{1}\\ \vdots\\ \operatorname*{div}\nabla\psi_{n}% \end{array} \right) . \] To simplify the notation, for any $\psi:\Omega\subset\mathbb{R}^{n}% \rightarrow\mathbb{R}^{n}$, we set% \[ \nabla\psi^{t}:=\left( \nabla\psi\right) ^{t},\qquad\nabla\psi^{s}:=\left( \nabla\psi\right) ^{s},\qquad\nabla\psi^{a}:=\left( \nabla\psi\right) ^{a}. \] \begin{lemma} \label{lemma 1}For every $p>1$ there exists a constant $\gamma_{n,p}\geq1$ such that \begin{equation} \int_{Q}\left\vert \nabla\psi\right\vert ^{p}\,dx\leq\gamma_{n,p}\int _{Q}\left\vert \nabla\psi^{a}\right\vert ^{p}\,dx \label{451}% \end{equation} for every $Q$-periodic function $\psi:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ of class $C^{\infty}$ with $\operatorname*{div}\psi=0$. \end{lemma} \begin{proof} Since $\operatorname*{div}\nabla\psi^{t}=\nabla\left( \operatorname*{div}% \psi\right) =0$ we have% \begin{equation} \Delta\psi=2\operatorname*{div}\left( \frac{\nabla\psi-\nabla\psi^{t}}% {2}\right) =2\operatorname*{div}\left( \nabla\psi^{a}\right) . \label{901}% \end{equation} Hence (\ref{451}) follows from standard $L^{p}$ estimates for periodic solutions of the Poisson equation (see \cite{GT}). \end{proof} Next we study the behavior of auxiliary functions of the type% \begin{equation} g\left( x\right) :=\left( \mu^{2}+\left\vert x\right\vert ^{2}\right) ^{\frac{p}{2}} \label{450}% \end{equation} defined on an arbitrary Hilbert space $X$. \begin{lemma} \label{Lemma 2}For every $p>1$ there exist two constants $\kappa_{p}$ and $K_{p}$, with $0<\kappa_{p}\leq1\leq K_{p}$, such that the following inequalities hold:% \begin{gather} \int_{0}^{1}\left( \mu^{2}+\left\vert x+ty\right\vert ^{2}\right) ^{\frac{p-2}{2}}(1-t)\,dt\geq\kappa_{p}\left( \mu^{2}+\left\vert x\right\vert ^{2}+\left\vert y\right\vert ^{2}\right) ^{\frac{p-2}{2}},\label{prima}\\ \int_{0}^{1}\left( \mu^{2}+\left\vert x+ty\right\vert ^{2}\right) ^{\frac{p-2}{2}}dt\leq K_{p}\left( \mu^{2}+\left\vert x\right\vert ^{2}+\left\vert y\right\vert ^{2}\right) ^{\frac{p-2}{2}} \label{seconda}% \end{gather} for every $x,$ $y\in X$ and every constant $\mu\geq0.$ \end{lemma} \begin{proof} Let us prove (\ref{prima}). If $1
2$, then we consider first the case where $\left\vert y\right\vert ^{2}\leq4\left( \mu^{2}+\left\vert x\right\vert ^{2}\right) $. For $0\leq t\leq1/2$ we have% \begin{align*} \left( \mu^{2}+\left\vert x+ty\right\vert ^{2}\right) ^{\frac{1}{2}} & \geq\left( \mu^{2}+\left\vert x\right\vert ^{2}\right) ^{\frac{1}{2}% }-t\left\vert y\right\vert \geq\left( 1-2t\right) \left( \mu^{2}+\left\vert x\right\vert ^{2}\right) ^{\frac{1}{2}}\\ & \geq5^{-\frac{1}{2}}\left( 1-2t\right) \left( \mu^{2}+\left\vert x\right\vert ^{2}+\left\vert y\right\vert ^{2}\right) ^{\frac{1}{2}}, \end{align*} hence \[ \left( \mu^{2}+\left\vert x+ty\right\vert ^{2}\right) ^{\frac{p-2}{2}}% \geq5^{\frac{2-p}{2}}\left( 1-2t\right) ^{p-2}\left( \mu^{2}+\left\vert x\right\vert ^{2}+\left\vert y\right\vert ^{2}\right) ^{\frac{p-2}{2}}. \] We deduce that (\ref{prima}) holds provided \[ \kappa_{p}\leq5^{\frac{2-p}{2}}\int_{0}^{\frac{1}{2}}\left( 1-2t\right) ^{p-2}(1-t)\,dt=\frac{5^{\frac{2-p}{2}}}{4}\frac{2p-1}{p\left( p-1\right) }. \] If $p>2$ and $\left\vert y\right\vert ^{2}>4\left( \mu^{2}+\left\vert x\right\vert ^{2}\right) $, then for $1/2\leq t\leq1$ we have% \begin{align*} \left( \mu^{2}+\left\vert x+ty\right\vert ^{2}\right) ^{\frac{1}{2}} & \geq t\left\vert y\right\vert -\left( \mu^{2}+\left\vert x\right\vert ^{2}\right) ^{\frac{1}{2}}\geq\left( t-\frac{1}{2}\right) \left\vert y\right\vert \\ & \geq2\cdot5^{-\frac{1}{2}}\left( t-\frac{1}{2}\right) \left( \mu ^{2}+\left\vert x\right\vert ^{2}+\left\vert y\right\vert ^{2}\right) ^{\frac{1}{2}}. \end{align*} Therefore% \[ \left( \mu^{2}+\left\vert x+ty\right\vert ^{2}\right) ^{\frac{p-2}{2}}% \geq5^{\frac{2-p}{2}}\left( 2t-1\right) ^{p-2}\left( \mu^{2}+\left\vert x\right\vert ^{2}+\left\vert y\right\vert ^{2}\right) ^{\frac{p-2}{2}}. \] We conclude that (\ref{prima}) is verified with \[ \kappa_{p}\leq5^{\frac{2-p}{2}}\int_{\frac{1}{2}}^{1}\left( 2t-1\right) ^{p-2}(1-t)\,dt=\frac{5^{\frac{2-p}{2}}}{4}\frac{1}{p\left( p-1\right) }. \] This concludes the proof of (\ref{prima}). Let us prove (\ref{seconda}). If $p\geq2$ then (\ref{seconda}) holds with $K_{p}=2^{\frac{p-2}{2}}$, since \[ \left( \mu^{2}+\left\vert x+ty\right\vert ^{2}\right) ^{\frac{p-2}{2}}% \leq2^{\frac{p-2}{2}}\left( \mu^{2}+\left\vert x\right\vert ^{2}+\left\vert y\right\vert ^{2}\right) ^{\frac{p-2}{2}}. \] In the case $1
1$, there exist two constants $\theta_{p\text{ }% }>0$ and $\Theta_{p}>0$ such that for every $\mu\geq0$\ the function $g$ defined in $\left( \ref{450}\right) $ satisfies the following inequalities% \begin{align*} \theta_{p}\left( \mu^{2}+\left\vert x\right\vert ^{2}+ \left\vert y\right\vert ^{2}\right) ^{\frac{p-2}{2}}\left\vert y\right\vert ^{2} & \leq g\left( x+y\right) - g\left( x\right) -\nabla g\left( x\right) \cdot y\\ & \leq\Theta_{p}\left( \mu^{2}+\left\vert x\right\vert ^{2}+\left\vert y\right\vert ^{2}\right) ^{\frac{p-2}{2}}\left\vert y\right\vert ^{2}% \end{align*} for every $x$, $y\in X$. \end{lemma} \begin{proof} By continuity it is enough to prove the statement when $0$ does not belong to the segment joining $x$ and $x+y.$ In this case the function \[ h\left( t\right) :=\left( \mu^{2}+\left\vert x+ty\right\vert ^{2}\right) ^{\frac{p}{2}}% \] belongs to $C^{\infty}\left( \left[ 0,1\right] \right) $, and Taylor's formula with integral remainder yields \[ h\left( 1\right) -h\left( 0\right) -h^{\prime}\left( 0\right) =\int _{0}^{1}h^{\prime\prime}\left( t\right) \left( 1-t\right) \,dt. \] By direct computation we see that% \begin{align*} p\left( \left( p-1\right) \wedge1\right) & \left( \mu^{2}+\left\vert x+ty\right\vert ^{2}\right) ^{\frac{p-2}{2}}\left\vert y\right\vert ^{2} \leq h^{\prime\prime}\left( t\right) \\ & \leq p\left( \left( p-1\right) \vee1\right) \left( \mu^{2}+\left\vert x+ty\right\vert ^{2}\right) ^{\frac{p-2}{2}}\left\vert y\right\vert ^{2}. \end{align*} The conclusion follows from Lemma \ref{Lemma 2}. \end{proof} In the proof of Theorem \ref{theorem 2} we will need the following extension of Lemma \ref{lemma 8} to the family of functions \[ g_{\beta}\left( x,y\right) :=\left( \mu^{2}+\left\vert x\right\vert ^{2}+\beta^{2}\left\vert y\right\vert ^{2}\right) ^{\frac{p}{2}},\qquad \beta\geq0, \] defined on the product of two Hilbert spaces $X$ and $Y$. \begin{lemma} \label{lemma 9}Let $p>1$, $\beta\geq0$ and $\mu\geq0$. Then% \begin{align} g_{\beta} & \left( x+\xi,y+\eta\right) -g_{\beta}\left( x,y\right) -\nabla_{\!x}g_{\beta}\left( x,y\right) \cdot\xi-\nabla_{\!y}g_{\beta }\left( x,y\right) \cdot\eta\nonumber\\ \geq & \theta_{p}\left( \mu^{2}+\left\vert x\right\vert ^{2}+\left\vert \xi\right\vert ^{2}+\beta^{2}\left\vert y\right\vert ^{2}+\beta^{2}\left\vert \eta\right\vert ^{2}\right) ^{\frac{p-2}{2}}\left( \left\vert \xi\right\vert ^{2}+\beta^{2}\left\vert \eta\right\vert ^{2}\right) \nonumber \end{align} for every $x$, $\xi\in X$, $y$, $\eta\in Y$, where $\theta_{p}$ is the first constant in Lemma \ref{lemma 8}. Therefore, if $p\geq2$, we have \begin{align*} & g_{\beta}\left( x+\xi,y+\eta\right) -g_{\beta}\left( x,y\right) -\nabla_{\!x}g_{\beta}\left( x,y\right) \cdot\xi-\nabla_{\!y}g_{\beta }\left( x,y\right) \cdot\eta\\ & \geq\theta_{p}\left( \mu^{2}+\left\vert x\right\vert ^{2}+\left\vert \xi\right\vert ^{2}\right) ^{\frac{p-2}{2}}\left\vert \xi\right\vert ^{2}+\frac{\theta_{p}\beta^{2}}{2}\left( \mu^{2}+\left\vert x\right\vert ^{2}\right) ^{\frac{p-2}{2}}\left\vert \eta\right\vert ^{2}+\frac{\theta _{p}\beta^{p}}{2}\left\vert \eta\right\vert ^{p}% \end{align*} for every $x$, $\xi\in X$, $y$, $\eta\in Y$. \end{lemma} \begin{proof} Observing that $g_{\beta}\left( x,y\right) =g_{1}\left( x,\beta y\right) $, the inequality can be obtained by applying Lemma \ref{lemma 8} to the Hilbert space $X\times Y$. \end{proof} We continue with some technical lemmas which are used in the proofs of Theorems \ref{theorem 2} and \ref{theorem 1}. \begin{lemma} \label{lemma10}Let $X$ be a Hilbert space and let $1
\mu^{2}+b^{2}$, then from (\ref{960}) with $\left( \mu^{2}+b^{2}\right) $ replaced by $\left( \mu^{2}+a^{2}% +b^{2}\right) $ we obtain% \begin{align*} b^{p} & \leq2\varepsilon^{\frac{p-2}{p}}\left( \mu^{2}+a^{2}+b^{2}\right) ^{\frac{p-2}{2}}b^{2}+\frac{\varepsilon}{2}\left( \mu^{2}+a^{2}+b^{2}\right) ^{\frac{p}{2}}\\ & \leq2\varepsilon^{\frac{p-2}{p}}\left( \mu^{2}+a^{2}+b^{2}\right) ^{\frac{p-2}{2}}b^{2}+\varepsilon a^{p}, \end{align*} which proves (\ref{400}). \end{proof} The next lemma shows that condition (\ref{3}) in Theorem \ref{theorem 2} can be obtained from a suitable bound on the second derivatives of $f$. This is trivial in the case $p\geq2$, but requires some work in the case $1
1$, $C>0$, and $\mu\geq0$ such that% \begin{equation} \left\vert \nabla^{2}f\left( x\right) \right\vert \leq C\left( \mu ^{2}+\left\vert x\right\vert ^{2}\right) ^{\frac{p-2}{2}} \label{12}% \end{equation} for every $x\in X\setminus\left\{ 0\right\} $. Then \begin{equation} \left\vert \nabla f\left( x+y\right) -\nabla f\left( x\right) \right\vert \leq K_{p}C\left( \mu^{2}+\left\vert x\right\vert ^{2}+\left\vert y\right\vert ^{2}\right) ^{\frac{p-2}{2}}\left\vert y\right\vert \label{13}% \end{equation} for every $x$, $y\in X$, where $K_{p}$ is the second constant in Lemma \ref{Lemma 2}. \end{lemma} \begin{proof} By continuity it is enough to prove the statement when $0$ does not belong to the segment joining $x$ and $x+y.$ In this case by (\ref{12}) we have% \[ \left\vert \nabla f\left( x+y\right) -\nabla f\left( x\right) \right\vert \leq C\left\vert y\right\vert \int_{0}^{1}\left( \mu^{2}+\left\vert x+ty\right\vert ^{2}\right) ^{\frac{p-2}{2}}dt, \] and the conclusion follows from Lemma\nolinebreak\ \ref{Lemma 2}. \end{proof} The estimate given by the following lemma will be crucial in the proof of Theorem \ref{theorem 2}. \begin{lemma} \label{Lemma 4}Let $X$ be a Hilbert space and let $f\in C^{1}\left( X\right) $. Assume that there exist $p>1$ and $\mu\geq0$ such that \begin{equation} \left\vert \nabla f\left( x+y\right) -\nabla f\left( x\right) \right\vert \leq\left( \mu^{2}+\left\vert x\right\vert ^{2}+\left\vert y\right\vert ^{2}\right) ^{\frac{p-2}{2}}\left\vert y\right\vert \label{50}% \end{equation} for every $x$, $y\in X$. If $1
0$ there exists $c_{\varepsilon,p}>0$, depending only on $\varepsilon$ and $p$, such that \begin{align} \left\vert f\left( x+y+z\right) -f\left( x+y\right) -\nabla f\left( x\right) \cdot z\right\vert & \label{14}\\ \leq\varepsilon\left( \mu^{2}+\left\vert x\right\vert ^{2}+\left\vert y\right\vert ^{2}\right) ^{\frac{p-2}{2}} & \left\vert y\right\vert ^{2}+c_{\varepsilon,p}\left( \mu^{2}+\left\vert z\right\vert ^{2}\right) ^{\frac{p-2}{2}}\left\vert z\right\vert ^{2}\nonumber \end{align} for every $x$, $y$, $z\in X$. If $p\geq2$, then for every $\varepsilon>0$ there exists $c_{\varepsilon,p}% >0$, depending only on $\varepsilon$ and $p$, such that% \begin{align} \left\vert f\left( x+y+z\right) -f\left( x+y\right) -\nabla f\left( x\right) \cdot z\right\vert & \label{61}\\ \leq\varepsilon\left( \mu^{2}+\left\vert x\right\vert ^{2}+\left\vert y\right\vert ^{2}\right) ^{\frac{p-2}{2}} & \left\vert y\right\vert ^{2}+c_{\varepsilon,p}\left( \mu^{2}+\left\vert x\right\vert ^{2}\right) ^{\frac{p-2}{2}}\left\vert z\right\vert ^{2}+c_{\varepsilon,p}\left\vert z\right\vert ^{p}\nonumber \end{align} for every $x$, $y$, $z\in X$. \end{lemma} \begin{proof} Let us consider first the case $1
0$ we have% \begin{align} \left( \mu^{2}+\left| x\right| ^{2}+\left| y\right| ^{2}\right) ^{\frac{p-2}{2}} & \left| y\right| \left| z\right| \leq\varepsilon \left( \mu^{2}+\left| x\right| ^{2}+\left| y\right| ^{2}\right) ^{\frac{p-2}{2}}\left| y\right| ^{2}\nonumber\\ & +\frac{1}{4\varepsilon}\left( \mu^{2}+\left| x\right| ^{2}+\left| y\right| ^{2}\right) ^{\frac{p-2}{2}}\left| z\right| ^{2}\label{53}\\ \leq & \varepsilon\left( \mu^{2}+\left| x\right| ^{2}+\left| y\right| ^{2}\right) ^{\frac{p-2}{2}}\left| y\right| ^{2}+\frac{1}{4\varepsilon}% \mu^{p-2}\left| z\right| ^{2}\nonumber\\ \leq & \varepsilon\left( \mu^{2}+\left| x\right| ^{2}+\left| y\right| ^{2}\right) ^{\frac{p-2}{2}}\left| y\right| ^{2}+\frac{1}{\varepsilon }\left( \mu^{2}+\left| z\right| ^{2}\right) ^{\frac{p-2}{2}}\left| z\right| ^{2}.\nonumber \end{align} If $\left| z\right| >\mu$, let $q:=p/\left( p-1\right) $ be the conjugate exponent of $p$. Since $\frac{p-2}{2}+\frac{1}{2}-\frac{1}{q}=\frac{p-2}{2q}$ we have \begin{align} \left( \mu^{2}+\left| x\right| ^{2}+\left| y\right| ^{2}\right) ^{\frac{p-2}{2}} & \left| y\right| \left| z\right| =\left( \mu ^{2}+\left| x\right| ^{2}+\left| y\right| ^{2}\right) ^{\frac{p-2}{2}% }\left| y\right| ^{1-\frac{2}{q}}\left| y\right| ^{\frac{2}{q}}\left| z\right| \nonumber\\ & \leq\left( \mu^{2}+\left| x\right| ^{2}+\left| y\right| ^{2}\right) ^{\frac{p-2}{2q}}\left| y\right| ^{\frac{2}{q}}\left| z\right| \label{54}\\ & \leq\varepsilon\left( \mu^{2}+\left| x\right| ^{2}+\left| y\right| ^{2}\right) ^{\frac{p-2}{2}}\left| y\right| ^{2}+\frac{1}{p\left( q\varepsilon\right) ^{p-1}}\left| z\right| ^{p}\nonumber\\ & \leq\varepsilon\left( \mu^{2}+\left| x\right| ^{2}+\left| y\right| ^{2}\right) ^{\frac{p-2}{2}}\left| y\right| ^{2}+k_{\varepsilon,p}\left( \mu^{2}+\left| z\right| ^{2}\right) ^{\frac{p-2}{2}}\left| z\right| ^{2}\nonumber \end{align} for some constant $k_{\varepsilon,p}$ depending only on $\varepsilon$ and $p$. Let us consider now the case $p\geq2$. By the Mean Value Theorem and by the Cauchy Inequality we have% \begin{align*} |f(x & +y+z)-f\left( x+y\right) -\nabla f\left( x\right) \cdot z|\\ \leq & \left\vert f\left( x+y+z\right) -f\left( x+y\right) -\nabla f\left( x+y\right) \cdot z\right\vert \\ & +\left\vert \nabla f\left( x+y\right) \cdot z-\nabla f\left( x\right) \cdot z\right\vert \\ \leq & \left( \mu^{2}+2\left\vert x\right\vert ^{2}+2\left\vert y\right\vert ^{2}+\left\vert z\right\vert ^{2}\right) ^{\frac{p-2}{2}}\left\vert z\right\vert ^{2}+\left( \mu^{2}+\left\vert x\right\vert ^{2}+\left\vert y\right\vert ^{2}\right) ^{\frac{p-2}{2}}\left\vert y\right\vert \left\vert z\right\vert \\ \leq & 6^{\frac{p-2}{2}}\left( \mu^{2}+\left\vert x\right\vert ^{2}\right) ^{\frac{p-2}{2}}\left\vert z\right\vert ^{2}+6^{\frac{p-2}{2}}\left\vert y\right\vert ^{p-2}\left\vert z\right\vert ^{2}+3^{\frac{p-2}{2}}\left\vert z\right\vert ^{p}\\ & +\frac{\varepsilon}{2}\left( \mu^{2}+\left\vert x\right\vert ^{2}+\left\vert y\right\vert ^{2}\right) ^{\frac{p-2}{2}}\left\vert y\right\vert ^{2}+k_{\varepsilon,p}\left( \mu^{2}+\left\vert x\right\vert ^{2}\right) ^{\frac{p-2}{2}}\left\vert z\right\vert ^{2}+k_{\varepsilon ,p}\left\vert y\right\vert ^{p-2}\left\vert z\right\vert ^{2}% \end{align*} for some constant $k_{\varepsilon,p}$ depending only on $\varepsilon$ and $p$. The conclusion follows from Young' inequality with exponents $p/\left( p-2\right) $ and $p/2$. \end{proof} The following lemma states an elementary property of strictly $2$-quasiconvex functions. \begin{lemma} \label{lemma 11}Assume that $f:\mathbb{M}_{\operatorname*{sym}}^{n\times n}\rightarrow\mathbb{R}$ satisfies the strict $2$-quasiconvexity condition (a) of Theorem \ref{theorem 2} for some constants $p>1$, $\mu\geq0$, $\nu>0$, and let $g:\mathbb{M}_{\operatorname*{sym}}^{n\times n}\rightarrow\mathbb{R}$ be the function defined by% \begin{equation} g(A):=\left( \mu^{2}+\left\vert A\right\vert ^{2}\right) ^{\frac{p}{2}}. \label{950}% \end{equation} Then the function $f_{\lambda}:=f-\lambda g$ is $2$-quasiconvex for $\lambda\leq\nu/\Theta_{p}$, where $\Theta_{p}$ is the second constant in Lemma \ref{lemma 8}. \end{lemma} \begin{proof} Let $A\in\mathbb{M}_{\operatorname*{sym}}^{n\times n}$ and $\phi\in C_{c}% ^{2}\left( Q\right) $. Since, by periodicity,% \[ \int_{Q}\nabla g(A)\cdot\nabla^{2}\phi\,dx=0, \] we have% \begin{align*} \int_{Q} & \left[ f_{\lambda}\left( A+\nabla^{2}\phi\right) -f_{\lambda }\left( A\right) \right] \,dx=\int_{Q}\left[ f\left( A+\nabla^{2}% \phi\right) -f\left( A\right) \right] \,dx\\ & -\lambda\int_{Q}\left[ g\left( A+\nabla^{2}\phi\right) -g\left( A\right) +\nabla g(A)\cdot\nabla^{2}\phi\right] \,dx\\ & \geq\left( \nu-\lambda\Theta_{p}\right) \int_{Q}\left( \mu^{2}+\left| A\right| ^{2}+\left| \nabla^{2}\phi\right| ^{2}\right) ^{\frac{p-2}{2}% }\left| \nabla^{2}\phi\right| ^{2}\,dx\geq0, \end{align*} which concludes the proof. \end{proof} In the proof of Theorem \ref{theorem 2} we need the following generalization of Lemma \ref{lemma 1}. \begin{lemma} \label{lemma 5}For every $p>1$ there exists a constant $\tau_{n,p}\geq1$ such that \begin{equation} \int_{Q}\left( \mu^{2}+\left\vert \nabla\psi^{s}\right\vert ^{2}\right) ^{\frac{p-2}{2}}\left\vert \nabla\psi^{s}\right\vert ^{2}\,dx\leq\tau _{n,p}\int_{Q}\left( \mu^{2}+\left\vert \nabla\psi^{a}\right\vert ^{2}\right) ^{\frac{p-2}{2}}\left\vert \nabla\psi^{a}\right\vert ^{2}\,dx \label{16}% \end{equation} for every constant $\mu\geq0$ and every $Q$-periodic function $\psi :\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ of class $C^{\infty}$ with $\operatorname*{div}\psi=0$. \end{lemma} \begin{proof} Let $\mu$ and $\psi$ be as in the statement of the lemma. In the case $p\geq2$ by Lemma \ref{lemma 1} we have% \[ \int_{Q}\left\vert \nabla\psi\right\vert ^{p}\,dx\leq\gamma_{n,p}\int _{Q}\left\vert \nabla\psi^{a}\right\vert ^{p}\,dx, \] and so \begin{align*} \int_{Q} & \left( \mu^{2}+\left\vert \nabla\psi\right\vert ^{2}\right) ^{\frac{p-2}{2}}\left\vert \nabla\psi\right\vert ^{2}\,dx\\ & \leq2^{\frac{p-2}{2}}\left\{ \mu^{p-2}\int_{Q}\left\vert \nabla \psi\right\vert ^{2}\,dx+\int_{Q}\left\vert \nabla\psi\right\vert ^{p}\,dx\right\} \\ & \leq2^{\frac{p-2}{2}}\left\{ \mu^{p-2}\gamma_{n,2}\int_{Q}\left\vert \nabla\psi^{a}\right\vert ^{2}\,dx+\gamma_{n,p}\int_{Q}\left\vert \nabla \psi^{a}\right\vert ^{p}\,dx\right\} \\ & \leq\tau_{n,p}\int_{Q}\left( \mu^{2}+\left\vert \nabla\psi^{a}\right\vert ^{2}\right) ^{\frac{p-2}{2}}\left\vert \nabla\psi^{a}\right\vert ^{2}\,dx \end{align*} with $\tau_{n,p}:=2^{\frac{p-2}{2}}\left( \gamma_{n,2}+\gamma_{n,p},\right) $. We consider now the case $1
\mu\right\} $, and let $\Psi_{\mu}:=1_{E_{\mu}% }\nabla\psi^{a}$ and $\Psi^{\mu}:=1_{E^{\mu}}\nabla\psi^{a}$, where $1_{E}$ is the characteristic function of the set $E$. Note that $\Psi_{\mu}$ and $\Psi^{\mu}$ are periodic vector-fields. Let $\psi_{\mu}$ and $\psi^{\mu}$ be periodic solutions of the equations \[ \Delta\psi_{\mu}=2\operatorname*{div}\Psi_{\mu}\qquad\text{and}\qquad \Delta\psi^{\mu}=2\operatorname*{div}\Psi^{\mu}. \] {}From the first equation we get \begin{equation} \int_{Q}\left\vert \nabla\psi_{\mu}\right\vert ^{2}\,dx\leq4\int_{Q}\left\vert \Psi_{\mu}\right\vert ^{2}\,dx. \label{30}% \end{equation} Standard $L^{p}$ estimates for periodic solutions of the Poisson equation (see \cite{GT}) yield a constant $\tilde{\gamma}_{n,p}\geq4$ such that \begin{equation} \int_{Q}\left\vert \nabla\psi^{\mu}\right\vert ^{p}\,dx\leq\tilde{\gamma }_{n,p}\int_{Q}\left\vert \Psi^{\mu}\right\vert ^{p}\,dx. \label{31}% \end{equation} {}From (\ref{30}) we obtain \begin{align} \int_{Q}\left( \mu^{2}+\left\vert \nabla\psi_{\mu}\right\vert ^{2}\right) ^{\frac{p-2}{2}} & \left\vert \nabla\psi_{\mu}\right\vert ^{2}\,dx\leq \mu^{p-2}\int_{Q}\left\vert \nabla\psi_{\mu}\right\vert ^{2}\,dx\nonumber\\ & \leq4\mu^{p-2}\int_{Q}\left\vert \Psi_{\mu}\right\vert ^{2}\,dx\label{40}\\ & \leq8\int_{Q\cap E_{\mu}}\left( \mu^{2}+\left\vert \nabla\psi ^{a}\right\vert ^{2}\right) ^{\frac{p-2}{2}}\left\vert \nabla\psi ^{a}\right\vert ^{2}\,dx,\nonumber \end{align} where the last inequality follows from the fact that $\left\vert \Psi_{\mu }\right\vert =\left\vert \nabla\psi^{a}\right\vert 1_{E_{\mu}}\leq\mu$. {}From (\ref{31}) we obtain \begin{align} \int_{Q}\left( \mu^{2}+\left\vert \nabla\psi^{\mu}\right\vert ^{2}\right) ^{\frac{p-2}{2}} & \left\vert \nabla\psi^{\mu}\right\vert ^{2}\,dx\leq \int_{Q}\left\vert \nabla\psi^{\mu}\right\vert ^{p}\,dx\nonumber\\ & \leq\tilde{\gamma}_{n,p}\int_{Q}\left\vert \Psi^{\mu}\right\vert ^{p}\,dx\label{41}\\ & \leq2\tilde{\gamma}_{n,p}\int_{Q\cap E^{_{\mu}}}\left( \mu^{2}+\left\vert \nabla\psi^{a}\right\vert ^{2}\right) ^{\frac{p-2}{2}}\left\vert \nabla \psi^{a}\right\vert ^{2}\,dx,\nonumber \end{align} where the last inequality follows from the fact that $\left\vert \Psi^{_{\mu}% }\right\vert =\left\vert \nabla\psi^{a}\right\vert 1_{E^{_{\mu}}}\geq \mu1_{E^{_{\mu}}}$. By (\ref{901}) we have \[ \Delta\left( \psi_{\mu}+\psi^{\mu}\right) =2\operatorname*{div}\nabla \psi^{a}=\Delta\psi, \] and since $\psi_{\mu}+\psi^{\mu}-\psi$ is a periodic function we deduce that $\nabla\psi=\nabla\psi_{\mu}+\nabla\psi^{\mu}$. Finally, from Lemma \ref{lemma10} and using (\ref{40}) and (\ref{41}) we obtain% \[ \int_{Q}\left( \mu^{2}+\left\vert \nabla\psi\right\vert ^{2}\right) ^{\frac{p-2}{2}}\left\vert \nabla\psi\right\vert ^{2}\,dx\leq\tau_{n,p}% \int_{Q}\left( \mu^{2}+\left\vert \nabla\psi^{a}\right\vert ^{2}\right) ^{\frac{p-2}{2}}\left\vert \nabla\psi^{a}\right\vert ^{2}\,dx, \] with $\tau_{n,p}=2\tilde{\gamma}_{n,p}$. Since $\left\vert \nabla\psi ^{s}\right\vert \leq\left\vert \nabla\psi\right\vert $ and the mapping $t\mapsto\left( \mu^{2}+t\right) ^{\frac{p-2}{2}}t$ is nondecreasing,\ inequality (\ref{16}) follows. \end{proof} \section{Proofs\label{section proofs}} \begin{proof} [Proof of Theorem \ref{theorem 2}]We begin by observing that (\ref{3}) gives \begin{equation} \left| f\left( A\right) \right| \leq k_{f}\left( 1+\left| A\right| ^{p}\right) \qquad\forall A\in\mathbb{M}_{\operatorname*{sym}}^{n\times n} \label{18}% \end{equation} for a suitable constant $k_{f}$ depending on $f$. \noindent\textbf{Step 1:} We first consider the case $1
0$, to be chosen at the end of the proof, let $G:\mathbb{M}^{n\times n}\rightarrow\mathbb{R}$ be the function defined by% \begin{equation} G(A):=f(A^{s})+\beta g(A^{a}), \label{19}% \end{equation} and let $F$ be its $1$-quasiconvexification, i.e., (see, e.g., \cite{D}) \begin{equation} F(A)=\inf\left\{ \int_{Q}G(A+\nabla\varphi(x))\,dx:\,\varphi\in C_{\operatorname*{per}}^{\infty}(Q;\mathbb{R}^{n})\right\} , \label{15}% \end{equation} for all $A\in\mathbb{M}^{n\times n}$. We want to prove that for every $\varepsilon>0$ there exists $\beta>0$ such that \begin{equation} \int_{Q}\left[ G\left( A+\nabla\varphi\right) -G\left( A\right) \right] \,dx\geq-\varepsilon\left( \mu^{2}+\left| A^{a}\right| ^{2}\right) ^{\frac{p-2}{2}}\left| A^{a}\right| ^{2} \label{20}% \end{equation} for every $A\in\mathbb{M}^{n\times n}$ and for every $\varphi\in C_{\operatorname*{per}}^{\infty}(Q;\mathbb{R}^{n})$. In view of (\ref{15}) this will imply that for every $A\in\mathbb{M}^{n\times n}$ we have% \begin{equation} G(A)-\varepsilon\left( \mu^{2}+\left| A^{a}\right| ^{2}\right) ^{\frac{p-2}{2}}\left| A^{a}\right| ^{2}\leq F(A)\leq G(A) \label{23}% \end{equation} which yields (\ref{5}). Inequality (\ref{6}) follows from (\ref{18}), (\ref{19}) and (\ref{23}). Let us prove (\ref{20}). Fix $\varphi\in C_{\operatorname*{per}}^{\infty }(Q;\mathbb{R}^{n})$ and consider the periodic Helmholtz decomposition \[ \varphi=\nabla\phi+\psi \] given by Lemma \ref{lemma 6}. Following the argument used by M\"{u}ller and \v{S}ver\'{a}k in the proof of Lemma 4.2 in \cite{MS}, we have% \begin{align} \int_{Q}[G & \left( A+\nabla\varphi\right) -G\left( A\right) ]\,dx\nonumber\\ = & \int_{Q}\left[ f\left( A^{s}+\nabla^{2}\phi+\nabla\psi^{s}\right) -f\left( A^{s}+\nabla^{2}\phi\right) \right] \,dx\nonumber\\ & +\int_{Q}\left[ f\left( A^{s}+\nabla^{2}\phi\right) -f\left( A^{s}\right) \right] \,dx\label{17}\\ & +\beta\int_{Q}\left[ \left( \mu^{2}+\left\vert A^{a}+\nabla\psi ^{a}\right\vert ^{2}\right) ^{\frac{p}{2}}-\left( \mu^{2}+\left\vert A^{a}\right\vert ^{2}\right) ^{\frac{p}{2}}\right] \,dx\nonumber\\ = & \!:I_{1}+I_{2}+I_{3}.\nonumber \end{align} Since $\nabla f\left( A^{s}\right) $ is a symmetric matrix we have $\nabla f\left( A^{s}\right) \cdot\nabla\psi^{s}=\nabla f\left( A^{s}\right) \cdot\nabla\psi$, and therefore, by periodicity, \[ \int_{Q}\nabla f\left( A^{s}\right) \cdot\nabla\psi^{s}\,dx=0. \] Hence \[ I_{1}=\int_{Q}\left[ f\left( A^{s}+\nabla^{2}\phi+\nabla\psi^{s}\right) -f\left( A^{s}+\nabla^{2}\phi\right) -\nabla f\left( A^{s}\right) \cdot\nabla\psi^{s}\right] \,dx. \] By Lemma \ref{Lemma 4} we have% \begin{align*} I_{1} & \geq-\nu\int_{Q}\left( \mu^{2}+\left\vert A^{s}\right\vert ^{2}+\left\vert \nabla^{2}\phi\right\vert ^{2}\right) ^{\frac{p-2}{2}% }\left\vert \nabla^{2}\phi\right\vert ^{2}\,dx\\ & -c_{\nu,p,L}\int_{Q}\left( \mu^{2}+\left\vert \nabla\psi^{s}\right\vert ^{2}\right) ^{\frac{p-2}{2}}\left\vert \nabla\psi^{s}\right\vert ^{2}\,dx, \end{align*} while the strict $2$-quasiconvexity of $f$ (condition (a)) yields% \[ I_{2}\geq\nu\int_{Q}\left( \mu^{2}+\left\vert A^{s}\right\vert ^{2}% +\left\vert \nabla^{2}\phi\right\vert ^{2}\right) ^{\frac{p-2}{2}}\left\vert \nabla^{2}\phi\right\vert ^{2}\,dx, \] and so, using Lemma \ref{lemma 5},\ we obtain% \begin{align} I_{1}+I_{2} & \geq-c_{\nu,p,L}\int_{Q}\left( \mu^{2}+\left\vert \nabla \psi^{s}\right\vert ^{2}\right) ^{\frac{p-2}{2}}\left\vert \nabla\psi ^{s}\right\vert ^{2}\,dx\label{24}\\ & \geq-c_{\nu,p,L}\,\tau_{n,p}\int_{Q}\left( \mu^{2}+\left\vert \nabla \psi^{a}\right\vert ^{2}\right) ^{\frac{p-2}{2}}\left\vert \nabla\psi ^{a}\right\vert ^{2}\,dx.\nonumber \end{align} Since $\nabla g(A^{a})$ is an antisymmetric matrix we have $\nabla g(A^{a})\cdot\nabla\psi^{a}=\nabla g(A^{a})\cdot\nabla\psi$, and therefore, by periodicity, \[ \int_{Q}\nabla g(A^{a})\cdot\nabla\psi^{a}\,dx=0. \] Hence, by Lemma \ref{lemma 8} and Lemma \ref{lemma10}, for every $0<\delta<1$ we obtain% \begin{align*} I_{3} & =\beta\int_{Q}\left[ g\left( A^{a}+\nabla\psi^{a}\right) -g\left( A^{a}\right) -\nabla g(A^{a})\cdot\nabla\psi^{a}\right] \,dx\\ & \geq\beta\theta_{p}\int_{Q}\left( \mu^{2}+\left\vert A^{a}\right\vert ^{2}+\left\vert \nabla\psi^{a}\right\vert ^{2}\right) ^{\frac{p-2}{2}% }\left\vert \nabla\psi^{a}\right\vert ^{2}\,dx\\ & \geq\beta\theta_{p}\delta^{\frac{2-p}{2}}\int_{Q}\left( \mu^{2}+\left\vert \nabla\psi^{a}\right\vert ^{2}\right) ^{\frac{p-2}{2}}\left\vert \nabla \psi^{a}\right\vert ^{2}\,dx-\beta\theta_{p}\delta\left( \mu^{2}+\left\vert A^{a}\right\vert ^{2}\right) ^{\frac{p-2}{2}}\left\vert A^{a}\right\vert ^{2}. \end{align*} Choosing $\beta>0$ and $0<\delta<1$ so that \[ \beta\theta_{p}\delta^{\frac{2-p}{2}}\geq c_{\nu,p,L}\,\tau_{n,p},\qquad \beta\theta_{p}\delta\leq\varepsilon \] we obtain \[ I_{1}+I_{2}+I_{3}\geq-\varepsilon\left( \mu^{2}+\left\vert A^{a}\right\vert ^{2}\right) ^{\frac{p-2}{2}}\left\vert A^{a}\right\vert ^{2}, \] which, together with (\ref{17}), yields (\ref{20}). \noindent\textbf{Step 2:} Let us consider now the case $p\geq2$. Let $\lambda:=\nu/\Theta_{p}$, where $\Theta_{p}$ is the second constant in Lemma \ref{lemma 8}. Given a constant $\beta>0$, to be chosen at the end of the proof, let $F:\mathbb{M}^{n\times n}\rightarrow\mathbb{R}$ be the function defined by% \begin{equation} F(A):=f(A^{s})-\lambda\left( \mu^{2}+\left\vert A^{s}\right\vert ^{2}\right) ^{\frac{p}{2}}+\lambda\left( \mu^{2}+\left\vert A^{s}\right\vert ^{2}% +\beta^{2}\left\vert A^{a}\right\vert ^{2}\right) ^{\frac{p}{2}}. \label{900}% \end{equation} It is clear that (\ref{5}) holds, while (\ref{6}) follows from (\ref{18}). It remains to prove that, for some $\beta>0$, the function\ $F$ is 1-quasiconvex, i.e., \begin{equation} \int_{Q}\left[ F\left( A+\nabla\varphi\right) -F\left( A\right) \right] \,dx\geq0 \label{200}% \end{equation} for every $A\in\mathbb{M}^{n\times n}$ and for every $\varphi\in C_{\operatorname*{per}}^{\infty}(Q;\mathbb{R}^{n})$. Let $f_{\lambda}$ be the 2-quasiconvex function defined in Lemma \ref{lemma 11}, and let \[ g_{\beta}(A)=\hat{g}_{\beta}\left( A^{s},A^{a}\right) :=\left( \mu ^{2}+\left| A^{s}\right| ^{2}+\beta^{2}\left| A^{a}\right| ^{2}\right) ^{\frac{p}{2}}, \] so that \[ F(A)=f_{\lambda}(A^{s})+\lambda g_{\beta}(A). \] Let us prove (\ref{200}). Fix a $Q$-periodic function $\varphi:\mathbb{R}% ^{n}\rightarrow\mathbb{R}^{n}$ of class $C^{\infty}$ and consider the periodic Helmholtz decomposition \[ \varphi=\nabla\phi+\psi \] given by Lemma \ref{lemma 6}. Then we have% \begin{align} \int_{Q}[F & \left( A+\nabla\varphi\right) -F\left( A\right) ]\,dx\nonumber\\ = & \int_{Q}\left[ f_{\lambda}\left( A^{s}+\nabla\varphi^{s}\right) -f_{\lambda}\left( A^{s}+\nabla\varphi^{s}-\nabla\psi^{s}\right) \right] \,dx\nonumber\\ & +\int_{Q}\left[ f_{\lambda}\left( A^{s}+\nabla^{2}\phi\right) -f_{\lambda}\left( A^{s}\right) \right] \,dx\label{170}\\ & +\lambda\int_{Q}\left[ g_{\beta}\left( A+\nabla\varphi\right) -g_{\beta }\left( A\right) \right] \,dx\nonumber\\ = & \!:I_{1}+I_{2}+I_{3}.\nonumber \end{align} Since $\nabla f_{\lambda}\left( A^{s}\right) $ is a symmetric matrix, we have $\nabla f_{\lambda}\left( A^{s}\right) \cdot\nabla\psi^{s}=\nabla f_{\lambda}\left( A^{s}\right) \cdot\nabla\psi$, and therefore, by periodicity, \[ \int_{Q}\nabla f_{\lambda}\left( A^{s}\right) \cdot\nabla\psi^{s}\,dx=0. \] Hence \[ I_{1}=-\int_{Q}\left[ f_{\lambda}\left( A^{s}+\nabla\varphi^{s}-\nabla \psi^{s}\right) -f_{\lambda}\left( A^{s}+\nabla\varphi^{s}\right) +\nabla f_{\lambda}\left( A^{s}\right) \cdot\nabla\psi^{s}\right] \,dx. \] Since the function $g$ defined in (\ref{950}) clearly satisfies condition (\ref{12}) , by Lemma \ref{Lemma 3} and (\ref{3}) it follows that (\ref{3}) still holds for the function $f_{\lambda}$ for a suitable constant $M>0$ in place of $L$. We are now in position to apply Lemma \ref{Lemma 4} to obtain a constant $\sigma=\sigma_{p,M}$ such that \begin{align*} I_{1} & \geq-\lambda\theta_{p}\int_{Q}\left( \mu^{2}+\left\vert A^{s}\right\vert ^{2}+\left\vert \nabla\varphi^{s}\right\vert ^{2}\right) ^{\frac{p-2}{2}}\left\vert \nabla\varphi^{s}\right\vert ^{2}\,dx\\ & -\sigma\left( \mu^{2}+\left\vert A^{s}\right\vert ^{2}\right) ^{\frac{p-2}{2}}\int_{Q}\left\vert \nabla\psi^{s}\right\vert ^{2}% \,dx-\sigma\int_{Q}\left\vert \nabla\psi^{s}\right\vert ^{p}\,dx, \end{align*} and so, using Lemma \ref{lemma 1},\ we obtain% \begin{align} I_{1} & \geq-\lambda\theta_{p}\int_{Q}\left( \mu^{2}+\left\vert A^{s}\right\vert ^{2}+\left\vert \nabla\varphi^{s}\right\vert ^{2}\right) ^{\frac{p-2}{2}}\left\vert \nabla\varphi^{s}\right\vert ^{2}\,dx\label{201}\\ & -\sigma\gamma_{n,2}\left( \mu^{2}+\left\vert A^{s}\right\vert ^{2}\right) ^{\frac{p-2}{2}}\int_{Q}\left\vert \nabla\psi^{a}\right\vert ^{2}% \,dx-\sigma\gamma_{n,p}\int_{Q}\left\vert \nabla\psi^{a}\right\vert ^{p}\,dx.\nonumber \end{align} On the other hand, the $2$-quasiconvexity of $f_{\lambda}$ yields \begin{equation} I_{2}\geq0. \label{202}% \end{equation} Since, by periodicity, \[ \int_{Q}\nabla g_{\beta}(A)\cdot\nabla\varphi\,dx=0, \] by Lemma \ref{lemma 9} we have% \begin{align} I_{3} & =\lambda\int_{Q}\left[ g_{\beta}\left( A+\nabla\varphi\right) -g_{\beta}\left( A\right) -\nabla g_{\beta}(A)\cdot\nabla\varphi\right] \,dx\nonumber\\ & \geq\lambda\theta_{p}\int_{Q}\left( \mu^{2}+\left\vert A^{s}\right\vert ^{2}+\left\vert \nabla\varphi^{s}\right\vert ^{2}\right) ^{\frac{p-2}{2}% }\left\vert \nabla\varphi^{s}\right\vert ^{2}\,dx\label{203}\\ & +\frac{\lambda\theta_{p}\beta^{2}}{2}\left( \mu^{2}+\left\vert A^{s}\right\vert ^{2}\right) ^{\frac{p-2}{2}}\int_{Q}\left\vert \nabla \psi^{a}\right\vert ^{2}\,dx+\frac{\lambda\theta_{p}\beta^{p}}{2}\int _{Q}\left\vert \nabla\psi^{a}\right\vert ^{p}\,dx.\nonumber \end{align} Choosing $\beta>0$ so that \[ \frac{\lambda\theta_{p}\beta^{2}}{2}\geq\sigma\gamma_{n,2},\qquad\frac {\lambda\theta_{p}\beta^{p}}{2}\geq\sigma\gamma_{n,p}, \] by (\ref{201}), (\ref{202}), and (\ref{203}), we obtain \[ I_{1}+I_{2}+I_{3}\geq0, \] which together with (\ref{170}) yields (\ref{200}). \end{proof} \begin{proof} [Proof of Theorem \ref{theorem 1}]Since the function $t\mapsto f\left( A+ta\otimes b+tb\otimes a\right) $ is convex on $\mathbb{R}$ for every $A\in\mathbb{M}_{\operatorname*{sym}}^{n\times n}$ and every $a$, $b\in\mathbb{R}^{n}$ (see, e.g., [\cite{FM}]), from the growth condition (b) it follows that there exists a constant $L>0$ depending only on $M$ and $p$ such that \begin{equation} \left\vert f\left( A+B\right) -f\left( A\right) \right\vert \leq L\left( 1+\left\vert A\right\vert ^{p-1}+\left\vert B\right\vert ^{p-1}\right) \left\vert B\right\vert \label{114}% \end{equation} for every $A$, $B\in\mathbb{M}_{\operatorname*{sym}}^{n\times n}$. Given a constant $\beta>0$, to be chosen at the end of the proof, let $G:\mathbb{M}% ^{n\times n}\rightarrow\mathbb{R}$ be the function defined by% \begin{equation} G(A):=f(A^{s})+\beta\left\vert A^{a}\right\vert ^{p}, \label{110}% \end{equation} and let $F$ be its $1$-quasiconvexification. We want to prove that there exist two increasing sequences of positive numbers $\left\{ \beta_{k}\right\} $ and $\left\{ \lambda_{k}\right\} $, depending only on $k$, $p$, $\mu$, $\nu$, $M$, but not on the specific function $f$,\ such that the corresponding functions $G_{k}$ satisfy% \begin{equation} \int_{Q}\left[ G_{k}\left( A+\nabla\varphi\right) -G_{k}\left( A\right) \right] \,dx\geq-\frac{1}{k}\left\vert A^{s}\right\vert ^{p}-\lambda _{k}\left\vert A^{a}\right\vert ^{p}-\frac{1}{k} \label{111}% \end{equation} for every $A\in\mathbb{M}^{n\times n}$ and for every $Q$-periodic function $\varphi:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ of class $C^{\infty}$. This will imply (see, e.g., \cite{D}) that for every $A\in\mathbb{M}^{n\times n}$ we have% \begin{equation} G_{k}(A)-\frac{1}{k}\left\vert A^{s}\right\vert ^{p}-\lambda_{k}\left\vert A^{a}\right\vert ^{p}-\frac{1}{k}\leq F_{k}(A)\leq G_{k}(A) \label{112}% \end{equation} which yields (\ref{1}) and (\ref{7}) since $G_{k}(A)=f\left( A\right) $ whenever $A\in\mathbb{M}_{\operatorname*{sym}}^{n\times n}$. Let us prove (\ref{111}). Fix a $Q$-periodic function $\varphi:\mathbb{R}% ^{n}\rightarrow\mathbb{R}^{n}$ of class $C^{\infty}$ and consider the periodic Helmholtz decomposition \[ \varphi=\nabla\phi+\psi \] given by Lemma \ref{lemma 6}. Then we have% \begin{align} \int_{Q}[G & \left( A+\nabla\varphi\right) -G\left( A\right) ]\,dx\nonumber\\ = & \int_{Q}\left[ f\left( A^{s}+\nabla^{2}\phi+\nabla\psi^{s}\right) -f\left( A^{s}+\nabla^{2}\phi\right) \right] \,dx\nonumber\\ & +\int_{Q}\left[ f\left( A^{s}+\nabla^{2}\phi\right) -f\left( A^{s}\right) \right] \,dx\label{113}\\ & +\beta\int_{Q}\left[ \left| A^{a}+\nabla\psi^{a}\right| ^{p}-\left| A^{a}\right| ^{p}\right] \,dx\nonumber\\ = & \!:I_{1}+I_{2}+I_{3}.\nonumber \end{align} By (\ref{114}) and by Cauchy's inequality, for every $\delta>0$ there exists a constant $c_{\delta,p,L}>0$ such that \begin{align*} I_{1} & \geq-L\int_{Q}\left( 1+\left| A^{s}+\nabla^{2}\phi\right| ^{p-1}+\left| \nabla\psi^{s}\right| ^{p-1}\right) \left| \nabla\psi ^{s}\right| \,dx\\ & \geq-\delta-\delta\left| A^{s}\right| ^{p}-\delta\int_{Q}\left| \nabla^{2}\phi\right| ^{p}\,dx-c_{\delta,p,L}\int_{Q}\left| \nabla\psi ^{s}\right| ^{p}\,dx. \end{align*} Hence, using Lemma \ref{lemma 1} we obtain \[ I_{1}\geq-\delta-\delta\left| A^{s}\right| ^{p}-\delta\int_{Q}\left| \nabla^{2}\phi\right| ^{p}\,dx-c_{\delta,p,L}\gamma_{n,p}\int_{Q}\left| \nabla\psi^{a}\right| ^{p}\,dx. \] \ If $p\geq2$ then we have \begin{align*} I_{1} & \geq-\delta-\delta\left| A^{s}\right| ^{p} -\delta\int_{Q}\left( \mu^{2}+\left| A^{s}\right| ^{2}+\left| \nabla^{2}\phi\right| ^{2}\right) ^{\frac{p-2}{2}}\left| \nabla^{2}\phi\right| ^{2}\,dx\\ & -c_{\delta,p,L}\gamma_{n,p}\int_{Q}\left( \left| A^{a}\right| ^{2}+\left| \nabla\psi^{a}\right| ^{2}\right) ^{\frac{p-2}{2}}\left| \nabla\psi^{a}\right| ^{2}\,dx. \end{align*} If $1
0$ so that $\beta_{k}\theta_{p}\geq\lambda_{k}$, from
(\ref{401}), (\ref{402}), and (\ref{403}), we obtain
\[
I_{1}+I_{2}+I_{3}\geq-\frac{1}{k}-\frac{1}{k}\left| A^{s}\right|
^{p}-\lambda_{k}\left| A^{a}\right| ^{p},
\]
which together with (\ref{113}) gives (\ref{111}).
\end{proof}
\begin{remark}
\label{remark1}\emph{It is clear from the proof of Theorem \ref{theorem 1}
that if }$f$\emph{ is nonnegative the we may take }$F_{k}$\emph{ to be also
nonnegative.}
\end{remark}
\section{Lower semicontinuity\label{section lower}}
The proof of Theorem \ref{theorem4} relies on the so-called Decomposition
Lemma (see \cite{FMP}).
\begin{lemma}
[Decomposition Lemma]\label{lemma decomposition}Let $\Omega$ be a bounded open
set in $\mathbb{R}^{n}$, let $p>1$, and let $\left\{ u_{k}\right\} $ be a
sequence weakly converging to a function $u$ in $W^{1,p}\left( \Omega
;\mathbb{R}^{n}\right) $. Then there exists a subsequence (not relabeled) and
a sequence $\left\{ v_{k}\right\} $ weakly converging to $u$ in
$W^{1,p}\left( \Omega;\mathbb{R}^{n}\right) $ such that $v_{k}=u$ in a
neighborhood of $\partial\Omega$, $\left\{ \left\vert \nabla v_{k}\right\vert
^{p}\right\} $\ is equi-integrable, and $\mathcal{L}^{n}\left( \left\{
u_{k}\neq v_{k}\right\} \right) \rightarrow0$.
\end{lemma}
The following simple lemma may be found in \cite{FLP}, however we include its
proof for the convenience of the reader.
\begin{lemma}
\label{lemma lower} Let $D\subset{\mathbb{R}}^{m}$ be an open set and let
\[
f:D\times\mathbb{M}^{d\times n}\rightarrow{\mathbb{R}}%
\]
be a lower semicontinuous function such that for every $v\in D$ the function
$f(v,\cdot)$ is continuous. Then for every $\bar{v}\in D$, $\varepsilon>0$,
and $L>0$ there exists $\delta=\delta(\bar{v},\varepsilon,L)\in(0,1)$ such
that
\[
f(\bar{v},A)\leq f(v,A)+\varepsilon
\]
for every $(v,A)\in D\times{\mathbb{M}}^{d\times n}$, with $|v-\bar{v}%
|\leq\delta$ and $|A|\leq L$.
\end{lemma}
\begin{proof}
Assume, for contradiction, that there exist $\bar{v}\in D$, $L>0$,
$\bar{\varepsilon}>0$, and a sequence
\[
\{(v_{k},A_{k})\}\subset D\times\overline{B_{d\times n}(0,L)},
\]
such that
\begin{equation}
\bar{\varepsilon}+f(v_{k},A_{k})