# Relaxation of multiple integrals below the growth exponent

• ### Irene Fonseca

Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213 USA
• ### Jan Malý

Faculty of Mathematics and Physics-KMA, Charles University, Sokolovská 83, 18600 Praha 8, Czech Republic

## Abstract

The integral representation of the relaxed energies

$\begin{matrix} \mathscr{F}^{q,p}\left(u,\Omega \right): =\inf_{\{u_n\}}\biggl\{\liminf_{n\to\infty}\int _{\Omega }F\left(x,u_{n},\nabla u_{n}\right)dx:u_{n} \in W^{1,q}\left(\Omega ,ℝ^{d}\right), \\ u_{n}\rightarrow u\text{ weakly in }W^{1,q}\left(\Omega ,ℝ^{d}\right) \biggr\}, \\ \mathscr{F}_{\text{loc}}^{q,p}\left(u,\Omega \right): = \inf_{\{u_n\}}\biggl\{\liminf_{n\to\infty}\int _{\Omega }F\left(x,u_{n},\nabla u_{n}\right)dx:u_{n} \in W_{\text{loc}}^{1,q}\left(\Omega ,ℝ^{d}\right), \\ u_{n}\rightarrow u\text{ weakly in }W^{1,q}\left(\Omega ,ℝ^{d}\right) \biggr\} \\ \end{matrix}$
$\begin{matrix} \begin{matrix} E:u\mapsto \int _{\Omega }F\left(x,u,\nabla u\right)dx, \\ \end{matrix} & u \in W^{1,q}\left(\Omega ,ℝ^{d}\right), \\ \end{matrix}$

where 0 $\leq$ F (ϰ,ζ,ξ) $\leq$ $C$ (1+|ζ|$^r$ + |ξ|$^q$ and max $\left\{1,r\frac{N - 1}{N + r},q\frac{N - 1}{N}\right\} > p \leq q$, is studied. In particular, $W^{1,q}$-sequential weak lower semicontinuity of E(·) is obtained in the case where F = F(ξ) is a quasiconvex function and p > q(N − 1)/N.

# Résumé

On étudie la représentation intégrale d’énergies relaxées

$\begin{matrix} \mathscr{F}^{q,p}\left(u,\Omega \right): =\inf_{\{u_n\}}\biggl\{\liminf_{n\to\infty}\int _{\Omega }F\left(x,u_{n},\nabla u_{n}\right)dx:u_{n} \in W^{1,q}\left(\Omega ,ℝ^{d}\right), \\ u_{n}\rightarrow u\text{ faible dans }W^{1,q}\left(\Omega ,ℝ^{d}\right) \biggr\}, \\ \mathscr{F}_{\text{loc}}^{q,p}\left(u,\Omega \right): = \inf_{\{u_n\}}\biggl\{\liminf_{n\to\infty}\int _{\Omega }F\left(x,u_{n},\nabla u_{n}\right)dx:u_{n} \in W_{\text{loc}}^{1,q}\left(\Omega ,ℝ^{d}\right), \\ u_{n}\rightarrow u\text{ faible dans }W^{1,q}\left(\Omega ,ℝ^{d}\right) \biggr\} \\ \end{matrix}$

de la fonctionnelle

$\begin{matrix} E:u\mapsto \int _{\Omega }F\left(x,u,\nabla u\right)dx, & u \in W^{1,q}\left(\Omega ,ℝ^{d}\right), \\ \end{matrix}$

où 0 $\leq$ F (ϰ,ζ,ξ) $\leq$ $C$ (1+|ζ|$^r$ + |ξ|$^q$ et max $\left\{1,r\frac{N - 1}{N + r},q\frac{N - 1}{N}\right\}> p \leq q$. En particulier, la $W^{1,q}$-semicontinuité inférieure séquentielle faible de E(·) est obtenue dans le cas où F = F(ξ) est une fonction quasiconvexe et p > q(N − 1)/N.