JournalsaihpcVol. 14, No. 3pp. 309–338

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
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Abstract

The integral representation of the relaxed energies

Fq,p(u,Ω):=inf⁡{un}{lim inf⁡n→∞∫ΩF(x,un,∇un)dx:un∈W1,q(Ω,Rd),un→u weakly in W1,q(Ω,Rd)},Flocq,p(u,Ω):=inf⁡{un}{lim inf⁡n→∞∫ΩF(x,un,∇un)dx:un∈Wloc1,q(Ω,Rd),un→u weakly in W1,q(Ω,Rd)} \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}
E:u↦∫ΩF(x,u,∇u)dx,u∈W1,q(Ω,Rd),\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 CC (1+|ζ|r^r + |ξ|q^q and max {1,rN−1N+r,qN−1N}>p≤q\left\{1,r\frac{N - 1}{N + r},q\frac{N - 1}{N}\right\} > p \leq q, is studied. In particular, W1,qW^{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

Fq,p(u,Ω):=inf⁡{un}{lim inf⁡n→∞∫ΩF(x,un,∇un)dx:un∈W1,q(Ω,Rd),un→u faible dans W1,q(Ω,Rd)},Flocq,p(u,Ω):=inf⁡{un}{lim inf⁡n→∞∫ΩF(x,un,∇un)dx:un∈Wloc1,q(Ω,Rd),un→u faible dans W1,q(Ω,Rd)}\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

E:u↦∫ΩF(x,u,∇u)dx,u∈W1,q(Ω,Rd),\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 CC (1+|ζ|r^r + |ξ|q^q et max {1,rN−1N+r,qN−1N}>p≤q\left\{1,r\frac{N - 1}{N + r},q\frac{N - 1}{N}\right\}> p \leq q. En particulier, la W1,qW^{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.

Cite this article

Irene Fonseca, Jan Malý, Relaxation of multiple integrals below the growth exponent. Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997), no. 3, pp. 309–338

DOI 10.1016/S0294-1449(97)80139-4