Relaxation of multiple integrals below the growth exponent

Irene Fonseca; Jan Malý

doi:10.1016/s0294-1449(97)80139-4

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

F^{q, p} (u, Ω) := in f_{{u_{n}}} {lim inf_{n \to \infty} \int_{Ω} F (x, u_{n}, \nabla u_{n}) d x : u_{n} \in W^{1, q} (Ω, R^{d}), u_{n} \to u weakly in W^{1, q} (Ω, R^{d})}, F_{loc}^{q, p} (u, Ω) := in f_{{u_{n}}} {lim inf_{n \to \infty} \int_{Ω} F (x, u_{n}, \nabla u_{n}) d x : u_{n} \in W_{loc}^{1, q} (Ω, R^{d}), u_{n} \to u weakly in W^{1, q} (Ω, R^{d})}

E : u \mapsto \int_{Ω} F (x, u, \nabla u) d x, u \in W^{1, q} (Ω, R^{d}),

where $0 \leq F (ϰ, ζ, ξ) \leq C (1 + ∣ ζ ∣^{r} + ∣ ξ ∣^{q})$ and $max {1, r \frac{N - 1}{N + r}, q \frac{N - 1}{N}} < p \leq q$ , is studied. In particular, $W^{1, p}$ -sequential weak lower semicontinuity of $E (\cdot)$ 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

F^{q, p} (u, Ω) := in f_{{u_{n}}} {lim inf_{n \to \infty} \int_{Ω} F (x, u_{n}, \nabla u_{n}) d x : u_{n} \in W^{1, q} (Ω, R^{d}), u_{n} \to u faible dans W^{1, q} (Ω, R^{d})}, F_{loc}^{q, p} (u, Ω) := in f_{{u_{n}}} {lim inf_{n \to \infty} \int_{Ω} F (x, u_{n}, \nabla u_{n}) d x : u_{n} \in W_{loc}^{1, q} (Ω, R^{d}), u_{n} \to u faible dans W^{1, q} (Ω, R^{d})}

de la fonctionnelle

E : u \mapsto \int_{Ω} F (x, u, \nabla u) d x, u \in W^{1, q} (Ω, R^{d}),

où $0 \leq F (ϰ, ζ, ξ) \leq C (1 + ∣ ζ ∣^{r} + ∣ ξ ∣^{q})$ et $max {1, r \frac{N - 1}{N + r}, q \frac{N - 1}{N}} < p \leq q$ . En particulier, la $W^{1, p}$ -semicontinuité inférieure séquentielle faible de $E (\cdot)$ 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