Omogeneizzazione di funzionali debolmente quasi periodici

Braides, Andrea

Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 81 (1987) no. 1, pp. 29-33

Résumé

Let $f = f(x,z)$ be quasiconvex in $z$, almost periodic in $x$ in the weak sense of Besicovitch and satisfy the estimate $$|z|^{p} \le f(x,z) \le \Lambda (1+|z|^{p}).$$ Then $f$ can be homogenized; that is there exists a function $\Psi$ depending only on $z$ such that the functionals $$\int_{\Omega} f \left( \frac{x}{\epsilon},Du(x) \right) \, dx \qquad u \in H^{1,p} (\Omega;\mathbb{R}^{m})$$ converge, as $\epsilon$ goes to $0$ (in the sense of $\Gamma$-convergence) to $$\int_{\Omega} \Psi (Du(x)) \, dx.$$ Moreover an asymptotic formula for $\Psi$ can be given.