The limit of Dirichlet systems for variable monotone operators in general perforated domains.
Consider \(u_n\) the solution of the problem \[ -\text{div\,} a_n (x,Du) = f_n \text{ in }{\mathcal D}' (\Omega_n; {\mathbb R}^M), \quad u \in W^{1,p}_0(\Omega_n; {\mathbb R}^M), \] where \(\Omega_n\) is a sequence of open sets with \(\Omega_n \subset \Omega\) for every \(n\), \(\Omega\) a fixed bounded open set in \({\mathbb R}^N\) (\(M\) and \(N\) positive integers), \(f_n\) a sequence converging in \(W^{-1, p'}(\Omega; {\mathbb R}^M)\) and \(a_n: \Omega \times {\mathcal M}_{M \times N} \to {\mathcal M}_{M \times N}\) a sequence of Carathéodory functions satisfying for some \(p \geqslant 2\) (indeed something different, but equivalent to (i)--(ii)--(iii)) (i) \(a_n(x,0) = 0\), (ii) \(( a_n(x,\xi_1) - a_n(x,\xi_2)):(\xi_1 - \xi_2) \geqslant \alpha | \xi_1 - \xi_2| ^p\) (\(\alpha > 0\)), (iii) \(| a_n(x,\xi_1) - a_n(x,\xi_2)| \leqslant \beta (r(x) + | \xi_1| + | \xi_2| )^{p(p-1-\sigma)/(p-\sigma)} | \xi_1 - \xi_2| ^{\sigma/(p-\sigma)}\) (\(\beta > 0\), \(\sigma \in (0,1]\), \(r \in L^p(\Omega)\)), for every \(n \in {\mathbb N}\), for every \(\xi_1, \xi_2 \in {\mathcal M}_{M \times N}\) and for a. e. \(x \in \Omega\). The authors prove the existence of a subsequence of \(a_n\) (still denoted by \(a_n\)), a function \(a\) satisfying conditions analogous to (i)--(ii)--(iii), a measure \(\mu\) (belonging to a subclass of the non-negative Borel measures) and a function \(F: \Omega \times {\mathbb R}^M \to {\mathbb R}^M\) (independent of \(f_n\)) such that \(u_n \to u\) in \(W^{1,p}_0(\Omega; {\mathbb R}^M)\)-weak, where \(u\) is the solution in \(W^{1,p}_0(\Omega; {\mathbb R}^M) \cap L^p_{\mu}(\Omega; {\mathbb R}^M\)) of the problem \[ \int_{\Omega} a(x, Du) : Dv \, dx + \int_{\Omega} F(x,u) v\,d\mu = \langle f,v \rangle \text{ for every } v \in W^{1,p}_0(\Omega; {\mathbb R}^M) \cap L^p_{\mu}(\Omega; {\mathbb R}^M). \] If \(\Omega_n \equiv \Omega\) for every \(n\) then \(\mu \equiv 0\). Finally a corrector result is also given.
scientific article; zbMATH DE number 1975942
scientific article