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Asymptotic behaviour and correctors for Dirichlet problems in perforated domains with homogeneous monotone operators. (English) Zbl 0899.35007

The paper deals with the asymptotic behaviour (as \(n\to +\infty)\) of the solution of nonlinear elliptic equations of the form \[ -\text{div} \bigl(a (x,Du_n) \bigr) =f\quad \text{in } \Omega_n, \qquad u_n\in W^{1,p} (\Omega_n), \tag{1} \] where \((\Omega_n)\) is a sequence of open subsets of a given domain \(\Omega \subset \mathbb{R}^N\), \(p>1\), \(f\in W^{-1,p'} (\Omega)\), and \(u_n\) are extended by zero outside \(\Omega_n\). The ellliptic operator \(A(u)=-\text{div} (a(x,Du))\) is supposed to be monotone, and more precisely:
(i) \(a:\Omega \times \mathbb{R}^N \to \mathbb{R}^N\) is a Borel function satisfying the homogeneity condition \(a(x,tz) =| t |^{p-2} ta(x,z)\);
(ii) there exist two positive constants \(c_0\), \(c_1\) such that \[ \text{if }p \geq 2\quad \begin{cases} \bigl(a(x,z_1) -a(x,z_2), z_1-z_2\bigr) \geq c_0| z_1-z_2 |^p\\ \bigl| a(x, z_1) -a(x,z_2) \bigr |\leq c_1| z_1-z_2 |\bigl(| z_1| +| z_2 |\bigr)^{p-2} \end{cases} \]
\[ \text{if }1<p\leq 2\quad \begin{cases} \bigl(a(x,z_1) -a(x,z_2), z_1-z_2 \bigr)\geq c_0 | z_1-z_2 |^2 \bigl(| z_1| +| z_2| \bigr)^{p-2}\\ \bigl| a(x,z_1) -a(x,z_2) \bigr | \leq c_1| z_1-z_2 |^{p-1}. \end{cases} \] The main theorem of the paper is the following compactness result: For every sequence \((\Omega_n)\) there exists a subsequence \((\Omega_{n_k})\) and a nonnegative Borel measure \(\mu\) such that for every \(f\in W^{-1,p'} (\Omega)\) the solutions \(u_{n_k}\) of (1) converge weakly in \(W_0^{1,p} (\Omega)\) to the solution \(u\) of the problem \[ -\text{div} \bigl(a(x,Du) \bigr)+ | u|^{p-2} u\mu=f \quad \text{in } \Omega, \qquad u\in W^{1,p} (\Omega) \cap L^p_\mu (\Omega). \] Moreover the measure \(\mu\) vanishes on all sets of \((1,p)\) capacity zero.
Reviewer: G.Buttazzo (Pisa)

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35J65 Nonlinear boundary value problems for linear elliptic equations
49J45 Methods involving semicontinuity and convergence; relaxation
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