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Homogenization of bounded solutions of elliptic equations with quadratic growth in periodically perforated domains. (English) Zbl 0944.35009

Let \(\Omega\subset\mathbb{R}^n\) be a bounded domain and \(\Omega_\varepsilon= \Omega-T_\varepsilon\), where \(T_\varepsilon\) is a closed set of \(\varepsilon\)-periodic holes of size \(\varepsilon\). The authors study the homogeneization of the nonlinear problem \[ u_\varepsilon\in H^1( \Omega_\varepsilon) \cap L^\infty (\Omega_\varepsilon), \]
\[ -\text{div} \bigl( A(x/ \varepsilon) Du_\varepsilon\bigr) +\gamma u_\varepsilon=H(x/ \varepsilon, u_\varepsilon,Du_\varepsilon) \text{ in }\Omega_\varepsilon, \]
\[ (A(x/ \varepsilon) Du_\varepsilon) \cdot\nu=0 \text{ on }\partial T_\varepsilon \]
\[ u_\varepsilon= 0\text{ on }\partial \Omega, \] where \(H(y,s, \xi)\) is 1-periodic in \(y\) and has quadratic growth with respect to \(\xi\). It is proved that the linear part of the limit problem is the homogenized matrix of the linear problem and that the nonlinear part is given by \(H^0(u,Du)\), where \(H^0\) is defined by \[ H^0(s,\xi)= \int_{]0,1 [-T}{_nH}\bigl(y,s,C(y) \xi\bigr) dy, \;(s,\xi)\in \mathbb{R}\times \mathbb{R}^n; \] \(C\) is the corrector matrix of the linear problem and \(T\) is the reference hole. According to the authors, they followed some ideas of A. Bensoussan, L. Boccardo and F. Murat [Appl. Math. Optim. 26, 253-272 (1992; Zbl 0795.35008)].

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35J60 Nonlinear elliptic equations

Citations:

Zbl 0795.35008
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