Donato, Patrizia; Gaudiello, Antonio; Sgambati, Luciana Homogenization of bounded solutions of elliptic equations with quadratic growth in periodically perforated domains. (English) Zbl 0944.35009 Asymptotic Anal. 16, No. 3-4, 223-243 (1998). 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)]. Reviewer: Ibrahim Aganović (Zagreb) Cited in 5 Documents MSC: 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 35J60 Nonlinear elliptic equations Keywords:nonlinear elliptic equation Citations:Zbl 0795.35008 PDFBibTeX XMLCite \textit{P. Donato} et al., Asymptotic Anal. 16, No. 3--4, 223--243 (1998; Zbl 0944.35009)