Affiliations: UFR des Sciences‐Mathématiques, UPRESA CNRS 6085, Site Colbert, Université de Rouen, 76821 Mont Saint Aignan Cedex, France | Dipartimento di Matematica ed Appl.“R. Caccioppoli”, Complesso Monte S. Angelo, Edificio T, Università di Napoli Federico II, 80125 Napoli (via Cintia), Italy | D.I.I.M.A., Facoltá di Scienze, Via S. Allende, Universitá di Salerno, 84081 Baronissi, Salerno, Italy
Abstract: Consider the domain \varOmega_\varepsilon =\varOmega -T_\varepsilon obtained by removing a closed set T_\varepsilon of \varepsilon‐periodic holes of size \varepsilon from a bounded open set \varOmega. We study the homogenization of the nonlinear problem \cases{-{\rm div}(A({x/\varepsilon})Du_\varepsilon)+ \gamma u_\varepsilon = H( {x/\varepsilon}, u_\varepsilon , Du_\varepsilon )& $\hbox{in }\varOmega_\varepsilon ,$\cr (A({x/\varepsilon})Du_\varepsilon)\cdot\underline\nu =0 &$\hbox{on } \Ncurpartial T_\varepsilon ,$\cr u_\varepsilon =0&$\hbox{on } \Ncurpartial \varOmega,$\cr u_\varepsilon\in H^1(\varOmega_\varepsilon )\cap L^\infty(\varOmega_\varepsilon ),&\cr}
where H( y,s,\xi) is ]0,1[^n‐periodic in y and has quadratic growth with respect to \xi. We prove that the linear part of the limit problem is the homogenized matrix of the linear problem and the nonlinear part is given by H^0( u, Du), where H^0 is defined by H^0( s,\xi) =\int_{]0,1[^n-\overline T} H ( y,s,C(y)\xi)\, {\rm d}y \quad \forall( s,\xi)\in\NBbbR\times\NBbbR^n ,
C( {\cdot/\varepsilon}) being the corrector matrices of the linear problem and T the reference hole.
