Homogenization of Nonlinear Unilateral Problems

Abstract

In this paper we consider the unilateral problems

$$ \left\{ \begin{gathered} {u_{\varepsilon }} \in K\left( {{\psi_{\varepsilon }}} \right) \hfill \\ \left\langle {{A_{\varepsilon }}\left( {{u_{\varepsilon }}} \right) - f,v - {u_{\varepsilon }}} \right\rangle \geqslant 0 \hfill \\ \forall v \in K\left( {{\psi_{\varepsilon }}} \right) \hfill \\ \end{gathered} \right. $$

((*))

where the right hand side f is fixed in W^−1,P’(Ω) and where the A_ε(v) are monotone operators acting from (Inline 1) into W^−1,p’(Ω) defined by

$$ {A_{\varepsilon }}(v) = - div\left( {{a_{\varepsilon }}\left( {x,Dv} \right)} \right)\;, $$

while the unilateral convex sets K(ψ_ε) are defined by

$$ K\left( {{\psi_{\varepsilon }}} \right) = \left\{ {v \in W_0^{{1,p}}\left( \Omega \right):v \geqslant {\psi_{\varepsilon }}\;a.e.\;in\;\Omega } \right\}\;. $$