Homogenization of a class of nonlinear eigenvalue problems

Abstract

In this article we study the asymptotic behaviour of the eigenvalues of a family of nonlinear monotone elliptic operators of the form A(epsilon) = - div(a(epsilon) (x, del u)), which are sub-differentials of even, positively homogeneous convex functionals, under the assumption that the operators G-converge to art operator A(hom) = div(a(hom) (x, del)u). We show that any limit point lambda of a sequence of eigenvalues A, is an eigenvalue of the limit operator A(hom,) where lambda(epsilon) is an eigenvalue corresponding to the operator lambda(epsilon). We also show the convergence of the sequence of first eigenval ties lambda(1)(epsilon) to the corresponding first eigenvalue of the homogenized operator.