Homogenization of a class of nonlinear eigenvalue problems
L. Baffico, Carlos Conca, Rajesh Mahadevan · Proceedings of the Royal Society of Edinburgh Section A Mathematics · 2006
In this article we study the asymptotic behaviour of the eigenvalues of a family of nonlinear monotone elliptic operators of the form A ε = −div( a ε ( x , ∇ u )), which are sub-differentials of even, positively homogeneous convex functionals, under the assumption that the operators G -converge to an operator A hom = −div( a hom ( x , ∇ u )). We show that any limit point λ of a sequence of eigenvalues λ ε is an eigenvalue of the limit operator A hom , where λ ε is an eigenvalue corresponding to the operator A ε . We also show the convergence of the sequence of first eigenvalues to the corresponding first eigenvalue of the homogenized operator.