A minimization problem for an elliptic eigenvalue problem with nonlinear dependence on the eigenparameter

摘要

In this paper we examine an eigenvalue optimization problem. Given two nonlinear functions alpha(lambda) and beta(lambda), find a subset D of the unit ball of measure A for which the first Dirichlet eigenvalue of the operator -div((alpha(lambda)chi D + beta(lambda)chi D-c)del u) = lambda u is as small as possible. This sort of nonlinear eigenvalue problems arises in the study of some quantum dots taking into account an electron effective mass. We establish the existence of a solution, and we propose a numerical algorithm to obtain an approximate description of the optimizer. (C) 2016 Elsevier Ltd. All rights reserved.