A minimization problem for an elliptic eigenvalue problem with nonlinear dependence on the eigenparameter
In this paper we examine an eigenvalue optimization problem. Given two nonlinear functions α(λ) and β(λ), find a subset D of the unit ball of measure A for which the first Dirichlet eigenvalue of the operator -div((α(λ)χD+β(λ)χDc)∇ u)=λu 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.
Nonlinear eigenvalue problem