AN OPTIMIZATION PROBLEM FOR THE FIRST STEKLOV EIGENVALUE OF A NONLINEAR PROBLEM

In this paper we study the first (nonlinear) Steklov eigenvalue, lambda, of the following problem: -Delta(p)u, + vertical bar u vertical bar(p-2)u+ alpha phi vertical bar u vertical bar(p-2)u = 0 in a bounded smooth domain Omega with vertical bar del u vertical bar(p-2) partial derivative u/partial derivative v = lambda vertical bar u vertical bar(p-2)u on the boundary partial derivative Omega. We analyze the dependence of this first eigenvalue with respect to the weight phi and with respect to the parameter alpha. We prove that for fixed alpha there exists an optimal phi(alpha) that minimizes lambda in the class of uniformly bounded measurable functions with fixed integral. Next, we study the limit of these minima as the parameter alpha goes to infinity and we find that the limit is the first Steklov eigenvalue in the domain with a hole where the eigenfunctions vanish.