Extremal Steklov–Neumann Eigenvalues

Abstract. Let [Formula: see text] be a bounded open planar domain with a smooth connected boundary, [Formula: see text], that has been partitioned into two disjoint components, [Formula: see text]. We consider the Steklov–Neumann eigenproblem on [Formula: see text], where a harmonic function is sought that satisfies the Steklov boundary condition on [Formula: see text] and the Neumann boundary condition on [Formula: see text]. We pose the extremal eigenvalue problems (EEPs) of minimizing/maximizing the [Formula: see text]th nontrivial Steklov–Neumann eigenvalue among boundary partitions of prescribed measure. We formulate a relaxation of these EEPs in terms of weighted Steklov eigenvalues where an [Formula: see text] density replaces the boundary partition. For these relaxed EEPs, we establish existence and prove optimality conditions. We also prove a homogenization result that allows us to use solutions to the relaxed EEPs to infer properties of solutions to the original EEPs. For a disk, we provide numerical and asymptotic evidence that the minimizing arrangement of [Formula: see text] for the [Formula: see text]th eigenvalue consists of [Formula: see text] connected components that are symmetrically arranged on the boundary. For a disk, for [Formula: see text], the constant density is a maximizer for the relaxed problem; we also provide numerical and asymptotic evidence that for [Formula: see text], the maximizing density for the relaxed problem is a nontrivial function; a sequence of rapidly oscillating Steklov/Neumann boundary conditions approach the supremum value.