Extremal Steklov-Neumann Eigenvalues

Abstract

Let $\hat{I}\text{c◯}$ be a bounded open planar domain with smooth connected boundary, $\hat{I}“$, that has been partitioned into two disjoint components, $\hat{I}“=\hat{I}{“}_S\bigsqcup \hat{I}{“}_N$. We consider the Steklov-Neumann eigenproblem on $\hat{I}\text{c◯}$, where a harmonic function is sought that satisfies the Steklov boundary condition on $\hat{I}{“}_S$ and the Neumann boundary condition on $\hat{I}{“}_N$. We pose the extremal eigenvalue problems (EEPs) of minimizing/maximizing the $k$-th non-trivial Steklov-Neumann eigenvalue among boundary partitions of prescribed measure. We formulate a relaxation of these EEPs in terms of weighted Steklov eigenvalues where an $L^{\infty }(\hat{I}“)$ 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 $\hat{I}{“}_S\bigsqcup \hat{I}{“}_N$ for the $k$-th eigenvalue consists of $k+1$ connected components that are symmetrically arranged on the boundary. For a disk, for $k=1$, the constant density is a maximizer for the relaxed problem; we also provide numerical and asymptotic evidence that for $k\ge 2$, the maximizing density for the relaxed problem is a non-trivial function; a sequence of rapidly oscillating Steklov/Neumann boundary conditions approach the supremum value.

23 pages, 6 figures, 2 pages appendix. We have corrected the variational characterization for $\hat{I}{»}_1^{-1}(\ddot{I})$