The exterior Steklov problem for Euclidean domains

TL;DR

This paper studies the Steklov eigenvalue problem in exterior Euclidean domains, establishing bounds and properties of eigenvalues, with implications for convex domains and comparisons to interior and 2D cases.

Contribution

It introduces an Escobar-type lower bound for the first exterior Steklov eigenvalue in convex domains, extending spectral theory in exterior problems.

Findings

Established equivalences between different formulations of the exterior Steklov problem.

Proved a sharp lower bound for the first eigenvalue involving boundary curvatures.

Showed the existence of convex domains with eigenvalues tending to infinity.

Abstract

We investigate the Steklov eigenvalue problem in an exterior Euclidean domain. First, we present several formulations of this problem and establish the equivalences between them. Next, we examine various properties of the exterior Steklov eigenvalues and eigenfunctions. One of our main findings is an Escobar-type lower bound for the first exterior Steklov eigenvalue on convex domains in dimensions three and higher. This bound is expressed in terms of the principal curvatures of the boundary and is sharp, with equality attained for a ball. Moreover, it implies the existence of a sequence of convex domains with fixed volume and the first exterior Steklov eigenvalues tending to infinity. This contrasts with the interior case, as well as with the two-dimensional exterior case, for which we show that an analogue of the Weinstock isoperimetric inequality holds.