The Steklov problem for exterior domains: asymptotic behavior and applications

Abstract

We investigate the spectral properties of the Steklov problem for the modified Helmholtz equation $(p-\Delta) u = 0$ in the exterior of a compact set, for which the positive parameter $p$ ensures exponential decay of the Steklov eigenfunctions at infinity. We obtain the small-$p$ asymptotic behavior of the eigenvalues and eigenfunctions and discuss their features for different space dimensions. These results find immediate applications to the theory of stochastic processes and unveil the long-time asymptotic behavior of probability densities of various first-passage times in exterior domains. Theoretical results are validated by solving the exterior Steklov problem by a finite-element method with a transparent boundary condition.