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.

References in corpus (21)

• Geometrical structure of Laplacian eigenfunctions
• Strong defocusing of molecular reaction times results from an interplay of geometry and reaction control
• Paradigm shift in diffusion-mediated surface phenomena
• Spectral theory of imperfect diffusion-controlled reactions on heterogeneous catalytic surfaces
• Theory of diffusion-influenced reactions in complex geometries
• Probability distribution of the boundary local time of reflected Brownian motion in Euclidean domains
• Surface Hopping Propagator: An Alternative Approach to Diffusion-Influenced Reactions
• Diffusion toward non-overlapping partially reactive spherical traps: fresh insights onto classic problems
• Fluctuations and correlations in chemical reaction kinetics and population dynamics
• Computational Methods For Extremal Steklov Problems
• Imperfect Diffusion-Controlled Reactions
• Statistics of boundary encounters by a particle diffusing outside a compact planar domain
• Joint distribution of multiple boundary local times and related first-passage time problems with multiple targets
• Depletion of Resources by a Population of Diffusing Species
• Statistics of diffusive encounters with a small target: Three complementary approaches
• Sloshing, Steklov and corners: Asymptotics of Steklov eigenvalues for curvilinear polygons
• Limiting absorption principle and perfectly matched layer method for Dirichlet Laplacians in quasi-cylindrical domains
• A numerical study of the Dirichlet-to-Neumann operator in planar domains
• Spectral properties of the Dirichlet-to-Neumann operator for spheroids
• Low energy scattering asymptotics for planar obstacles
• Sharp bounds for the first two eigenvalues of an exterior Steklov eigenvalue problem