Mixed Steklov-Neumann problem: asymptotic analysis and applications to diffusion-controlled reactions

TL;DR

This paper analyzes the asymptotic behavior of the mixed Steklov-Neumann spectral problem for the modified Helmholtz equation, providing formulas validated by numerical examples and applying results to diffusion-controlled reactions and first-passage processes.

Contribution

It introduces asymptotic formulas for eigenvalues and eigenfunctions of the Steklov-Neumann problem as the boundary subset shrinks, supported by numerical validation and applied to reaction time analysis.

Findings

Asymptotic formulas accurately predict eigenvalues and eigenfunctions.

Numerical results confirm high accuracy of asymptotic formulas even for larger boundary subsets.

Universal function quantifies multiple failed reaction attempts in diffusion processes.

Abstract

We consider the mixed Steklov-Neumann spectral problem for the modified Helmholtz equation in a bounded domain when the Steklov condition is imposed on a connected subset of the smooth boundary. In order to deduce the asymptotic behavior in the limit when the size of the subset goes to zero, we reformulate the original problem in terms of an integral operator whose kernel is the restriction of a suitable Green's function (or pseudo-Green's function) to the subset. Its singular behavior on the boundary yields the asymptotic formulas for the eigenvalues and eigenfunctions of the Steklov-Neumann problem. While this analysis remains at a formal level, it is supported by extensive numerical results for two basic examples: an arc on the boundary of a disk and a spherical cap on the boundary of a ball. Solving the original Steklov-Neumann problem numerically in these domains, we validate the asymptotic formulas and reveal their high accuracy, even when the subset is not small. A straightforward application of these spectral results to first-passage processes and diffusion-controlled reactions is presented. We revisit the small-target limit of the mean first-reaction time on perfectly or partially reactive targets. The effect of multiple failed reaction attempts is quantified by a universal function for the whole range of reactivities. Moreover, we extend these results to more sophisticated surface reactions that go beyond the conventional narrow escape problem.