New Insights into Diffusion-Controlled Reactions via Mixed Steklov-Neumann Problem

Recent research by Denis S. Grebenkov, titled "Mixed Steklov-Neumann problem: asymptotic analysis and applications to diffusion-controlled reactions," explores the mathematical framework surrounding diffusion-controlled reactions in complex media. This study, submitted to arXiv on August 30, 2024, delves into the mixed Steklov-Neumann problem, which is crucial for understanding first-passage processes.

The paper investigates scenarios where a small target or escape window, governed by the Steklov condition, is positioned on a reflecting boundary, characterized by the Neumann condition. Grebenkov examines two primary configurations: an arc-shaped target on the boundary of a disk and a spherical-cap-shaped target on the boundary of a ball.

A significant contribution of this research is the construction of an explicit kernel for an integral operator that determines the eigenvalues and eigenfunctions associated with these configurations. The study also presents the asymptotic behavior of these eigenvalues and eigenfunctions in the small-target limit, extending the results to other bounded domains with smooth boundaries.

Furthermore, the paper applies these findings to first-passage processes, specifically revisiting the small-target behavior of the mean first-reaction time on both perfectly and partially reactive targets. This work enhances the understanding of surface reactions and extends the conventional narrow escape problem, which is relevant in various fields, including chemical physics and statistical mechanics.

The full paper can be accessed through arXiv with the identifier arXiv:2409.00213.