The Asymptotic Analysis of Some PDE and Steklov Eigenvalue Problems with Partially Reactive Patches in 3-D

We consider steady-state diffusion in a three-dimensional bounded domain with a smooth reflecting boundary that is partially covered by small partially reactive patches. By using the method of matched asymptotic expansions, we investigate the competition of these patches for a diffusing particle and the crucial role of surface reactions on these targets. After a brief overview of former contributions to this field, we first illustrate our approach by considering the classical problems of the mean first-reaction time (MFRT) and the splitting probability for partially reactive patches characterized by a Robin boundary condition. For a spherical domain, we derive a three-term asymptotic expansion for the MFRT and splitting probabilities in the small-patch limit. This expansion is valid for arbitrary reactivities, and also accounts for the effect of the spatial configuration of patches on the boundary. Secondly, we consider more intricate surface reactions modeled by mixed Steklov-Neumann or Steklov-Neumann-Dirichlet problems. We provide the first derivation of the asymptotic behavior of the eigenvalues and eigenfunctions for these spectral problems in the small-patch limit for a spherical domain. Extensions of these asymptotic results to arbitrary domains and their physical applications are discussed.