Singular limit phenomenon in a nonlinear elliptic model arising in electrochemistry

Abstract

We study a singularly perturbed harmonic problem, with nonlinear Neumann boundary conditions of signed exponential type, arising in galvanic corrosion applications. We focus on the low-conductivity regime, which leads to a singular limit in which the solution presents a jump in the boundary. We prove convergence of the solution traces in the corresponding fractional Sobolev spaces, identify the interior limit solution, and obtain a sharp logarithmic energy expansion for smooth boundary junctions. The proofs rely mostly on trace Moser--Trudinger estimates for showing well-posedness and regularity, and on mollifier approximation techniques for the singular limit analysis. Numerical experiments test the predicted convergence rate.