Qualitative analysis and iterative approximation of the Laplace equation with exponential Robin boundary conditions

Abstract

This paper investigates an elliptic boundary value problem involving the Laplace equation on the unit disk in two-dimensional space, subject to a nonlinear exponential Robin boundary condition on a portion of the boundary. This model is physically motivated by the study of corrosion localized on metallic structures, where the flux law exhibits an exponential dependence on the electrochemical potential. We establish the existence and uniqueness of the solution in an appropriate Hilbert space under suitable smallness assumptions on the boundary data. Our mathematical approach is constructive: we define an iterative sequence of linearized problems and prove its strong convergence toward the solution of the original nonlinear system. A key technical contribution of this work is the derivation of explicit Sobolev embedding constants for the periodic setting on the unit disk, providing refined estimates necessary to control the exponential nonlinearity.