Concentration along geodescis for a nonlinear Steklov problem arising in corrosion modelling

TL;DR

This paper studies a nonlinear boundary value problem modeling corrosion, proving solutions concentrate along boundary geodesics as a key parameter approaches zero.

Contribution

It introduces a new existence result for solutions concentrating along boundary geodesics in a nonlinear Steklov problem relevant to corrosion modeling.

Findings

Solutions exist that concentrate along boundary geodesics as λ approaches zero.

The problem models corrosion processes with nonlinear boundary conditions.

Concentration phenomena are rigorously established in a three-dimensional setting.

Abstract

We consider the problem of finding pairs $(\lambda; u)$, with $\lambda > 0$ and $u$ a harmonic function in a three dimensional torus-like domain, satisfying the nonlinear boundary condition $\partial_{\nu} u = \lambda \sinh u$ on the boundary. This type of boundary condition arises in corrosion modelling (Butler Volmer condition). We prove existence of solutions which concentrate along some geodesics of the boundary as the parameter $\lambda$ goes to zero.