Concentration along Geodesics for a Nonlinear Steklov Problem Arising in Corrosion Modeling

Abstract

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