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

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.