Singular limits of a two-dimensional boundary value problem arising in corrosion modelling

Abstract

We consider the boundary value problem Du = 0 in Omega, partial derivative u/partial derivative v = 2 lambda sinh u on partial derivative Omega

where Omega is a smooth and bounded domain in R-2 and lambda > 0. We prove that for any integer k >= 1 there exist at least two solutions u (lambda) with the property that the boundary flux satisfies up to subsequences lambda -> 0,

2 lambda sinh(mu(lambda)) -> 2 pi Sigma(2k)(j=1)(-1)(j-1 delta)xi(,)(j)

where the xi (j) are points of partial derivative Omega ordered clockwise in j.