Non-unique solutions to boundary value problems for non-symmetric divergence form equations

Abstract

We calculate explicitly solutions to the Dirichlet and Neumann boundary value problems in the upper half plane, for a family of divergence form equations having non-symmetric coefficients with a jump discontinuity. It is shown that the boundary equation method and the Lax–Milgram method for constructing solutions may give two different solutions when the coefficients are sufficiently non-symmetric.