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

TL;DR

This paper explicitly solves boundary value problems for divergence form equations with non symmetric coefficients, revealing that different solution methods can yield distinct solutions under certain conditions.

Contribution

It demonstrates that the boundary equation method and Lax--Milgram method can produce different solutions for non symmetric divergence form equations with discontinuities.

Findings

Solutions explicitly calculated for specific boundary value problems.

Different solution methods can lead to non-uniqueness in solutions.

Non symmetry and discontinuities affect solution uniqueness.

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 with 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.