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

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