For every integer m≥4 there are smooth non-disk domains in R^2 on which Δu+λu=0 admits a solution with constant nonzero Dirichlet and Neumann data, the first such counterexamples to the Willms-Gladwell conjecture.
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Non-symmetric solutions to an overdetermined problem for the Helmholtz equation in the plane
For every integer m≥4 there are smooth non-disk domains in R^2 on which Δu+λu=0 admits a solution with constant nonzero Dirichlet and Neumann data, the first such counterexamples to the Willms-Gladwell conjecture.