For every bounded connected Lipschitz domain in the plane, the third Neumann eigenvalue is strictly less than the first Dirichlet eigenvalue, removing the simple-connectivity assumption from Rohleder's theorem.
Aviles,Symmetry theorems related to Pompeiu’s problem, Amer
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Infinitely many non-ball planar domains carry Neumann eigenfunctions that are constant on the boundary, so Schiffer's and Pompeiu's conjectures are false in R^2.
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Neumann--Dirichlet eigenvalue comparison at the first Dirichlet threshold in the plane
For every bounded connected Lipschitz domain in the plane, the third Neumann eigenvalue is strictly less than the first Dirichlet eigenvalue, removing the simple-connectivity assumption from Rohleder's theorem.
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Counterexamples to Schiffer's Conjecture
Infinitely many non-ball planar domains carry Neumann eigenfunctions that are constant on the boundary, so Schiffer's and Pompeiu's conjectures are false in R^2.