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.
Chakalov,Sur un probl `eme de D
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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.