A computer-assisted proof constructs a bounded simply connected non-circular planar domain with a nonconstant Neumann eigenfunction equal to 1 on the boundary, disproving Schiffer's and Pompeiu's conjectures.
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A computer-assisted counterexample to the planar Pompeiu and Schiffer conjectures
A computer-assisted proof constructs a bounded simply connected non-circular planar domain with a nonconstant Neumann eigenfunction equal to 1 on the boundary, disproving Schiffer's and Pompeiu's conjectures.