Any bounded domain admitting a weak solution to a degenerate overdetermined elliptic problem with constant normal derivative is a ball; for ring-shaped domains, both boundaries must be balls but need not be concentric.
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Symmetry in Serrin-type overdetermined problems
Any bounded domain admitting a weak solution to a degenerate overdetermined elliptic problem with constant normal derivative is a ball; for ring-shaped domains, both boundaries must be balls but need not be concentric.