Counterexamples to Symmetry for Partially Overdetermined Elliptic Problems

TL;DR

This paper presents counterexamples demonstrating that the classical symmetry results for elliptic overdetermined problems do not extend to cases where boundary conditions are only partially prescribed, clarifying the scope of previous positive results.

Contribution

It provides the first known counterexamples for partially overdetermined elliptic problems, clarifying the limitations of symmetry results in these cases.

Findings

Counterexamples show symmetry may fail in partially overdetermined problems.

Results justify assumptions used in previous positive symmetry theorems.

Clarifies the boundary conditions under which symmetry results hold or fail.

Abstract

We exhibit several counterexamples showing that the famous Serrin's symmetry result for semilinear elliptic overdetermined problems may not hold for partially overdetermined problems, that is when both Dirichlet and Neumann boundary conditions are prescribed only on part of the boundary. Our counterexamples enlighten subsequent positive symmetry results obtained by the first two authors for such partially overdetermined systems and justify their assumptions as well.