## Symmetry for an overdetermined boundary problem in a punctured domain.(English)Zbl 1152.35523

Summary: We prove that if there exists a positive nonconstant function $$u$$ which is $$p$$-harmonic ($$1<p\leqslant n$$) in a punctured domain $$\varOmega \setminus \{0\}\subset \mathbb R^n$$ and such that both $$u$$ and $$\frac{\partial u}{\partial v}$$ are constant on $$\partial \varOmega$$, then $$u$$ is radial and $$\partial \varOmega$$ is a round sphere. The proof is based on a combination of integral identities, maximum principles and the isoperimetric inequality.

### MSC:

 35R35 Free boundary problems for PDEs 35J60 Nonlinear elliptic equations 35J20 Variational methods for second-order elliptic equations
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### References:

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