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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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