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Radial symmetry and uniqueness for an overdetermined problem. (English) Zbl 0980.35117

Consider a function \(U\), harmonic in a ring-shaped domain and taking two constant (distinct) values on the two connected components of the boundary. If we know in advance that one of components is a sphere, and \(u\) satisfies some overdetermined condition on the other one, can we conclude that \(u\) is radial. This paper answers this question for certain overdetermined condition on the gradient of \(u\), generalizing some previous results. Conditions depending on the principal curvature of the boundary are also investigated. Existence and uniqueness of a radial solution to the overdetermined problem are discussed. Some extension to ellipsoidal domains, as well as to quasilinear elliptic equations are carried out.

MSC:

35N05 Overdetermined systems of PDEs with constant coefficients
35J25 Boundary value problems for second-order elliptic equations
58J70 Invariance and symmetry properties for PDEs on manifolds
35N10 Overdetermined systems of PDEs with variable coefficients
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