## 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

### Keywords:

harmonic function; extension to ellipsoidal domains
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### References:

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