Radial symmetry and uniqueness for an overdetermined problem (Q2709882)
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scientific article
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Radial symmetry and uniqueness for an overdetermined problem |
scientific article |
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12 March 2002
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harmonic function
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extension to ellipsoidal domains
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Radial symmetry and uniqueness for an overdetermined problem (English)
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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.
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