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Symmetry and overdetermined boundary value problems. (English) Zbl 0789.35118

Summary: A polarized version of Bochner’s identity is used to obtain symmetry results for an overdetermined boundary value problem on constant curvature manifolds. The identity is also used to prove a Rellich identity for Dirichlet eigenvalues of the Laplacian. Finally, a reflection argument which was developed by Alexandrov is used to obtain symmetry results for overdetermined boundary value problems on constant curvature manifolds.

MSC:

35N05 Overdetermined systems of PDEs with constant coefficients
53C20 Global Riemannian geometry, including pinching
31C12 Potential theory on Riemannian manifolds and other spaces
58J05 Elliptic equations on manifolds, general theory
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