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Partially overdetermined elliptic boundary value problems. (English) Zbl 1156.35049

Summary: We consider semilinear elliptic Dirichlet problems in bounded domains, overdetermined with a Neumann condition on a proper part of the boundary. Under different kinds of assumptions, we show that these problems admit a solution only if the domain is a ball. When these assumptions are not fulfilled, we discuss possible counterexamples to symmetry. We also consider Neumann problems overdetermined with a Dirichlet condition on a proper part of the boundary, and the case of partially overdetermined problems on exterior domains.

MSC:

35J70 Degenerate elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
35B50 Maximum principles in context of PDEs
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