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Overdetermined boundary problems with nonconstant Dirichlet and Neumann data

Miguel Domínguez-Vázquez, Alberto Enciso and Daniel Peralta-Salas

Vol. 16 (2023), No. 9, 1989–2003

We consider the overdetermined boundary problem for a general second-order semilinear elliptic equation on bounded domains of n , where one prescribes both the Dirichlet and Neumann data of the solution. We are interested in the case where the data are not necessarily constant and where the coefficients of the equation can depend on the position, so that the overdetermined problem does not generally admit a radial solution. Our main result is that, nevertheless, under minor technical hypotheses nontrivial solutions to the overdetermined boundary problem always exist.

overdetermined boundary value problems, semilinear elliptic equations
Mathematical Subject Classification
Primary: 35N25
Secondary: 35J61
Received: 18 August 2020
Revised: 23 March 2022
Accepted: 14 April 2022
Published: 11 November 2023
Miguel Domínguez-Vázquez
Department of Mathematics
Universidade de Santiago de Compostela
Alberto Enciso
Instituto de Ciencias Matemáticas
Consejo Superior de Investigaciones Científicas
Daniel Peralta-Salas
Instituto de Ciencias Matemáticas
Consejo Superior de Investigaciones Científicas

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