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Research Article | Open Access

An overdetermined problem for elliptic equations

Tynysbek Kalmenov( ), Nurbek Kakharman
Institute of Mathematics and Mathematical Modeling, Pushkin Str. 125, Almaty 050010, Kazakhstan
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Abstract

This paper is devoted to finding a necessary and sufficient condition for the solvability of the overdetermined problem for Poisson's equation with both the Dirichlet and Neumann conditions on the entire boundary. The proof is based on the boundary condition formula for the Newton potential. The obtained results are also extended to general second-order linear elliptic equations. As a byproduct, we present a characterization of the Schiffer property. It gives a definitive answer to the Schiffer problem.

Keywords

overdetermined problemSchiffer conjectureDirichlet conditionNeumann conditionNewton potential
CLC number: 35J25, 35N25

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AIMS Mathematics
Volume 9 Issue 8,
August 2024
Pages 20627-20640
DOI: 10.3934/math.20241002

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Cite this article:
Kalmenov T, Kakharman N. An overdetermined problem for elliptic equations. AIMS Mathematics, 2024, 9(8): 20627-20640. https://doi.org/10.3934/math.20241002

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Received: 27 March 2024
Revised: 07 June 2024
Accepted: 19 June 2024
Published: 15 August 2024
©2024 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0)