An overdetermined problem for elliptic equations

Tynysbek Kalmenov; Nurbek Kakharman; Tynysbek Kalmenov; Nurbek Kakharman

doi:10.3934/math.20241002

AIMS Mathematics

2024, Volume 9, Issue 8: 20627-20640. doi: 10.3934/math.20241002

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Research article

An overdetermined problem for elliptic equations

Tynysbek Kalmenov ^,,
Nurbek Kakharman

Institute of Mathematics and Mathematical Modeling, Pushkin Str. 125, Almaty 050010, Kazakhstan

Received: 27 March 2024 Revised: 07 June 2024 Accepted: 19 June 2024 Published: 25 June 2024
MSC : 35J25, 35N25

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.
- overdetermined problem,
- Schiffer conjecture,
- Dirichlet condition,
- Neumann condition,
- Newton potential
Citation: Tynysbek Kalmenov, Nurbek Kakharman. An overdetermined problem for elliptic equations[J]. AIMS Mathematics, 2024, 9(8): 20627-20640. doi: 10.3934/math.20241002

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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.

References

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