An overdetermined problem for elliptic equations

Tynysbek Sh. Kalmenov, Nurbek Kakharman

2024

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DOI 10.3934/math.20241002
Data confidence 70 medium
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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.