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Symmetry theorems for the overdetermined eigenvalue problems. (English) Zbl 1129.35051

Let \(\Omega\subset \mathbb{R}^n\) be a bounded domain with \(C^1\) boundary \(\partial \Omega\). Then the Schiffer conjecture asserts that the existence of a nontrivial solution \(u\) of the overdetermined Neumann eigenvalue problems \(\Delta u+\alpha u=0\) in \(\Omega,\alpha>0,u| _{\partial \Omega}=c=\text{const}, \frac{\partial u}{\partial \nu}| _{\partial \Omega}=0\) (\(\nu\) is the unit exterior normal to \(\partial \Omega\)) implies that \(\Omega\) is a ball. The author has proved that the Schiffer conjecture is true if and only if the third order interior normal derivative of the Neumann eigenfunction \(u\) is constant on \(\partial \Omega\) and \(u\) is symmetric about the center of the ball \(\Omega\). Also by a similar way here the following Berenstein conjecture [C. A. Berenstein, J. Anal. Math. 37, 128–144 (1980; Zbl 0449.35024)] is proved: for a bounded \(C^{2,\eta}\) domain \(\Omega\subset \mathbb{R}^n\) if there exists a nontrivial solution \(v\) of the eigenvalue problem \(\Delta v+\alpha v=0\) in \(\Omega,\alpha>0,v| _{\partial \Omega}=0,\frac{\partial v}{\partial v}| _{\partial \Omega}=0\), then \(\Omega\) is a ball.

MSC:

35N05 Overdetermined systems of PDEs with constant coefficients
35J25 Boundary value problems for second-order elliptic equations

Citations:

Zbl 0449.35024
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References:

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