## 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

Zbl 0449.35024
Full Text:

### References:

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