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From Neumann to Steklov and beyond, via Robin: The Weinberger way
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 143, Number 3, June 2021
- pp. 969-994
- 10.1353/ajm.2021.0024
- Article
- Additional Information
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abstract:
The second eigenvalue of the Robin Laplacian is shown to be maximal for the ball among domains of fixed volume, for negative values of the Robin parameter $\alpha$ in the regime connecting the first nontrivial Neumann and Steklov eigenvalues, and even somewhat beyond the Steklov regime. The result is close to optimal, since the ball is not maximal when $\alpha$ is sufficiently large negative, and the problem admits no maximiser when $\alpha$ is positive.