From Neumann to Steklov and beyond, via Robin: the Weinberger way

Pedro Freitas; Richard Laugesen

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arXiv · 1810.07461

From Neumann to Steklov and beyond, via Robin: the Weinberger way

Pedro Freitas·Richard Laugesen

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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.

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Pedro Freitas, Richard Laugesen. 2018-10-17. From Neumann to Steklov and beyond, via Robin: the Weinberger way. https://arxiv.org/abs/1810.07461

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