Establishes an asymptotic connection between compressed clamped-plate eigenvalues and Robin-Laplacian eigenvalues, then numerically shows that extremal domains develop boundary structure and the first eigenfunction gains more nodal domains as compression grows.
From Neumann to Steklov and beyond, via Robin: the Weinberger way
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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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2019 1verdicts
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On the behaviour of clamped plates under large compression
Establishes an asymptotic connection between compressed clamped-plate eigenvalues and Robin-Laplacian eigenvalues, then numerically shows that extremal domains develop boundary structure and the first eigenfunction gains more nodal domains as compression grows.