Parallel coordinates in three dimensions and sharp spectral isoperimetric inequalities
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In this article, it was shown that the ball maximizes the first Robin eigenvalue with negative boundary parameter among all convex domains of equal surface area under the weaker restriction that the boundary of the domain is diffeomorphic to the sphere and convex or axiconvex.Abstract:
In this paper we show how the method of parallel coordinates can be extended to three dimensions. As an application, we prove the conjecture of Antunes et al. (Adv Calc Var 10:357–380, 2017) that “the ball maximises the first Robin eigenvalue with negative boundary parameter among all convex domains of equal surface area” under the weaker restriction that the boundary of the domain is diffeomorphic to the sphere and convex or axiconvex. We also provide partial results in arbitrary dimensions.read more
Citations
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Journal ArticleDOI
From Neumann to Steklov and beyond, via Robin: the Weinberger way
TL;DR: The second eigenvalue of the Robin Laplacian is shown to be maximal for the ball among domains of fixed volume, for negative values of Robin parameter $\alpha$ in the regime connecting the first nontrivial Neumann and Steklov eigenvalues as mentioned in this paper.
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Spectral optimization for Robin Laplacian on domains admitting parallel coordinates
Pavel Exner,Vladimir Lotoreichik +1 more
TL;DR: In this article, the spectral optimization for the Robin Laplacian on a family of planar domains admitting parallel coordinates was studied and it was shown that if the curve length is kept fixed, the first eigenvalue referring to the fixed-width strip is for any value of the Robin parameter maximized by a circular annulus.
Book ChapterDOI
The Negative Spectrum of the Robin Laplacian
TL;DR: In this article, the negative spectrum of the Laplacian with a mixed attractive Robin boundary condition was studied. But the results were limited to the case where the boundary condition is unknown.
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A spectral isoperimetric inequality for Robin Laplacians on 2-manifolds and applications
TL;DR: In this paper, the problem of geometric optimization of the lowest eigenvalue for the Laplacian on a compact, simply-connected two-dimensional manifold with boundary subject to an attractive Robin boundary condition was considered.
References
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Handbook of Mathematical Functions
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A general comparison theorem with applications to volume estimates for submanifolds
Ernst Heintze,Hermann Karcher +1 more
TL;DR: Gauthier-Villars as mentioned in this paper implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions).