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- W2578885660 abstract "Let $$Omega $$ be an open set in Euclidean space $$mathbb {R}^m,, m=2,3,...$$ , and let $$v_{Omega }$$ denote the torsion function for $$Omega $$ . It is known that $$v_{Omega }$$ is bounded if and only if the bottom of the spectrum of the Dirichlet Laplacian acting in $${mathcal {L}}^2(Omega )$$ , denoted by $$lambda (Omega )$$ , is bounded away from 0. It is shown that the previously obtained bound $$Vert v_{Omega }Vert _{{mathcal {L}}^{infty }(Omega )}lambda (Omega )ge 1$$ is sharp: for $$min {2,3,...}$$ , and any $$epsilon >0$$ we construct an open, bounded and connected set $$Omega _{epsilon }subset mathbb {R}^m$$ such that $$Vert v_{Omega _{epsilon }}Vert _{{mathcal {L}}^{infty }(Omega _{epsilon })} lambda (Omega _{epsilon })<1+epsilon $$ . An upper bound for $$v_{Omega }$$ is obtained for planar, convex sets in Euclidean space $$mathbb {R}^2$$ , which is sharp in the limit of elongation. For a complete, non-compact, m-dimensional Riemannian manifold M with non-negative Ricci curvature, and without boundary it is shown that $$v_{Omega }$$ is bounded if and only if the bottom of the spectrum of the Dirichlet–Laplace–Beltrami operator acting in $${mathcal {L}}^2(Omega )$$ is bounded away from 0." @default.
- W2578885660 created "2017-01-26" @default.
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- W2578885660 date "2017-04-26" @default.
- W2578885660 modified "2023-10-11" @default.
- W2578885660 title "Spectral Bounds for the Torsion Function" @default.
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- W2578885660 doi "https://doi.org/10.1007/s00020-017-2371-0" @default.
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