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 \(m\in \{2,3,...\}\), 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.
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van den Berg, M. Spectral Bounds for the Torsion Function. Integr. Equ. Oper. Theory 88, 387–400 (2017). https://doi.org/10.1007/s00020-017-2371-0
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DOI: https://doi.org/10.1007/s00020-017-2371-0