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- W2962939398 abstract "Let $$Omega $$ be an open set in Euclidean space with finite Lebesgue measure $$vert Omega vert $$ . We obtain some properties of the set function $$F:Omega mapsto {mathbb {R}}^+$$ defined by $$begin{aligned} F(Omega )=frac{T(Omega )lambda _1(Omega )}{vert Omega vert } , end{aligned}$$ where $$T(Omega )$$ and $$lambda _1(Omega )$$ are the torsional rigidity and the first eigenvalue of the Dirichlet Laplacian respectively. We improve the classical Pólya bound $$F(Omega )le 1,$$ and show that $$begin{aligned} F(Omega )le 1- nu _m T(Omega )|Omega |^{-1-frac{2}{m}}, end{aligned}$$ where $$nu _m$$ depends only on m. For any $$m=2,3,ldots $$ and $$epsilon in (0,1)$$ we construct an open set $$Omega _{epsilon }subset {mathbb {R}}^m$$ such that $$F(Omega _{epsilon })ge 1-epsilon $$ ." @default.
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- W2962939398 date "2016-11-09" @default.
- W2962939398 modified "2023-10-11" @default.
- W2962939398 title "On Pólya’s Inequality for Torsional Rigidity and First Dirichlet Eigenvalue" @default.
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- W2962939398 doi "https://doi.org/10.1007/s00020-016-2334-x" @default.
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