Let Ω be an open set in Euclidean space with finite Lebesgue measure |Ω|. We obtain some properties of the set function F : Ω → R^+ defined by F(Ω) = T(Ω)λ_1(Ω)/ |Ω| where T(Ω) and λ_1(Ω) are the torsional rigidity and the first eigenvalue of the Dirichlet Laplacian respectively. We improve the classical Pólya bound F (Ω) ≤ 1, and show that F(Ω) ≤ 1 − ν_mT(Ω)|Ω|^(−1− 2/m) where ν_m depends only on m. For any m = 2,3,... and ε ∈ (0,1) we construct an open set Ω_ε ⊂R^m such that F(Ω_ε)≥1−ε.
On Pólya’s Inequality for Torsional Rigidity and First Dirichlet Eigenvalue / van Den Berg, M.; Ferone, V.; Nitsch, C.; Trombetti, C.. - In: INTEGRAL EQUATIONS AND OPERATOR THEORY. - ISSN 0378-620X. - 86:4(2016), pp. 579-600. [10.1007/s00020-016-2334-x]
On Pólya’s Inequality for Torsional Rigidity and First Dirichlet Eigenvalue
Ferone V.;Nitsch C.;Trombetti C.
2016
Abstract
Let Ω be an open set in Euclidean space with finite Lebesgue measure |Ω|. We obtain some properties of the set function F : Ω → R^+ defined by F(Ω) = T(Ω)λ_1(Ω)/ |Ω| where T(Ω) and λ_1(Ω) are the torsional rigidity and the first eigenvalue of the Dirichlet Laplacian respectively. We improve the classical Pólya bound F (Ω) ≤ 1, and show that F(Ω) ≤ 1 − ν_mT(Ω)|Ω|^(−1− 2/m) where ν_m depends only on m. For any m = 2,3,... and ε ∈ (0,1) we construct an open set Ω_ε ⊂R^m such that F(Ω_ε)≥1−ε.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
