On the { L}^{ p} norm of the torsion function

M Van Den Berg,T Kappeler

doi:10.1007/s11587-018-0412-x

Abstract

Bounds are obtained for the L^p norm of the torsion function v_{varOmega }, i.e. the solution of -varDelta v=1,, vin H_0^1(varOmega ), in terms of the Lebesgue measure of varOmega and the principal eigenvalue lambda _1(varOmega ) of the Dirichlet Laplacian acting in L^2(varOmega ). We show that these bounds are sharp for 1le ple 2.

Highlights

Let Ω be a non-empty open set in Euclidean space Rm with boundary ∂Ω
A classical inequality, e.g. [14], asserts that the function F1 defined on the open sets in Rm with finite Lebesgue measure
(ii) To prove (10) we observe that since Ω has finite Lebesgue measure the spectrum of the Dirichlet Laplacian acting in L2(Ω) is discrete, and consists of an increasing sequence of eigenvalues

Summary

Introduction

Let Ω be a non-empty open set in Euclidean space Rm with boundary ∂Ω It is well-known [2,3] that if the bottom of the Dirichlet Laplacian defined by. [14], asserts that the function F1 defined on the open sets in Rm with finite Lebesgue measure. (ii) For Ω open in Rm with λ1(Ω) > 0, F∞(Ω) = vΩ L∞(Ω)λ1(Ω) It follows from the Faber–Krahn inequality that if |Ω| < ∞ λ1(Ω) > 0. Proof of Theorem 1 (i) To prove (9) for 1 ≤ p ≤ q < ∞ we use Hölder’s inequality to obtain that vΩp ≤. (ii) To prove (10) we observe that since Ω has finite Lebesgue measure the spectrum of the Dirichlet Laplacian acting in L2(Ω) is discrete, and consists of an increasing sequence of eigenvalues.

Proof of Theorem 3

A two-parameter family of inequalities