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

Abstract

Bounds are obtained for the Lp norm of the torsion function vΩ, i.e. the solution of −Δv=1,v∈H10(Ω), in terms of the Lebesgue measure of Ω and the principal eigenvalue λ1(Ω) of the Dirichlet Laplacian acting in L2(Ω). We show that these bounds are sharp for 1≤p≤2.