Absolute continuity of the best Sobolev constant of a bounded domain

Abstract

Let $λ_{q}:=\inf{\Vert\nabla u\Vert_{L^{p}(Ω)}^{p}/\Vertu\Vert_{L^{q}(Ω)}^{p}:u\in W_{0}^{1,p}(Ω)\setminus{0}} $, where $\Omega$ is a bounded and smooth domain of ${\mathbb{R}}^N,$ $1<p<N$ and $1\le q\le p^{\star }$ We prove that the function $q\mapsto {\lambda }_q$ is absolutely continuous in the closed interval $[1,p^{\star }].$

References in corpus (1)

• Solutions of the Cheeger problem via torsion functions

Cited by in corpus (3)

• Asymptotics for the best Sobolev constants and their extremal functions
• On a singular minimizing problem
• The Cheeger constant as limit of Sobolev-type constants