Absolute continuity of the best Sobolev constant of a bounded domain

TL;DR

This paper proves that the function defining the best Sobolev constant for a bounded domain varies absolutely continuously with respect to the parameter q within a specific range, extending understanding of Sobolev inequalities.

Contribution

It establishes the absolute continuity of the best Sobolev constant function with respect to q, a novel result in the analysis of Sobolev inequalities.

Findings

The function λ_q is absolutely continuous on [1, p*].

The result applies to smooth bounded domains in ℝ^N.

Provides new insights into the stability of Sobolev constants.

Abstract

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