On the rate of change of the best constant in the Sobolev inequality

TL;DR

This paper investigates how the optimal constant in the Sobolev inequality varies as the domain expands, introducing a novel reversed Hölder inequality that could have broader applications.

Contribution

It provides the first estimate of the rate of change of the Sobolev inequality's best constant with respect to domain expansion and proves a new reversed Hölder inequality.

Findings

Derived an estimate for the rate of change of the Sobolev constant

Proved a new reversed Hölder inequality

Potential applications in analysis of domain-dependent inequalities

Abstract

We estimate the rate of change of the best constant in the Sobolev inequality of a Euclidean domain which moves outward. Along the way we prove an inequality which reverses the usual Holder inequality, which may be of independent interest.