Two isoperimetric inequalities for the Sobolev constant

TL;DR

This paper establishes two new isoperimetric inequalities related to the Sobolev embedding constant and its extremal functions, extending classical results and recent inequalities in complex analysis and spectral theory.

Contribution

It introduces two novel isoperimetric inequalities for the Sobolev constant and extremal functions, generalizing classical and recent inequalities in analysis.

Findings

Proves a variation of the Schwarz Lemma for Sobolev constants.

Generalizes an isoperimetric inequality for the Laplacian's first eigenfunction.

Links Sobolev inequalities with classical geometric inequalities.

Abstract

In this note we prove two isoperimetric inequalities for the sharp constant in the Sobolev embedding and its associated extremal function. The first such inequality is a variation on the classical Schwarz Lemma from complex analysis, similar to recent inequalities of Burckel, Marshall, Minda, Poggi-Corradini, and Ransford, while the second generalises an isoperimetric inequality for the first eigenfunction of the Laplacian due to Payne and Rayner.