On principal frequencies and isoperimetric ratios in convex sets

TL;DR

This paper establishes bounds on the Poincaré-Sobolev constant for convex sets, generalizes Pólya's result, and derives sharp Buser's inequalities for the p-Laplacian across all dimensions and p-values.

Contribution

It extends classical inequalities to convex sets for all p, providing new bounds and revealing a subtle shape optimization phenomenon as the integrability exponent varies.

Findings

Bound on Poincaré-Sobolev constant in convex sets

Sharp Buser's inequality for p-Laplacian in all dimensions

Identification of a shape optimization phenomenon

Abstract

On a convex set, we prove that the Poincar\'e-Sobolev constant for functions vanishing at the boundary can be bounded from above by the ratio between the perimeter and a suitable power of the $N-$dimensional measure. This generalizes an old result by P\'olya. As a consequence, we obtain the sharp {\it Buser's inequality} (or reverse Cheeger inequality) for the $p-$Laplacian on convex sets. This is valid in every dimension and for every $1<p<+\infty$. We also highlight the appearing of a subtle phenomenon in shape optimization, as the integrability exponent varies.