# On principal frequencies and isoperimetric ratios in convex sets

created by brasco on 07 Jun 2018
modified on 26 Jun 2018

[BibTeX]

Preprint

Inserted: 7 jun 2018
Last Updated: 26 jun 2018

Pages: 24
Year: 2018

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.