Calculus of Variations and Geometric Measure Theory

L. Brasco

On principal frequencies and isoperimetric ratios in convex sets

created by brasco on 07 Jun 2018
modified on 11 Mar 2019


Accepted Paper

Inserted: 7 jun 2018
Last Updated: 11 mar 2019

Journal: Ann. Fac. Sci. Toulouse Math.
Pages: 25
Year: 2019


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.

Keywords: p-Laplacian, shape optimization, convex sets, Cheeger constant, Buser's inequality