Calculus of Variations and Geometric Measure Theory
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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


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