Calculus of Variations and Geometric Measure Theory

L. Briani - G. Buttazzo - F. Prinari

On a class of Cheeger inequalities

created by buttazzo on 25 Nov 2021
modified by prinari on 31 Jan 2023


Published Paper

Inserted: 25 nov 2021
Last Updated: 31 jan 2023

Journal: Annali di Matematica Pura ed Applicata
Year: 2021


We study a general version of the Cheeger inequality by considering the shape functional ${\mathcal F}_{p,q}(\Omega)=\lambda_p^{1/p}(\Omega)/\lambda_q(\Omega)^{1/q}$. The infimum and the supremum of ${\mathcal F}_{p,q}$ are studied in the class of {\it all} domains $\Omega$ of ${\mathbb R}^d$ and in the subclass of {\it convex} domains. In the latter case the issue concerning the existence of an optimal domain for ${\mathcal F}_{p,q}$ is discussed.

Keywords: p-Laplacian, shape optimization, Cheeger constant, principal eigenvalue