Università del Salento

Variations on a theme by Cheeger

ABSTRACT
We study a general version of the Cheeger inequality by considering the shape functional $F_{p,q}(\Omega)=\lambda_p^{1/p}(\Omega)/\lambda_q^{1/q}(\Omega)$, where $\lambda_p(\Omega)$ denotes the principal eigenvalue of the Dirichlet $p$-Laplacian. The infimum and the supremum of $F_{p,q}$ 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 $F_{p,q}$ is deeply discussed.

Joint work with:
Luca Briani - Department of Mathematics, Technical University of Munich (TUM), Germany;
Francesca Prinari - Department of Agriculture, Food and Environment, University of Pisa, Italy.

Ref. Prof. Antonio Leaci

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