*Preprint*

**Inserted:** 25 nov 2021

**Last Updated:** 25 nov 2021

**Year:** 2021

**Abstract:**

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

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