Laboratoire IMATH

We study a generalized form of the Cheeger inequality by considering the shape functional $F_p,q(\Omega)=\lambda_p^1/p(\Omega)/\lambda_q^1/q(\Omega)$, where the original Cheeger case corresponds to $p=2$ and $q=1$. Here $\lambda_p(\Omega)$ denotes the principal eigenvalue of the Dirichlet $p$-Laplacian. The infimum and the supremum of $F_p,q$ are discussed, together with the existence of optimal domains. Some open problems will be illustrated as well.