Principal eigenvalue estimates via the supremum of torsion

Abstract

We show that the reciprocal of the principal eigenvalue of some operators is comparable to the supremum of the solution to associated generalized torsion problems or the expected exit time for stochastic processes. As a result, we extend estimates, known for the Laplacian on simply connected two-dimensional domains, to general n-dimensional domains, to symmetric stable processes and to the p-Laplacian. Our proofs rely on probabilistic estimates and interpretations of the eigenvalues and the torsion functions.