Inequalities for Exit times and Eigenvalues of Balls

This first result of this paper is about the Laplace transform of u(X T ) where u is harmonic on some bounded domain Ω, X t is Brownian motion and T is the exit time from Ω. The following results focus on exit times from balls and Faber–Krahn and reverse Faber–Krahn type inequalities for balls. We also study the behaviour of the first Dirichlet eigenvalue for complex balls under complex interpolation. The method of proof heavily relays on the log-concavity of gaussian measures.