A local Faber–Krahn inequality and applications to Schrödinger equations

Abstract

ABSTRACTWe prove a local Faber–Krahn inequality for solutions u to the Dirichlet problem for Δ+V on an arbitrary domain Ω in ℝn. Suppose a solution u assumes a global maximum at some point x0∈Ω and u(x0)>0. Let T(x0) be the smallest time at which a Brownian motion, started at x0, has exited the domain Ω with probability ≥1∕2. For nice (e.g., convex) domains, but we make no assumption on the geometry of the domain. Our main result is that there exists a ball B of radius such thatprovided that n≥3. In the case n = 2, the above estimate fails and we obtain a substitute result. The Laplacian may be replaced by a uniformly elliptic operator in divergence form. This result both unifies and strenghtens a series of earlier results.