We prove a local Faber-Krahn inequality for solutions $u$ to the Dirichlet problem for $Delta + V$ on an arbitrary domain $Omega$ in $mathbb{R}^n$. Suppose a solution $u$ assumes a global maximum at some point $x_0 in Omega$ and $u(x_0)>0$. Let $T(x_0)$ be the smallest time at which a Brownian motion, started at $x_0$, has exited the domain $Omega$ with probability $ge 1/2$. For nice (e.g., convex) domains, $T(x_0) asymp d(x_0,partialOmega)^2$ but we make no assumption on the geometry of the domain. Our main result is that there exists a ball $B$ of radius $asymp T(x_0)^{1/2}$ such that $$ | V |_{L^{frac{n}{2}, 1}(Omega cap B)} ge c_n > 0, $$ provided that $n ge 3$. 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.
Download