A Local Faber-Krahn inequality and Applications to Schr\"odinger's Equation

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, $T(x_0) \asymp d(x_0,\partial\Omega)^2$ 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.