The Hopf lemma for the Schr\"odinger operator

Abstract

We prove the Hopf boundary point lemma for solutions of the Dirichlet problem involving the Schr\"odinger operator $- \Delta + V$ with a nonnegative potential $V$ which merely belongs to $L_{\mathrm{loc}}^1(\Omega)$. More precisely, if $u \in W_0^{1, 2}(\Omega) \cap L^2(\Omega; V \mathrm{d}x)$ satisfies $- \Delta u + V u = f$ on $\Omega$ for some nonnegative datum $f \in L^\infty(\Omega)$, $f \not\equiv 0$, then we show that at every point $a \in \partial\Omega$ where the classical normal derivative $\partial u(a) / \partial n$ exists and satisfies the Poisson representation formula, one has $\partial u(a) / \partial n > 0$ if and only if the boundary value problem $$ \begin{cases} \begin{aligned} - \Delta v + V v &= 0 && \text{in $\Omega$,} \\ v &= \nu && \text{on $\partial\Omega$,} \end{aligned} \end{cases} $$ involving the Dirac measure $\nu = \delta_a$ has a solution. More generally, we characterize the nonnegative finite Borel measures $\nu$ on $\partial\Omega$ for which the boundary value problem above has a solution in terms of the set where the Hopf lemma fails.