Hopf potentials for Schroedinger operators

Luigi Orsina, Augusto C. Ponce

Published 2017-02-15Version 1

We establish the Hopf boundary point lemma for the Schr\"odinger operator $-\Delta + V$ involving singular potentials $V \in L^{1}_{\mathrm{loc}}(\Omega)$: we have that either the Hopf lemma holds for every nonnegative supersolution which vanishes on $\partial\Omega$ or it fails for every such a supersolution, almost everywhere with respect to the surface measure on $\partial\Omega$. The proof relies on the existence of nontrivial solutions of the nonhomogeneous Dirichlet problem with boundary datum in $L^{\infty}(\partial\Omega)$.