Hopf potentials for the Schr\"odinger operator

TL;DR

This paper proves a boundary point lemma for the Schrödinger operator with potentials in L^1_loc, showing that if one supersolution has a negative normal derivative at the boundary, then all do, under certain conditions.

Contribution

It extends the Hopf boundary point lemma to Schrödinger operators with minimal regularity potentials, using solutions of nonhomogeneous Dirichlet problems.

Findings

Established the Hopf boundary point lemma for Schrödinger operators with L^1_loc potentials.

Proved that the boundary derivative property for one supersolution extends to all supersolutions.

Utilized existence results for nonhomogeneous Dirichlet problems with boundary data in L^∞.

Abstract

We establish the Hopf boundary point lemma for the Schr\"odinger operator $-\Delta + V$ involving potentials $V$ that merely belong to the space $L^{1}_{loc}(\Omega)$. More precisely, we prove that among all supersolutions $u$ of $-\Delta + V$ which vanish on the boundary $\partial\Omega$ and are such that $V u \in L^{1}(\Omega)$, if there exists one supersolution which satisfies $\partial u/\partial n < 0$ almost everywhere on $\partial\Omega$ with respect to the outward unit vector $n$, then such a property holds for every nontrivial supersolution in the same class. We rely on the existence of nontrivial solutions of the nonhomogeneous Dirichlet problem with boundary datum in $L^{\infty}(\partial\Omega)$.

Full text

See pages 1-36 of Hopf-final-4.pdf