Hopf potentials for the Schrödinger operator

Abstract

We establish the Hopf boundary point lemma for the Schrödinger operator $-\Delta +V$ involving potentials $V$ that merely belong to the space $L_{loc}^1(\Omega )$. More precisely, we prove that among all supersolutions $u$ of $-\Delta +V$ which vanish on the boundary $\partial \Omega$ and are such that $Vu\in L^1(\Omega )$, 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 $L^{\infty }(\partial \Omega )$.

Arxiv title has been modified to coincide with the paper as published

References in corpus (3)

• Inverse Problem for a Curved Quantum Guide
• Linear diffusion with singular absorption potential and/or unbounded convective flow: the weighted space approach
• Schroedinger operators involving singular potentials and measure data

Cited by in corpus (4)

• On the nonexistence of Green's function and failure of the strong maximum principle
• The Hopf lemma for the Schrödinger operator
• An Agmon-Allegretto-Piepenbrink principle for Schroedinger operators
• A representation formula for the distributional normal derivative