Hopf potentials for the Schrödinger operator

Orsina, Luigi; Ponce, Augusto C.

doi:10.2140/apde.2018.11.2015

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Hopf potentials for the Schrödinger operator

Luigi Orsina and Augusto C. Ponce

Vol. 11 (2018), No. 8, 2015–2047

DOI: 10.2140/apde.2018.11.2015

Abstract

We establish the Hopf boundary point lemma for the Schrödinger operator $- Δ + V$ involving potentials $V$ that merely belong to the space $L_{loc}^{1} (Ω)$ . More precisely, we prove that among all nonnegative supersolutions $u$ of $- Δ + V$ which vanish on the boundary $\partial Ω$ and are such that $V u \in L^{1} (Ω)$ , if there exists one supersolution that satisfies $\partial u ∕ \partial n < 0$ almost everywhere on $\partial Ω$ 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 $L^{\infty} (\partial Ω)$ .

Keywords

Hopf lemma, boundary point lemma, Schrödinger operator, weak normal derivative

Mathematical Subject Classification 2010

Primary: 35B05, 35B50

Secondary: 31B15, 31B35

Milestones

Received: 3 June 2017

Accepted: 9 April 2018

Published: 6 June 2018

Authors

Luigi Orsina
Dipartimento di Matematica “Sapienza” Università di Roma Roma Italy

Augusto C. Ponce
Institut de Recherche en Mathématique et Physique Université catholique de Louvain Louvain-la-Neuve Belgium

Vol. 11, No. 8, 2018