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Hopf potentials for the Schrödinger operator

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Ponce, Augusto [UCL] Orsina, Luigi [I-Rome]

We establish the Hopf boundary point lemma for the Schrödinger operator $-\Delta+V$ involving potentials $V$ that merely belong to the space $L^1_loc(\Omega)$. More precisely, we prove that among all nonnegative supersolutions $u$ of $-\Delta+V$ which vanish on the boundary $\partial\Omega$ and are such that $Vu\inL^1(\Omega)$, if there exists one supersolution that 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 $L^\infty(\partial\Omega)$.

Document type Article de périodique (Journal article) – Article de recherche
Access type Accès libre
Publication date 2018
Language Anglais
Journal information "Analysis & PDE" - Vol. 11, no.8, p. 2015-2047
Peer reviewed yes
Publisher Analysis & Partial Differential Equations. Mathematical Sciences Publishers Department of Mathematics. (University of California, USA)
issn 2157-5045
Publication status Publié
Affiliation UCL - SST/IRMP - Institut de recherche en mathématique et physique
Keywords Hopf lemma ; boundary point lemma ; Schrödinger operator ; weak normal derivative
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Bibliographic reference Ponce, Augusto ; Orsina, Luigi. Hopf potentials for the Schrödinger operator. In: Analysis & PDE, Vol. 11, no.8, p. 2015-2047
Permanent URL http://hdl.handle.net/2078.1/200928