The Hopf Lemma for the Schrödinger Operator
| Autor: | Ponce Augusto C., Wilmet Nicolas |
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| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: | |
| Zdroj: | Advanced Nonlinear Studies, Vol 20, Iss 2, Pp 459-475 (2020) |
| Druh dokumentu: | article |
| ISSN: | 1536-1365 2169-0375 |
| DOI: | 10.1515/ans-2020-2078 |
| Popis: | We prove the Hopf boundary point lemma for solutions of the Dirichlet problem involving the Schrödinger operator -Δ+V{-\Delta+V} with a nonnegative potential V which merely belongs to Lloc1(Ω){L_{\mathrm{loc}}^{1}(\Omega)}. More precisely, if u∈W01,2(Ω)∩L2(Ω;Vdx){u\in W_{0}^{1,2}(\Omega)\cap L^{2}(\Omega;V\mathop{}\!\mathrm{d}{x})} satisfies -Δu+Vu=f{-\Delta u+Vu=f} on Ω for some nonnegative datum f∈L∞(Ω){f\in L^{\infty}(\Omega)}, f≢0{f\not\equiv 0}, then we show that at every point a∈∂Ω{a\in\partial\Omega} where the classical normal derivative ∂u(a)∂n{\frac{\partial u(a)}{\partial n}} exists and satisfies the Poisson representation formula, one has ∂u(a)∂n>0{\frac{\partial u(a)}{\partial n}>0} if and only if the boundary value problem |
| Databáze: | Directory of Open Access Journals |
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