The Hopf Lemma for the Schrödinger Operator

Abstract

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;V⁢d⁢x){u\in W_{0}^{1,2}(\Omega)\cap L^{2}(\Omega;V\mathop{}\!\mathrm{d}{x})} satisfies -Δ⁢u+V⁢u=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

Keywords

• schrödinger operator
• hopf lemma
• boundary value problem
• singular potential
• measure datum
• 35j10
• 35d05
• 35g15
• 35c15