The Hopf Lemma for the Schrödinger Operator
Augusto C. Ponce, Nicolas Wilmet · Advanced Nonlinear Studies · 2020
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 L loc 1 ( Ω ) {L_{\mathrm{loc}}^{1}(\Omega)} . More precisely, if u ∈ W 0 1 , 2 ( Ω ) ∩ L 2 ( 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 ot\equiv 0} , then we show that at every point a ∈ ∂ Ω {a\in\partial\Omega} where the class