The Hopf Lemma for the Schrödinger Operator

Abstract

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\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 { - Δ ⁢ v + V ⁢ v = 0 in  Ω , v = ν on  ∂ ⁡ Ω , \begin{dcases}\begin{aligned} \displaystyle-\Delta v+Vv&\displaystyle=0&&% \displaystyle\phantom{}\text{in ${\Omega}$,}\\ \displaystyle v&\displaystyle=\nu&&\displaystyle\phantom{}\text{on ${\partial% \Omega}$,}\end{aligned}\end{dcases} involving the Dirac measure ν = δ a {\nu=\delta_{a}} has a solution. More generally, we characterize the nonnegative finite Borel measures ν on ∂ ⁡ Ω {\partial\Omega} for which the boundary value problem above has a solution in terms of the set where the Hopf lemma fails.