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.

中文翻译：

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薛定谔算子的 Hopf 引理

摘要 我们证明了涉及 Schrödinger 算子的狄利克雷问题的解的 Hopf 边界点引理 - Δ + V {-\Delta+V} 具有仅属于 L loc 1 ⁢ ( Ω ) {L_{\mathrm 的非负势 V {loc}}^{1}(\Omega)} 。更准确地说，如果 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})} 满足 - Δ ⁢ u + V ⁢ u = f {-\Delta u+Vu=f} on Ω 对于一些非负数据 f ∈ L ∞ ⁢ ( Ω ) {f\in L^{\infty}(\Omega)} , f ≢ 0 {f\not\equiv 0} ，那么我们证明在每个点 a ∈ ∂ ⁡ Ω { a\in\partial\Omega} 其中经典正态导数∂ ⁡ u ⁢ ( a ) ∂ ⁡ n {\frac{\partial u(a)}{\partial n}} 存在并且满足泊松表示公式，有∂ ⁡ u ⁢ ( a ) ∂ ⁡ n > 0 {\frac{\partial u(a)}{\partial n}> 0} 当且仅当边界值问题 { - 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} 涉及狄拉克测度 ν = δ a {\nu=\delta_{a}} 有一个解。更一般地，我们刻画了 ∂ ⁡ Ω {\partial\Omega} 上的非负有限 Borel 测度 ν，上面的边界值问题在 Hopf 引理失败的集合方面有一个解。\displaystyle\phantom{}\text{on ${\partial% \Omega}$,}\end{aligned}\end{dcases} 涉及狄拉克测度 ν = δ a {\nu=\delta_{a}}一个解法。更一般地，我们刻画了 ∂ ⁡ Ω {\partial\Omega} 上的非负有限 Borel 测度 ν，上面的边界值问题在 Hopf 引理失败的集合方面有一个解。\displaystyle\phantom{}\text{on ${\partial% \Omega}$,}\end{aligned}\end{dcases} 涉及狄拉克测度 ν = δ a {\nu=\delta_{a}}一个解法。更一般地，我们刻画了 ∂ ⁡ Ω {\partial\Omega} 上的非负有限 Borel 测度 ν，上面的边界值问题在 Hopf 引理失败的集合方面有一个解。