论文标题
Schrödinger操作员的Hopf引理
The Hopf lemma for the Schrödinger operator
论文作者
论文摘要
我们证明了涉及Schrödingeroperator $-Δ+ V $的Dirichlet问题解决方案的HOPF边界点引理，其非负势$ V $仅属于$ L _ {\ Mathrm {loc}}}}^1（ω）$。更准确地说，如果$ u \ in w_0^{1，2}（ω）\ cap l^2（ω; v \ mathrm {d} x）$满足$ - ΔU + v u = f $ on $ fub on $ fub on $ f $ in $ fon in $ nonnegative act $ f \ in l^\ infty（$ f \ equy）$ f \ quant $ f \ quant $ f \ quant $ f \ quant $ a copt in $ f \ a $ f \ a copt \ quant $ f \ a $ f \ a \partΩ$在存在经典的常规衍生$ \ partial u（a） / \ partial n $中存在并满足Poisson表示公式，一个人具有$ \ partial u（a） / \ partial n> 0 $，并且仅当边界价值问题$ $ $ $ $ $ $ $ $ $ begin \ begin {case} case}} $ nign {allign {alligned} $ n an a a al prinds} $ untign} - \\ v＆=ν&& \ text {on $ \partialΩ$，} \ end {aligned} \ end {cases} $$涉及dirac量度$ν=δ_a$具有解决方案。更一般而言，我们表征了$ \partialΩ$上的非负有有限的borel测量$ν$，上面的边界价值问题在HOPF LEMMA失败的集合方面具有解决方案。
We prove the Hopf boundary point lemma for solutions of the Dirichlet problem involving the Schrödinger operator $- Δ+ V$ with a nonnegative potential $V$ which merely belongs to $L_{\mathrm{loc}}^1(Ω)$. More precisely, if $u \in W_0^{1, 2}(Ω) \cap L^2(Ω; V \mathrm{d}x)$ satisfies $- Δu + V u = f$ on $Ω$ for some nonnegative datum $f \in L^\infty(Ω)$, $f \not\equiv 0$, then we show that at every point $a \in \partialΩ$ where the classical normal derivative $\partial u(a) / \partial n$ exists and satisfies the Poisson representation formula, one has $\partial u(a) / \partial n > 0$ if and only if the boundary value problem $$ \begin{cases} \begin{aligned} - Δv + V v &= 0 && \text{in $Ω$,} \\ v &= ν&& \text{on $\partialΩ$,} \end{aligned} \end{cases} $$ involving the Dirac measure $ν= δ_a$ has a solution. More generally, we characterize the nonnegative finite Borel measures $ν$ on $\partialΩ$ for which the boundary value problem above has a solution in terms of the set where the Hopf lemma fails.
