محفوظ في:
| المؤلفون الرئيسيون: | , |
|---|---|
| التنسيق: | Preprint |
| منشور في: |
2020
|
| الموضوعات: | |
| الوصول للمادة أونلاين: | https://arxiv.org/abs/2001.03341 |
| الوسوم: |
إضافة وسم
لا توجد وسوم, كن أول من يضع وسما على هذه التسجيلة!
|
جدول المحتويات:
- 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.