The Hopf lemma for the Schrödinger operator
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2020
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| _version_ | 1866913676800294912 |
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| author | Ponce, Augusto C. Wilmet, Nicolas |
| author_facet | Ponce, Augusto C. Wilmet, Nicolas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2001_03341 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | The Hopf lemma for the Schrödinger operator Ponce, Augusto C. Wilmet, Nicolas Analysis of PDEs Primary: 35J10, 35D05, 35G15, Secondary: 35C15 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. |
| title | The Hopf lemma for the Schrödinger operator |
| topic | Analysis of PDEs Primary: 35J10, 35D05, 35G15, Secondary: 35C15 |
| url | https://arxiv.org/abs/2001.03341 |