On the nonexistence of Green's function and failure of the strong maximum principle
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2018
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| _version_ | 1866912217902874624 |
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| author | Orsina, Luigi Ponce, Augusto C. |
| author_facet | Orsina, Luigi Ponce, Augusto C. |
| contents | Given any Borel function $V : Ω\to [0, +\infty]$ on a smooth bounded domain $Ω\subset \mathbb{R}^{N}$, we establish that the strong maximum principle for the Schrödinger operator $-Δ+ V$ in $Ω$ holds in each Sobolev-connected component of $Ω\setminus Z$, where $Z \subset Ω$ is the set of points which cannot carry a Green's function for $- Δ+ V$. More generally, we show that the equation $- Δu + V u = μ$ has a distributional solution in $W_{0}^{1, 1}(Ω)$ for a nonnegative finite Borel measure $μ$ if and only if $μ(Z) = 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1808_07267 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | On the nonexistence of Green's function and failure of the strong maximum principle Orsina, Luigi Ponce, Augusto C. Analysis of PDEs Functional Analysis 35J10, 35B05, 35B50 (Primary), 31B15, 31B35, 31C15 (Secondary) Given any Borel function $V : Ω\to [0, +\infty]$ on a smooth bounded domain $Ω\subset \mathbb{R}^{N}$, we establish that the strong maximum principle for the Schrödinger operator $-Δ+ V$ in $Ω$ holds in each Sobolev-connected component of $Ω\setminus Z$, where $Z \subset Ω$ is the set of points which cannot carry a Green's function for $- Δ+ V$. More generally, we show that the equation $- Δu + V u = μ$ has a distributional solution in $W_{0}^{1, 1}(Ω)$ for a nonnegative finite Borel measure $μ$ if and only if $μ(Z) = 0$. |
| title | On the nonexistence of Green's function and failure of the strong maximum principle |
| topic | Analysis of PDEs Functional Analysis 35J10, 35B05, 35B50 (Primary), 31B15, 31B35, 31C15 (Secondary) |
| url | https://arxiv.org/abs/1808.07267 |