Hölder regularity and Liouville Theorem for the Schrödinger equation with certain critical potentials, and applications to Dirichlet problems
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| Format: | Preprint |
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2024
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| _version_ | 1866909336364646400 |
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| author | Li, Bo Li, Ji Wu, Liangchuan |
| author_facet | Li, Bo Li, Ji Wu, Liangchuan |
| contents | Let $(X,d,μ)$ be a metric measure space satisfying a doubling property with the upper/lower dimension $Q\ge n>1$, and admitting an $L^2$-Poincaré inequality. In this article, we establish the Hölder continuity and a Liouville-type theorem for the (elliptic-type) Schrödinger equation $$\mathbb L u(x,t)=-\partial^2_{t}u(x,t)+\mathcal L u(x,t)+V(x)u(x,t)=0,\quad x\in X,\, t\in\mathbb R, $$ where $\mathcal L$ is a non-negative operator generated by a Dirichlet form on $X$, and the non-negative potential $V$ is a Muckenhoupt weight belonging to the reverse Hölder class ${RH}_q(X)$ for some $q>\max\{Q/2,1\}$. Note that $Q/2$ is critical for the regularity theory of $-Δ+V$ on $\mathbb{R}^Q$ ($Q\ge3$) by Shen's work in 1995, which hints the critical index of $V$ for the regularity results above on $X\times \mathbb R$ may be $(Q+1)/2$. Our results show that this critical index is in fact $\max\{Q/2,1\}$. Our approach primarily relies on the controllable growth of $V$ and the elliptic theory for the operator $\mathbb L$/$-\partial^2_{t}+\mathcal{L}$ on $X\times \mathbb R$, rather than the analogs for $\mathcal L+V$/$\mathcal{L}$ on $X$, under the critical index setting. As applications, we further obtain some characterizations for solutions to the Schrödinger equation $-\partial^2_{t}u+\mathcal L u+Vu=0$ in $X\times \mathbb R_+$ with boundary values in BMO/CMO/Morrey spaces related to $V$, improving previous results to the critical index $q>\max\{Q/2,1\}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_03418 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hölder regularity and Liouville Theorem for the Schrödinger equation with certain critical potentials, and applications to Dirichlet problems Li, Bo Li, Ji Wu, Liangchuan Analysis of PDEs Classical Analysis and ODEs 35J10, 42B35, 43A85 Let $(X,d,μ)$ be a metric measure space satisfying a doubling property with the upper/lower dimension $Q\ge n>1$, and admitting an $L^2$-Poincaré inequality. In this article, we establish the Hölder continuity and a Liouville-type theorem for the (elliptic-type) Schrödinger equation $$\mathbb L u(x,t)=-\partial^2_{t}u(x,t)+\mathcal L u(x,t)+V(x)u(x,t)=0,\quad x\in X,\, t\in\mathbb R, $$ where $\mathcal L$ is a non-negative operator generated by a Dirichlet form on $X$, and the non-negative potential $V$ is a Muckenhoupt weight belonging to the reverse Hölder class ${RH}_q(X)$ for some $q>\max\{Q/2,1\}$. Note that $Q/2$ is critical for the regularity theory of $-Δ+V$ on $\mathbb{R}^Q$ ($Q\ge3$) by Shen's work in 1995, which hints the critical index of $V$ for the regularity results above on $X\times \mathbb R$ may be $(Q+1)/2$. Our results show that this critical index is in fact $\max\{Q/2,1\}$. Our approach primarily relies on the controllable growth of $V$ and the elliptic theory for the operator $\mathbb L$/$-\partial^2_{t}+\mathcal{L}$ on $X\times \mathbb R$, rather than the analogs for $\mathcal L+V$/$\mathcal{L}$ on $X$, under the critical index setting. As applications, we further obtain some characterizations for solutions to the Schrödinger equation $-\partial^2_{t}u+\mathcal L u+Vu=0$ in $X\times \mathbb R_+$ with boundary values in BMO/CMO/Morrey spaces related to $V$, improving previous results to the critical index $q>\max\{Q/2,1\}$. |
| title | Hölder regularity and Liouville Theorem for the Schrödinger equation with certain critical potentials, and applications to Dirichlet problems |
| topic | Analysis of PDEs Classical Analysis and ODEs 35J10, 42B35, 43A85 |
| url | https://arxiv.org/abs/2410.03418 |