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| FIELD | Math:Analysis |
|---|---|
| DATE | December 23 (Mon), 2019 |
| TIME | 14:00-15:00 |
| PLACE | 8101 |
| SPEAKER | 권현우 |
| HOST | Jeong, In-Jee |
| INSTITUTE | 공군사관학교 |
| TITLE | Elliptic equations with singular drifts on Lipschitz domains |
| ABSTRACT | We consider the Dirichlet and Neumann problems for second-order elliptic equations $$ -\triangle u + \mathrm{div} (u \mathbf{b})=f,\quad -\triangle v -\mathbf{b}\cdot \nabla v =g $$ in a bounded Lipschitz domain $\Omega$ in $\mathbb{R}^n$ ($n \geq 3$) with the first-order term given by a singular vector field $\mathbf{b}$. Under the assumption that $\mathbf{b}\in L^n(\Omega)^n$, we first establish existence and uniqueness of solutions in $L^p_{\alpha}(\Omega)$ for the Dirichlet and Neumann problems. Here $L_\alpha^p(\Omega)$ denotes the Sobolev space (or Bessel potential space) with the pair $(\alpha,p)$ satisfying certain conditions. These results extend the classical works of Jerison-Kenig (1995) and Fabes-Mendez-Mitrea (1998) for the Poisson equation. We also prove existence and uniqueness of solutions of the Dirichlet problem with boundary data in $L^2(\partial\Omega)$. This talk is based on joint work with Hyunseok Kim. |
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