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Seminar
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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