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Seminar
FIELD Math:Analysis
DATE January 15 (Fri), 2021
TIME 14:00-16:00
PLACE Online
SPEAKER Kown, Hyunwoo
HOST Jeong, In-Jee
INSTITUTE 공군사관학교/서강대학교
TITLE $W^{1,2+epsilon}$-results for elliptic equations with drifts in weak $L^n$-spaces
ABSTRACT We consider the following Dirichlet problems for elliptic equations with singular drift $\mathbf{b}$:

\[ \text{(a) } -\operatorname{div}(A \nabla u)+\operatorname{div}(u\mathbf{b})=f,\quad \text{(b) } -\operatorname{div}(A^T \nabla v)-\mathbf{b} \cdot \nabla v =g \quad \text{in } \Omega, \]

where $\Omega$ is a bounded Lipschitz domain in $\mathbb{R}^n$, $n\geq 2$. Assuming that $\mathbf{b}\in L^{n,\infty}(\Omega)^n$ has non-negative weak divergence in $\Omega$, we establish existence and uniqueness of weak solution in $W^{1,2+\varepsilon}_0(\Omega)$ of the problem (b) when $A$ is bounded and uniformly elliptic. As an application, we prove unique solvability of weak solution $u$ in $\bigcap_{q<2} W^{1,q}_0(\Omega)$ for the problem (a) for every $f\in \bigcap_{q<2} W^{-1,q}(\Omega)$.
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