Drawn by @cute_bae_am

## Contact

Department of Mathematics,
Republic of Korea Air Force Academy,
Postbox 335-2, 635, Danjae-ro,
Sangdang-gu, Cheongju-si Chungcheongbuk-do,
Republic of Korea

Email. willkwon[at]sogang[dot]ac[dot]kr
Phone. +82-43-290-6468

I’m Hyunwoo Kwon(a.k.a. Will Kwon). I am a full-time lecturer in the Department of Mathematics at Republic of Korea Air Force Academy. Previously, I did my master in mathematics at Sogang University. My advisor was Prof. Hyunseok Kim.

Here is my personal mathematics blog site.

Here is my CV.

The drawing was given by one of my students at Republic of Korea Air Force Academy @cute_bae_am.

## Research Interests

• Partial Differential Equations
• Harmonic Analysis (real-variable methods)

## Publications / Preprints

1. Elliptic equations in divergence form with drifts in $L^2$, to appear in Proc. Amer. Math. Soc. doi, arXiv:2104.01300 [math.AP]

Abstract. We consider the Dirichlet problem for second-order linear elliptic equations in divergence form

$-\mathrm{div} (A\nabla u) + \mathbf{b}\cdot \nabla u +\lambda u = f+\mathrm{div } \mathbf{F}\quad \text{in } \Omega\quad \text{and}\quad u=0\quad \text{on } \partial\Omega$,

in bounded Lipschitz domain $\Omega$ in $\mathbb{R}^2$, where $A:\mathbb{R}^2\rightarrow \mathbb{R}^{2^2}$, $\mathbf{b}: \Omega\rightarrow \mathbb{R}^2$, and $\lambda \geq 0$ are given. If $2<p<\infty$ and $A$ has a small mean oscillation in small balls, $\Omega$ has small Lipschitz constant, and $\mathrm{div } A, \mathbf{b}\in L^2(\Omega;\mathbb{R}^2)$, then we prove existence and uniqueness of weak solutions in $W_0^{1,p}(\Omega)$ of the problem. Similar result also holds for the dual problem.

2. Existence and uniqueness of weak solution in $W^{1,2+\varepsilon}$ for elliptic equations with drifts in weak-$L^{n}$ spaces, J. Math. Anal. Appl. 500(1) (2021) 125165, [Journal], arXiv:2011.07524 [math.AP]

Abstract. We consider the following Dirichlet problems for elliptic equations with singular drift $\mathbf{b}$:

$\text{(a)}\, −\mathrm{div}(A\nabla u)+\mathrm{div}(u\mathbf{b})=f,\qquad\text{(b)}\, −\mathrm{div}(A^T\nabla v)−\mathbf{b}\cdot\nabla v=g\quad \text{in } \Omega,$

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 $W^{1,2-}_0(\Omega)$ for the problem (a) for every $f\in W^{-1,2-}(\Omega)$.

3. Dirichlet and Neumann problems for elliptic equations with singular drifts on Lipschitz domain, with H. Kim, arXiv:1811.12619 [math.AP]

Abstract. We consider the Dirichlet and Neumann problems for second-order linear elliptic equations:

$−\triangle u+\mathrm{div}(u\mathbf{b})=f\quad \text{and}\quad −\triangle v−\mathbf{b}\cdot\nabla v=g$

in a bounded Lipschitz domain $\Omega$ in $\mathbb{R}^n$ ($n\geq 3$), where $\mathbf{b}:\Omega\rightarrow \mathbb{R}^n$ is a given vector field. 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 $L^p_{\alpha}(\Omega)$ 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 $L^2(\partial\Omega)$. Our results for the Dirichlet problems hold even for the case $n=2$.