# Research

From Newton’s mechanics, many scientists and mathematicians have devoted to studying differential equations to explain several phenomena in the nature. It is natural to ask whether we can solve boundary value problems of given partial differential equations. If we know an existence of solution of the problem, one can also ask some properties of solutions that have.

## Elliptic equations with singular drift terms

For several years, I have devoted to studying unique solvability of linear elliptic equations with singular drift terms in Sobolev spaces. To explain the results, let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$, $n\geq 2$. We consider the following Dirichlet problems of elliptic equations of second-order:

$(D)\qquad -\mathrm{div}(A \nabla u)+\mathrm{div}(u\mathbf{b})=f\quad \text{in } \Omega\qquad u=0\quad \text{on } \partial\Omega$

and

$(D')\qquad -\mathrm{div}(A \nabla v)-\mathbf{b}\cdot \nabla v =g\quad \text{in } \Omega\qquad v=0\quad \text{on } \partial\Omega$

Here $A=(a^{ij}):\mathbb{R}^n\rightarrow \mathbb{R}^{n\times n}$ denotes an $n\times n$ real matrix-valued function which is uniformly elliptic, that is, there exists $0<\delta<1$ such that \begin{equation}\label{eq:uniformly-elliptic} \delta|\xi|^2 \leq a^{ij}(x)\xi_i \xi_j\quad \text{and}\quad |a^{ij}(x)|\leq \delta^{-1}\quad \text{for all } x,\xi \in \mathbb{R}^n, \end{equation} $\mathbf{b}=(b^1,b^2,\dots,b^n) : \Omega\rightarrow \mathbb{R}^n$ is a given vector field. Here we follow the usual summation convention for repeated indices.

When $\mathbf{b}$ is zero or a bounded vector field, $L^p$-theory for these equations is well-known by several authors. One might ask $L^p$-theory for unbounded vector field, $\mathbf{b}\in L^q(\Omega;\mathbb{R}^n)$.

During the master course, I wrote a research article [KK18] with my academic advisor Prof. Hyunseok Kim. At the Republic of Korea Air Force Academy, I wrote two research articles [K20] and [K21].

1. In a joint work with Hyunseok Kim [KK18], when $A=I$ and $\mathbf{b} \in L^{n}(\Omega;\mathbb{R}^n)$, we proved $L^{\alpha,p}$-solvability for the problems $(D)$ on arbitrary bounded Lipschitz domains $\Omega$ in $\mathbb{R}^n$, $n\geq 3$ for appropriate $(\alpha,p)\in \mathscr{A}\cap \mathscr{B}$. Here $L^{\alpha,p}(\Omega)$ denotes the Bessel potential spaces over $\Omega$, which generalizes the classical Sobolev spaces $W^{k,p}(\Omega)$. Here $\mathscr{A}$ is the set of admissible pairs $(\alpha,p)$ such that the Dirichlet problem for the Poisson equation has a unique solution in $L^{\alpha,p}_0(\Omega) ={ u\in L^{\alpha,p}(\Omega) : u=0\text{ on } \partial\Omega}$. Also, $\mathscr{B}$ denotes the set of some pairs $(\alpha,p)$ for which $\mathbf{b} \cdot \nabla u \in L^{\alpha-2,p}(\Omega)$ for all $u\in L_0^{\alpha,p}(\Omega)$. Similar result also proved for the problem $(D’)$. This result extends the classical result of Jerison-Kenig [JK95]. In the same paper, Neumann problems were also considered.
2. When $A$ satisfies \eqref{eq:uniformly-elliptic} and $\mathbf{b} \in L^{n,\infty}(\Omega;\mathbb{R}^n)$, $n\geq 2$, $\mathrm{div} \mathbf{b} \geq 0$ in $\Omega$, I proved that if $q>2$, then there exists $\varepsilon>0$ such that if $g\in W^{-1,q}(\Omega)$, then there exists a unique $v\in W_0^{1,2+\varepsilon}(\Omega)$ of the problem $(D’)$. As an application, I proved that if $f\in \bigcap_{q<2} W^{-1,q}(\Omega)$, there exists a unique $u\in \bigcap_{q<2} W_0^{1,q}(\Omega)$ satisfying the problem $(D)$. The result can be found in [K20]. When $n=2$, this result extends Kim-Tsai [KT20] and Chernobai-Shilkin [CS19].
3. In [K21], I proved that if $2<p<\infty$, $A$ satisfies mean oscillation in small balls and $\mathrm{div} A \in L^{2}(\Omega;\mathbb{R}^2)$, $\Omega$ is a bounded Lipschitz domain in $\mathbb{R}^2$ which has a small Lipschitz constant, and $\mathbf{b} \in L^{2}(\Omega;\mathbb{R}^2)$, then the problem $(D’)$ has a unique solution in $W_0^{1,p}(\Omega)$. A similar result was also proved for the problem $(D)$. Results are new even if $A=I$. The results complement Kim-Kim [KK15] and Kang-Kim [KK17] when $n=2$. Further research should be done to remove an additional assumption on $\mathrm{div} A$.