# Elliptic equations in divergence form with drifts in $L^{2}$

Hyunwoo Kwon  (28187) Department of Mathematics, Republic of Korea Air Force
Academy, Postbox 335-2, 635, Danjae-ro Sangdang-gu, Cheongju-si
Chungcheongbuk-do, Republic of Korea  willkwon<EMAIL_ADDRESS>
# Elliptic equations in divergence form with drifts in $L^{2}$

Hyunwoo Kwon  (28187) Department of Mathematics, Republic of Korea Air Force
Academy, Postbox 335-2, 635, Danjae-ro Sangdang-gu, Cheongju-si
Chungcheongbuk-do, Republic of Korea  willkwon<EMAIL_ADDRESS>
###### Abstract.

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

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

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 $\operatorname{div}A,\,\mathbf{b}\in
L^{2}(\Omega;\mathbb{R}^{2})$, then we prove existence and uniqueness of weak
solutions in ${W}^{1,p}_{0}(\Omega)$ of the problem. Similar result also holds
for the dual problem.

###### Key words and phrases:

elliptic equations, singular drift terms

###### Key words and phrases:

elliptic equations, singular drift terms

###### 2010 Mathematics Subject Classification:

35J15, 35J25

###### 2010 Mathematics Subject Classification:

35J15, 35J25

##  1\. Introduction

This paper is devoted to complementing known results on $W^{1,p}$-estimates
for second-order linear elliptic equations with singular drifts terms. Let
$\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^{n}$, $n\geq 2$. For a
fixed constant $\lambda\geq 0$ and a given vector field
$\mathbf{b}=(b^{1},b^{2},\dots,b^{n}):\Omega\rightarrow\mathbb{R}^{n}$, we
consider the following Dirichlet problems of linear elliptic equations of
second-order:

$\left\\{\begin{aligned} -\operatorname{div}(A\nabla u)+\mathbf{b}\cdot\nabla
u+\lambda u&=f+\operatorname{div}\mathbf{F}&&\quad\text{in }\Omega,\\\
u&=0&&\quad\text{on }\partial\Omega.\end{aligned}\right.$ ($D$)

and

$\left\\{\begin{aligned} -\operatorname{div}(A^{T}\nabla
v)-\operatorname{div}(v\mathbf{b})+\lambda
v&=g+\operatorname{div}\mathbf{G}&&\quad\text{in }\Omega,\\\
v&=0&&\quad\text{on }\partial\Omega,\end{aligned}\right.$ ($D^{\prime}$)

Here $A=(a^{ij}):\mathbb{R}^{n}\rightarrow\mathbb{R}^{n\times n}$ denotes an
$n\times n$ real matrix-valued measurable function which is uniformly
elliptic, that is, there exists $0<\delta<1$ such that

$\delta|\xi|^{2}\leq\sum_{i,j=1}^{n}a^{ij}(x)\xi_{i}\xi_{j}\quad\text{and}\quad\max_{1\leq
i,j\leq n}|a^{ij}(x)|\leq\delta^{-1}\quad\text{for all
}x,\xi\in\mathbb{R}^{n}.$ (1.1)

$W^{1,p}$-estimates for the problems ($D$) and ($D^{\prime}$) were established
by several authors under various assumptions on the leading coefficients
$a^{ij}$ and the domains $\Omega$ when $\mathbf{b}=\mathbf{0}$ or more
generally $\mathbf{b}\in L^{\infty}(\Omega;\mathbb{R}^{n})$; see [16, 2, 4, 6,
9, 10, 8, 22] and references therein. Also, see the recent survey paper of
Dong [7]. In particular, Dong-Kim [10] proved $W^{1,p}$-estimates for the
problems ($D$) and ($D^{\prime}$) when the leading coefficients satisfy small
mean oscillations in small balls and $\mathbf{b}\in
L^{\infty}(\Omega;\mathbb{R}^{n})$ on a bounded Lipschitz domain with small
Lipschitz constant, see Assumptions 2.2 and 2.3 for precise statements. One
may ask whether we can obtain $W^{1,p}$-estimates for the problems ($D$) and
($D^{\prime}$) with unbounded drifts.

Suppose that $\mathbf{b}\in L^{q}(\Omega;\mathbb{R}^{n})$, where $n\leq
q<\infty$ if $n\geq 3$ and $2<q<\infty$ if $n=2$. In Ladyzhenskaya-Ural’tseva
[26, Chapter 3], they considered the following Dirichlet problem

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

where $A$ satisfies (1.1), $\mathbf{b},\mathbf{c}\in
L^{q}(\Omega;\mathbb{R}^{n})$, $d\in L^{q/2}(\Omega)$, and $\Omega$ is a
bounded Lipschitz domain in $\mathbb{R}^{n}$. It was shown that under some
restricted condition on the zeroth order term $d$, for every $f\in
L^{2\hat{n}/(\hat{n}+2)}(\Omega)$ and $\mathbf{F}\in
L^{2}(\Omega;\mathbb{R}^{n})$, there exists a unique weak solution
$u\in{W}^{1,2}_{0}(\Omega)$ of the problem. Here $\hat{n}=n$ if $n\geq 3$ and
$\hat{n}=2+\varepsilon$ if $n=2$. Stampacchia [28] also considered a similar
problem, but the result is similar to that of Ladyzhenskaya-Ural’tseva.

To the best knowledge of the author, Trudinger [29] first proved that given
$\lambda\geq 0$, $f\in L^{2}(\Omega)$, and $\mathbf{F}\in
L^{2}(\Omega;\mathbb{R}^{n})$, there exists a unique weak solution $u$ in
${W}^{1,2}_{0}(\Omega)$ for the problem ($D$). The key tools to prove the
theorem are the weak maximum principle and the Fredholm alternative theorem.
Later, Droniou [11] gave another proof by showing $W^{1,2}$-estimates for the
problem ($D^{\prime}$) and duality method. This result was extended by Kim-Kim
[19] who proved $W^{1,p}$-estimates for the problem ($D$) with $\lambda=0$
when $A$ is the identity matrix, $q^{\prime}<p<\infty$, and $\Omega$ is a
bounded $C^{1}$-domain. Later, Kang-Kim [17] proved $W^{1,p}$-estimates for
the problem ($D$) when $q^{\prime}<p<\infty$, $A$ has small mean oscillation,
and $\Omega$ is a bounded Lipschitz domain with small Lipschitz constant.
Similar results also hold for the problem ($D^{\prime}$). We also mention that
there are some recent results on the problems ($D$) and ($D^{\prime}$) when
the drift $\mathbf{b}$ is in weak $L^{n}$-space; see Moscariello [27], Kim-
Tsai [21], and the recent result of the author [25].

The purpose of this paper is to complement $W^{1,p}$-results on elliptic
equations with the drift $\mathbf{b}\in L^{2}(\Omega;\mathbb{R}^{2})$, which
were not considered in [19, 17, 18]. We remark that our result is new even if
$A$ is the identity matrix. The motivation for writing this paper is the
recent paper due to Krylov [24] who proved $W^{2,p}$-result for second-order
elliptic equations of non-divergence form with the drift $\mathbf{b}$ in
$L^{n}(\Omega;\mathbb{R}^{n})$, $n\geq 2$. More precisely, if $1<p<n$,
$\Omega$ is a bounded $C^{1,1}$-domain, $A$ has small mean oscillation (see
Assumption 2.2), $\mathbf{b}\in L^{n}(\Omega;\mathbb{R}^{n})$, and
$\lambda\geq 0$, then there exists a unique $u\in{W}^{1,p}_{0}(\Omega)\cap
W^{2,p}(\Omega)$ satisfying

$\sum_{i,j=1}^{n}a^{ij}D_{ij}u+\mathbf{b}\cdot\nabla u-\lambda
u=f\quad\text{in }\Omega.$

Related to our paper, this is the first result on solvability of elliptic
equations with $\mathbf{b}\in L^{2}(\Omega;\mathbb{R}^{2})$. Motivated by this
result, one may consider $W^{1,p}$-results for the problems ($D$) and
($D^{\prime}$) when $\mathbf{b}\in L^{2}(\Omega;\mathbb{R}^{2})$.

In this paper, it will be shown in Theorem 2.5 that if $2<p<\infty$, $A$ has a
small mean oscillation in small balls, $\operatorname{div}A,\mathbf{b}\in
L^{2}(\Omega;\mathbb{R}^{2})$, and $\Omega$ has small Lipschitz constant, then
for every $\lambda\geq 0$, $f\in L^{p}(\Omega)$, and $\mathbf{F}\in
L^{p}(\Omega;\mathbb{R}^{2})$, there exists a unique weak solution
$u\in{W}^{1,p}_{0}(\Omega)$ of the problem ($D$). By duality, we have a
similar result for the problem ($D^{\prime}$), see Section 2 for the precise
statements and the definition of $\operatorname{div}A\in
L^{2}(\Omega;\mathbb{R}^{2})$.

Our method to prove Theorem 2.5 is to use a functional analytic argument as in
[17, 20, 18]. A key tool is the following $\varepsilon$-inequality
(Proposition 3.1) inspired by Gerhardt [15]: suppose that $\Omega$ is a
bounded Lipschitz domain in $\mathbb{R}^{2}$ and $2<p<\infty$. Then for each
$\varepsilon>0$, there exists a constant
$C_{\varepsilon}=C_{\varepsilon}(\varepsilon,p,\mathbf{b},\Omega)>0$ such that

$\|{\mathbf{b}\cdot\nabla u}\|_{W^{-1,p}(\Omega)}\leq\varepsilon\|{\nabla
u}\|_{L^{p}(\Omega)}+C_{\varepsilon}\|{u}\|_{L^{p}(\Omega)}$ (1.2)

for all $u\in W^{1,p}(\Omega)$. Using this estimate, we prove that if
$2<p<\infty$, $A$ has small mean oscillations on small balls and $\Omega$ has
small Lipschitz constant, then for sufficiently large $\lambda_{1}$, the
following holds for $\lambda\geq\lambda_{1}$: if $f\in L^{p}(\Omega)$, and
$\mathbf{F}\in L^{p}(\Omega;\mathbb{R}^{2})$, then there exists a unique weak
solution $u\in{W}^{1,p}_{0}(\Omega)$ of the problem ($D$). Similar results
also hold for the problem ($D^{\prime}$). This result induces an operator
$\mathcal{L}_{p}+\lambda_{1}\mathcal{I}_{p}:W^{-1,p}(\Omega)\rightarrow{W}^{1,p}_{0}(\Omega)$
whose inverse can be regarded as a compact operator on $L^{p}(\Omega)$. Hence
by the Fredholm alternative theorem (see [3, Theorem 6.6] e.g.), it suffices
to prove the uniqueness of weak solutions in ${W}^{1,p}_{0}(\Omega)$ for the
problem ($D$), see Section 5 for the definition of
$\mathcal{L}_{p}+\lambda_{1}\mathcal{I}_{p}$ and the reduction. To show the
uniqueness of weak solutions in ${W}^{1,p}_{0}(\Omega)$ of the problem ($D$),
we use an Alexsandrov type maximum principle, which was recently proved by
Krylov [23, Corollary 3.1], see Theorem 4.2. To use this theorem in our
setting, we assume in addition that $\operatorname{div}A\in
L^{2}(\Omega;\mathbb{R}^{2})$ to convert an elliptic equation in divergence
form into an equation in non-divergence from. It seems to be open whether we
can remove the additional assumption $\operatorname{div}A\in
L^{2}(\Omega;\mathbb{R}^{2})$.

The organization of this paper is as follows. We introduce some notations and
state the main theorem in the next section. In Section 3, we prove
$W^{1,p}$-results for the problems ($D$) and ($D^{\prime}$) for sufficiently
large $\lambda$. Next, we prove the uniqueness of weak solutions of the
problem ($D$) in Section 4. Proof of the main theorem is presented in Section
5. For the reader’s convenience, we give all necessary details that can be
found in [17, 20, 18].

### Acknowledgements

The author would like to thank Prof. Doyoon Kim for introducing the recent
result of Krylov [23] to the author and for a valuable discussion. Also, the
author would like to thank the advisor Prof. Hyunseok Kim for comments on
previous drafts. Finally, the author would like to thank the anonymous referee
for the careful reading of the manuscript and for giving useful comments and
suggestions to improve the paper.

##  2\. Notation and Main result

In this section, we introduce several notations used in this article. Also, we
give the main theorem of this paper. We use “$:=$” to denote a definition. As
usual, $\mathbb{R}^{n}$ stands the standard Euclidean space of $n$-points and
$|\cdot|$ is the standard Euclidean norm on $\mathbb{R}^{n}$. For $r>0$ and
$x\in\mathbb{R}^{n}$, we write $B_{r}(x):=\\{y\in\mathbb{R}^{n}:|x-y|<r\\}$.
We also write $B_{r}:=B_{r}(0)$. For $x\in\mathbb{R}^{n}$, we write
$x=(x^{\prime},x_{n})$ where $x^{\prime}\in\mathbb{R}^{n-1}$ and
$B_{r}^{\prime}(x^{\prime}):=\\{y^{\prime}\in\mathbb{R}^{n-1}:|x^{\prime}-y^{\prime}|<r\\}$.
For $1\leq j,k\leq n$, we denote

$D_{j}u=\frac{\partial u}{\partial x_{j}},\quad
D_{kj}u=D_{j}D_{k}u=u_{x_{k}x_{j}}.$

We also use the notation $\nabla u:=(D_{1}u,\dots,D_{n}u)$ for the gradient of
$u$.

We denote by $X^{\prime}$ the dual space of a Banach space $X$. The dual
pairing of $X$ and $X^{\prime}$ is denoted by
$\left<\cdot,\cdot\right>_{X^{\prime},X}$ or simply
$\left<\cdot,\cdot\right>$. For $k\in\mathbb{N}\cup\\{\infty\\}$, let
$C_{c}^{k}(\Omega)$ be the space of all functions in $C^{k}(\mathbb{R}^{n})$
with compact supports in $\Omega$ and let $C^{k}(\overline{\Omega})$ the space
of the restrictions to $\overline{\Omega}$ of all functions in
$C^{k}(\mathbb{R}^{n})$. For $k\in\mathbb{N}$ and $1\leq p<\infty$,
$L^{p}(\Omega)$ and $W^{k,p}(\Omega)$ denote the standard $L^{p}$-space on
$\Omega$ with Lebesgue measure and the Sobolev spaces on $\Omega$,
respectively. We define ${W}^{1,p}_{0}(\Omega)$ the closure of
$C_{c}^{\infty}(\Omega)$ in $W^{1,p}(\Omega)$. For $1<p<\infty$, we write
$p^{\prime}:=p/(p-1)$ the conjugate exponent to $p$. For such $p$, we define
$W^{-1,p}(\Omega):=({W}^{1,p^{\prime}}_{0}(\Omega))^{\prime}$. For $1\leq
p<n$, $p^{*}:=\frac{np}{n-p}$ the Sobolev exponent. For $1\leq p<\infty$, by
$L^{p}(\Omega;\mathbb{R}^{m})$, we denote the set of all
$\mathbb{R}^{m}$-valued measurable functions $\mathbf{u}=(u^{1},\dots,u^{m})$
on $\Omega$ satisfying

$\|{\mathbf{u}}\|_{L^{p}(\Omega)}:=\left(\int_{\Omega}|\mathbf{u}(x)|^{p}\,d{x}\right)^{1/p}<\infty.$

Similarly, $C_{c}^{\infty}(\Omega;\mathbb{R}^{m})$ denotes the set of all
$\mathbb{R}^{m}$-valued measurable functions $\Phi=(\phi^{1},\dots,\phi^{m})$
on $\Omega$ satisfying $\phi^{i}\in C_{c}^{\infty}(\Omega)$ for all $1\leq
i\leq m$.

For open sets $U$ and $V$, we write $V\Subset U$ if $\overline{V}$ is compact
and $\overline{V}\subset U$. For $1\leq p<\infty$ and
$k\in\mathbb{N}\cup\\{0\\}$, we write $W^{k,p}_{\mathrm{loc}}(\Omega)$ if
$u:\Omega\rightarrow\mathbb{R}$ satisfy $u\in W^{k,p}(\Omega^{\prime})$ for
any $\Omega^{\prime}\Subset\Omega$. Similarly, a vector field
$\mathbf{u}:\Omega\rightarrow\mathbb{R}^{m}$ is in
$L^{p}_{\mathrm{loc}}(\Omega;\mathbb{R}^{m})$ if $\mathbf{u}\in
L^{p}(\Omega^{\prime};\mathbb{R}^{m})$ for any $\Omega^{\prime}\Subset\Omega$.
For a measurable function $f$ on $E\subset\mathbb{R}^{n}$, we write

$(f)_{E}=\frac{1}{|E|}\int_{E}f\,d{x}=\fint_{E}f\,d{x},$

where $|E|$ denotes the $n$-dimensional Lebesgue measure of $E$. Finally, by
$C=C(p_{1},\dots,p_{k})$, we denote a generic positive constant depending only
on the parameters $p_{1},\dots,p_{k}$.

We define weak solutions of the problem ($D$) and ($D^{\prime}$) as follows.

######  Definition 2.1.

Let $\lambda\geq 0$, $1<p<\infty$, and
$\mathbf{b}:\Omega\rightarrow\mathbb{R}^{n}$ be a given measurable vector
field.

  1. (1)

Given $f\in L^{p}(\Omega)$ and $\mathbf{F}\in L^{p}(\Omega;\mathbb{R}^{n})$,
we say that $u\in{W}^{1,p}_{0}(\Omega)$ is a _weak solution_ of ($D$) if
$\mathbf{b}\cdot\nabla u\in L^{1}_{\mathrm{loc}}(\Omega)$ and

$\int_{\Omega}A\nabla
u\cdot\nabla\phi\,d{x}+\int_{\Omega}(\mathbf{b}\cdot\nabla
u)\phi\,d{x}+\lambda\int_{\Omega}u\phi\,d{x}=\int_{\Omega}f\phi\,d{x}-\int_{\Omega}\mathbf{F}\cdot\nabla\phi\,d{x}$
(2.1)

for all $\phi\in C_{c}^{\infty}(\Omega)$.

  2. (2)

Given $g\in L^{p^{\prime}}(\Omega)$ and $\mathbf{G}\in
L^{p^{\prime}}(\Omega;\mathbb{R}^{n})$, we say that
$v\in{W}^{1,p^{\prime}}_{0}(\Omega)$ is a _weak solution_ of ($D^{\prime}$) if
$v\mathbf{b}\in L^{1}_{\mathrm{loc}}(\Omega;\mathbb{R}^{n})$ and

$\int_{\Omega}(A^{T}\nabla
v+v\mathbf{b})\cdot\nabla\psi\,d{x}+\lambda\int_{\Omega}v\psi\,d{x}=\int_{\Omega}g\psi\,d{x}-\int_{\Omega}\mathbf{G}\cdot\nabla\psi\,d{x}$
(2.2)

for all $\psi\in C_{c}^{\infty}(\Omega)$.

Let $\mathbf{b}\in L^{n}(\Omega;\mathbb{R}^{n})$. If $n^{\prime}\leq
p<\infty$, then by Hölder’s inequality, we have $\mathbf{b}\cdot\nabla v\in
L^{1}(\Omega)$ for any $v\in W^{1,p}(\Omega)$. If $1<p\leq n$, then it follows
from Hölder’s inequality and Sobolev’s embedding theorem that

$\|{v\mathbf{b}}\|_{L^{1}(\Omega)}\leq\|{\mathbf{b}}\|_{L^{n}(\Omega)}\|{v}\|_{L^{n^{\prime}}(\Omega)}\leq
C\|{\mathbf{b}}\|_{L^{n}(\Omega)}\|{v}\|_{W^{1,p}(\Omega)}$

for all $v\in W^{1,p}(\Omega)$. Hence if we have an unbounded drift
$\mathbf{b}\in L^{n}(\Omega;\mathbb{R}^{n})$, then the range of $p$ is limited
to ensure the well-definedness of weak solutions in ${W}^{1,p}_{0}(\Omega)$
for problems ($D$) and ($D^{\prime}$), respectively.

We impose the following regularity assumption on the leading coefficients:

######  Assumption 2.2 ($\gamma$).

There exists a constant $R_{0}\in(0,1]$ such that

$\max_{1\leq i,j\leq
n}\fint_{B_{r}(x)}\left|a^{ij}(y)-(a^{ij})_{B_{r}(x)}\right|\,d{y}\leq\gamma$

for any $x\in\mathbb{R}^{n}$ and $0<r\leq R_{0}$.

Note that if $A$ is in VMO (see e.g. [22]), then Assumption 2.2 $(\gamma)$ is
satisfied for any $\gamma>0$.

Next, we impose the following regularity assumption on the boundary of the
domain $\Omega$:

######  Assumption 2.3 ($\theta$).

There is a constant $R_{0}\in(0,1]$ such that for any
$x_{0}\in\partial\Omega$, there exists a Lipschitz function
$\sigma:\mathbb{R}^{n-1}\rightarrow\mathbb{R}$ such that

$\Omega\cap B_{R_{0}}(x_{0})=\\{x\in B_{r}(x_{0}):x_{n}>\sigma(x^{\prime})\\}$

and

$\sup_{x^{\prime},y^{\prime}\in
B_{R_{0}}^{\prime}(x_{0}^{\prime}),x^{\prime}\neq
y^{\prime}}\frac{|\sigma(x^{\prime})-\sigma(y^{\prime})|}{|x^{\prime}-y^{\prime}|}\leq\theta$

in some coordinate system.

It is easy to check that every $C^{1}$-domain satisfies Assumption 2.3
$(\theta)$ for any $\theta>0$.

To state our main theorem, we introduce the definition of weak
$L^{2}$-divergence for a bounded measurable matrix-valued function.

######  Definition 2.4.

A bounded measurable matrix-valued function
$A:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2\times 2}$ has _weak
$L^{2}$-divergence in $\Omega$_ if there exists a vector field $\mathbf{c}$ in
$L^{2}(\Omega;\mathbb{R}^{2})$ such that

$\int_{\Omega}\sum_{i,j=1}^{2}a^{ij}D_{i}\phi^{j}dx=-\int_{\Omega}\mathbf{c}\cdot\Phi\,d{x}$

for all $\Phi=(\phi^{1},\phi^{2})\in C_{c}^{\infty}(\Omega;\mathbb{R}^{2})$.
In this case, we write $\mathbf{c}=\operatorname{div}A$ and
$\operatorname{div}A\in L^{2}(\Omega;\mathbb{R}^{2})$.

Now we state the main theorem of this paper.

######  Theorem 2.5.

Let $\Omega$ be a bounded domain in $\mathbb{R}^{2}$, $2<p<\infty$, and
$\lambda\geq 0$. Suppose that $A$ satisfies (1.1), $\operatorname{div}A\in
L^{2}(\Omega;\mathbb{R}^{2})$, and $\mathbf{b}\in
L^{2}(\Omega;\mathbb{R}^{2})$. Then there exist constants
$\gamma=\gamma(\delta,p,\|{\operatorname{div}A}\|_{L^{2}(\Omega)},\|{\mathbf{b}}\|_{L^{2}(\Omega)})$
and $\theta=\theta(\delta,p)>0$ such that under Assumptions 2.2 $(\gamma)$ and
2.3 $(\theta)$, the following hold:

  1. (i)

For every $f\in L^{p}(\Omega)$ and $\mathbf{F}\in
L^{p}(\Omega;\mathbb{R}^{2})$, there exists a unique weak solution
$u\in{W}^{1,p}_{0}(\Omega)$ of ($D$). Moreover we have

$\|{\nabla u}\|_{L^{p}(\Omega)}+\lambda^{1/2}\|{u}\|_{L^{p}(\Omega)}\leq
C\left[\min\left(1,\lambda^{-1}\right)\|{f}\|_{L^{p}(\Omega)}+\|{\mathbf{F}}\|_{L^{p}(\Omega)}\right]$

for some constant $C$ independent of $u$, $f$, $\mathbf{F}$, and $\lambda$.

  2. (ii)

For every $g\in L^{p^{\prime}}(\Omega)$ and $\mathbf{G}\in
L^{p^{\prime}}(\Omega;\mathbb{R}^{2})$, there exists a unique weak solution
$v\in{W}^{1,p^{\prime}}_{0}(\Omega)$ of ($D^{\prime}$). Moreover we have

$\|{\nabla
v}\|_{L^{p^{\prime}}(\Omega)}+\lambda^{1/2}\|{v}\|_{L^{p^{\prime}}(\Omega)}\leq
C\left[\min\left(1,\lambda^{-1}\right)\|{g}\|_{L^{p^{\prime}}(\Omega)}+\|{\mathbf{G}}\|_{L^{p^{\prime}}(\Omega)}\right]$

for some constant $C$ independent of $v$, $g$, $\mathbf{G}$, and $\lambda$.

######  Remark.

(i) Theorem 2.5 complements the result of Kim-Kim [19] when
$a^{ij}=\delta^{ij}$ and $\mathbf{b}\in L^{2}(\Omega;\mathbb{R}^{2})$, where
$\delta^{ij}$ is the Kronecker delta.

(ii) The range $2<p<\infty$ in Theorem 2.5 is optimal. Define

$a^{ij}=\delta^{ij},\quad
u(x)=\ln|\ln|x||,\quad\text{and}\quad\mathbf{b}(x)=-\frac{x}{|x|^{2}\ln|x|}.$

Then $\mathbf{b}\in L^{2}(B_{1/e};\mathbb{R}^{2})$ and
$u\in{W}^{1,2}_{0}(B_{1/e})$ is a nontrivial weak solution satisfying the
problem ($D$) with $\lambda=0$, $f=0$, and $\mathbf{F}=\mathbf{0}$. Hence the
uniqueness of weak solutions in ${W}^{1,2}_{0}(\Omega)$ of the problem ($D$)
fails in general. More related examples can be found in Filonov [13] and
Filonov-Shilkin [14].

(iii) In Kang-Kim [17, Theorem 2.5], they obtained $W^{1,p}$-estimates for the
problems ($D$) and ($D^{\prime}$) when $a^{ij}$ has small mean oscillations in
small balls and $\mathbf{b}\in L^{q}(\Omega;\mathbb{R}^{2})$ with $q>2$. It
seems to be open whether we can remove additional assumption
$\operatorname{div}A\in L^{2}(\Omega;\mathbb{R}^{2})$ when $\mathbf{b}\in
L^{2}(\Omega;\mathbb{R}^{2})$.

##  3\. Solvability of the problems ($D$) and ($D^{\prime}$) for large
$\lambda$

In this section, we obtain $W^{1,p}$-estimates for the problems ($D$) and
($D^{\prime}$) for sufficiently large $\lambda$.

We first show basic estimates for the drift terms and the
$\varepsilon$-inequalities inspired by Gerhardt [15], which play crucial roles
in the proof of the main theorem of this paper.

######  Proposition 3.1.

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^{n}$, $n\geq 2$, and
$n^{\prime}<p<\infty$. Suppose that $\mathbf{b}\in
L^{n}(\Omega;\mathbb{R}^{n})$. Then there exists a constant
$C=C(n,p,\Omega)>0$ such that

$\int_{\Omega}|(v\mathbf{b})\cdot\nabla u|\,d{x}\leq
C\|{\mathbf{b}}\|_{L^{n}(\Omega)}\|{u}\|_{W^{1,p}(\Omega)}\|{v}\|_{W^{1,p^{\prime}}(\Omega)}$
(3.1)

for all $u\in W^{1,p}(\Omega)$ and $v\in W^{1,p^{\prime}}(\Omega)$.

For each $\varepsilon>0$, there exists a constant
$C_{\varepsilon}=C_{\varepsilon}(\varepsilon,n,p,\mathbf{b},\Omega)>0$ such
that

$\mathbf{b}\cdot\nabla u\in
W^{-1,p}(\Omega)\quad\text{and}\quad\|{\mathbf{b}\cdot\nabla
u}\|_{W^{-1,p}(\Omega)}\leq\varepsilon\|{u}\|_{W^{1,p}(\Omega)}+C_{\varepsilon}\|{u}\|_{L^{p}(\Omega)}$
(3.2)

for all $u\in W^{1,p}(\Omega)$. Similarly, for each $\varepsilon>0$, there
exists a constant
$C_{\varepsilon}^{*}=C_{\varepsilon}^{*}(\varepsilon,n,p,\mathbf{b},\Omega)>0$
such that

$\operatorname{div}(v\mathbf{b})\in
W^{-1,p^{\prime}}(\Omega)\quad\text{and}\quad\|{\operatorname{div}(v\mathbf{b})}\|_{W^{-1,p^{\prime}}(\Omega)}\leq\varepsilon\|{v}\|_{W^{1,p^{\prime}}(\Omega)}+C_{\varepsilon}^{*}\|{v}\|_{L^{p^{\prime}}(\Omega)}$
(3.3)

for all $v\in W^{1,p^{\prime}}(\Omega)$.

###### Proof.

By Hölder’s inequality and Sobolev’s embedding theorem, we have

$\displaystyle\int_{\Omega}|(v\mathbf{b})\cdot\nabla u|\,d{x}$
$\displaystyle\leq\|{\mathbf{b}}\|_{L^{n}(\Omega)}\|{\nabla
u}\|_{L^{p}(\Omega)}\|{v}\|_{L^{(p^{\prime})^{*}}(\Omega)}$ $\displaystyle\leq
C(n,p,\Omega)\|{\mathbf{b}}\|_{L^{n}(\Omega)}\|{u}\|_{W^{1,p}(\Omega)}\|{v}\|_{W^{1,p^{\prime}}(\Omega)}.$

for all $u\in W^{1,p}(\Omega)$ and $v\in W^{1,p^{\prime}}(\Omega)$. This
proves (3.1).

Note that $\mathbf{b}\cdot\nabla u\in W^{-1,p}(\Omega)$ by the estimate (3.1).
To show $\varepsilon$-inequality (3.2), note that integration by part shows
that the identity

$\int_{\Omega}(\mathbf{b}\cdot\nabla
u)vdx=-\int_{\Omega}(\mathbf{b}\cdot\nabla
v)u\,d{x}-\int_{\Omega}(\operatorname{div}\mathbf{b})uv\,d{x}$ (3.4)

holds for any $\mathbf{b}\in C^{\infty}_{c}(\Omega;\mathbb{R}^{n})$, $u\in
C^{\infty}(\overline{\Omega})$, and $v\in C_{c}^{\infty}(\Omega)$. Since
$C^{\infty}(\overline{\Omega})$ is dense in $W^{1,p}(\Omega)$ (see [12,
Theorem 4.3]) and the estimate (3.1) holds, a standard density argument shows
that identity (3.4) holds for any $\mathbf{b}\in
C_{c}^{\infty}({\Omega};\mathbb{R}^{n})$, $u\in W^{1,p}(\Omega)$, and $v\in
C_{c}^{\infty}(\Omega)$.

Let $\varepsilon>0$ be given. Since $C^{\infty}_{c}(\Omega;\mathbb{R}^{n})$ is
dense in $L^{n}(\Omega;\mathbb{R}^{n})$, there exists
$\mathbf{b}_{\varepsilon}\in C^{\infty}_{c}(\Omega;\mathbb{R}^{n})$ such that

$\|{\mathbf{b}_{\varepsilon}-\mathbf{b}}\|_{L^{n}(\Omega)}<{\varepsilon}/{C},$

where $C$ is the same constant in (3.1). Fix $v\in C_{c}^{\infty}(\Omega)$.
Then by (3.4), we have

$\displaystyle\int_{\Omega}(\mathbf{b}\cdot\nabla u)v\,d{x}$
$\displaystyle=\int_{\Omega}[(\mathbf{b}-\mathbf{b}_{\varepsilon})\cdot\nabla
u]v\,d{x}+\int_{\Omega}(\mathbf{b}_{\varepsilon}\cdot\nabla u)vdx$
$\displaystyle=\int_{\Omega}[(\mathbf{b}-\mathbf{b}_{\varepsilon})\cdot\nabla
u]v\,d{x}-\int_{\Omega}(\mathbf{b}_{\varepsilon}\cdot\nabla
v)udx-\int_{\Omega}(\operatorname{div}\mathbf{b}_{\varepsilon})uv\,d{x}.$

By (3.1) and Hölder’s inequality, we get

$\displaystyle\int_{\Omega}|(\mathbf{b}\cdot\nabla u)v|\,d{x}$
$\displaystyle\leq
C\|{\mathbf{b}-\mathbf{b}_{\varepsilon}}\|_{L^{n}(\Omega)}\|{u}\|_{W^{1,p}(\Omega)}\|{v}\|_{W^{1,p^{\prime}}(\Omega)}$
$\displaystyle\mathrel{\phantom{=}}+\|{\mathbf{b}_{\varepsilon}}\|_{L^{\infty}(\Omega)}\|{u}\|_{L^{p}(\Omega)}\|{v}\|_{L^{p^{\prime}}(\Omega)}+\|{\operatorname{div}\mathbf{b}_{\varepsilon}}\|_{L^{\infty}(\Omega)}\|{u}\|_{L^{p}(\Omega)}\|{v}\|_{L^{p^{\prime}}(\Omega)}$
$\displaystyle\leq\left(\varepsilon\|{u}\|_{W^{1,p}(\Omega)}+C_{\varepsilon}\|{u}\|_{L^{p}(\Omega)}\right)\|{v}\|_{W^{1,p^{\prime}}(\Omega)},$

where
$C_{\varepsilon}=\|{\mathbf{b}_{\varepsilon}}\|_{L^{\infty}(\Omega)}+\|{\operatorname{div}\mathbf{b}_{\varepsilon}}\|_{L^{\infty}(\Omega)}$.
Since $v\in C_{c}^{\infty}(\Omega)$ was arbitrary chosen, this implies that

$\|{\mathbf{b}\cdot\nabla
u}\|_{W^{-1,p}(\Omega)}\leq\varepsilon\|{u}\|_{W^{1,p}(\Omega)}+C_{\varepsilon}\|{u}\|_{L^{p}(\Omega)},$

which proves (3.2).

To show (3.3), suppose that $u\in C^{\infty}_{c}(\Omega)$ and $v\in
W^{1,p^{\prime}}(\Omega)$. Since

$\int_{\Omega}(v\mathbf{b})\cdot\nabla
u\,d{x}=\int_{\Omega}[v(\mathbf{b}-\mathbf{b}_{\varepsilon})]\cdot\nabla
u\,d{x}+\int_{\Omega}(v\mathbf{b}_{\varepsilon})\cdot\nabla u\,d{x},$

it follows from (3.1) and Hölder’s inequality that

$\int_{\Omega}|(v\mathbf{b})\cdot\nabla
u|\,d{x}\leq(\varepsilon\|{v}\|_{W^{1,p^{\prime}}(\Omega)}+\|{\mathbf{b}_{\varepsilon}}\|_{L^{\infty}(\Omega)}\|{v}\|_{L^{p^{\prime}}(\Omega)})\|{u}\|_{W^{1,p}(\Omega)}.$
(3.5)

Since

$\left<\operatorname{div}(v\mathbf{b}),u\right>=-\int_{\Omega}(v\mathbf{b})\cdot\nabla
u\,d{x}$

for all $u\in C_{c}^{\infty}(\Omega)$, it follows from (3.5) that
$\operatorname{div}(v\mathbf{b})\in W^{-1,p^{\prime}}(\Omega)$ and the
estimate (3.3) holds. This completes the proof of Proposition 3.1. ∎

We use the following special case of Dong-Kim [10, Theorem 7].

######  Theorem 3.2.

Let $1<p<\infty$ and $\Omega$ be a bounded domain in $\mathbb{R}^{n}$, $n\geq
2$. Then there exist $\gamma=\gamma(n,p,\delta)>0$,
$\theta=\theta(n,p,\delta)>0$, and
$\lambda_{0}=\lambda_{0}(n,p,\delta,R_{0})\geq 1$ such that under Assumptions
2.2 $(\gamma)$ and 2.3 $(\theta)$, the following holds for any
$\lambda\geq\lambda_{0}$: for any $f\in L^{p}(\Omega;\mathbb{R}^{n})$ and
$\mathbf{F}\in L^{p}(\Omega;\mathbb{R}^{n})$, there exists a unique weak
solution $u\in{W}^{1,p}_{0}(\Omega)$ such that

$-\operatorname{div}(A\nabla u)+\lambda
u=f+\operatorname{div}\mathbf{F}\quad\text{in }\Omega.$

Moreover we have

$\lambda^{1/2}\|{u}\|_{L^{p}(\Omega)}+\|{\nabla u}\|_{L^{p}(\Omega)}\leq
C\left(\lambda^{-1/2}\|{f}\|_{L^{p}(\Omega)}+\|{\mathbf{F}}\|_{L^{p}(\Omega)}\right)$

for some constant $C=C(n,p,\delta,R_{0},\Omega)>0$.

######  Remark.

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$, $n\geq 2$, and
$1<p<\infty$. If $f_{0}\in W^{-1,p}(\Omega)$, then there exists
$\mathbf{F}_{0}\in L^{p}(\Omega;\mathbb{R}^{n})$ such that

$\operatorname{div}\mathbf{F}_{0}=f_{0}\quad\text{in
}\Omega\quad\text{and}\quad\|{\mathbf{F}_{0}}\|_{L^{p}(\Omega)}\leq
C(n,p,\Omega)\|{f_{0}}\|_{W^{-1,p}(\Omega)},$

(see e.g. [21, Lemma 3.9]). By Theorem 3.2, there exist
$\gamma=\gamma(n,p,\delta)>0$, $\theta=\theta(n,p,\delta,R_{0})>0$, and
$\lambda_{0}=\lambda_{0}(n,p,\delta,R_{0})\geq 1$ such that under Assumptions
2.2 $(\gamma)$ and 2.3 $(\theta)$, we have for $\lambda\geq\lambda_{0}$ and
for each $f\in L^{p}(\Omega)$, there exists a unique weak solution
$u\in{W}^{1,p}_{0}(\Omega)$ such that

$\int_{\Omega}(A\nabla
u)\cdot\nabla\phi\,d{x}+\lambda\int_{\Omega}u\phi\,d{x}=\int_{\Omega}f\phi\,d{x}-\int_{\Omega}\mathbf{F}_{0}\cdot\nabla\phi\,d{x}$

for all $\phi\in C_{c}^{\infty}(\Omega)$. Moreover we have

$\displaystyle\lambda^{1/2}\|{u}\|_{L^{p}(\Omega)}+\|{\nabla
u}\|_{L^{p}(\Omega)}$ $\displaystyle\leq
C(\lambda^{-1/2}\|{f}\|_{L^{p}(\Omega)}+\|{\mathbf{F}_{0}}\|_{L^{p}(\Omega)})$
(3.6) $\displaystyle\leq
C(\lambda^{-1/2}\|{f}\|_{L^{p}(\Omega)}+\|{f_{0}}\|_{W^{-1,p}(\Omega)})$

for some constant $C=C(n,p,\delta,R_{0},\Omega)>0$.

Now we present the main theorem of this section. From now on, we mainly focus
on the case $\mathbf{b}\in L^{2}(\Omega;\mathbb{R}^{2})$ since other cases
$\mathbf{b}\in L^{q}(\Omega;\mathbb{R}^{n})$ are already considered in Kang-
Kim [17] when $n\leq q<\infty$ if $n\geq 3$ and $2<q<\infty$ if $n=2$.

######  Theorem 3.3.

Let $2<p<\infty$ and $\Omega$ be a bounded domain in $\mathbb{R}^{2}$. Suppose
that $A$ satisfies (1.1) and $\mathbf{b}\in L^{2}(\Omega;\mathbb{R}^{2})$.
Then there exist $\gamma=\gamma(p,\delta)>0$, $\theta=\theta(p,\delta)>0$, and
$\lambda_{1}=\lambda_{1}(p,\delta,R_{0},\Omega,\mathbf{b})\geq 1$ such that
under Assumptions 2.2 $(\gamma)$ and 2.3 $(\theta)$, the following results
hold for any $\lambda\geq\lambda_{1}$:

  1. (i)

If $f\in L^{p}(\Omega)$ and $\mathbf{F}\in L^{p}(\Omega;\mathbb{R}^{2})$, then
there exists a unique weak solution $u\in{W}^{1,p}_{0}(\Omega)$ of ($D$).
Moreover

$\|{\nabla u}\|_{L^{p}(\Omega)}+\lambda^{1/2}\|{u}\|_{L^{p}(\Omega)}\leq
C\left[\lambda^{-1/2}\|{f}\|_{L^{p}(\Omega)}+\|{\mathbf{F}}\|_{L^{p}(\Omega)}\right]$

for some constant $C=C(p,\delta,R_{0},\Omega)>0$.

  2. (ii)

If $g\in L^{p^{\prime}}(\Omega)$ and $\mathbf{G}\in
L^{p^{\prime}}(\Omega;\mathbb{R}^{2})$, then there exists a unique weak
solution $v\in{W}^{1,p^{\prime}}_{0}(\Omega)$ of ($D^{\prime}$). Moreover

$\|{\nabla
v}\|_{L^{p^{\prime}}(\Omega)}+\lambda^{1/2}\|{v}\|_{L^{p^{\prime}}(\Omega)}\leq
C\left[\lambda^{-1/2}\|{g}\|_{L^{p^{\prime}}(\Omega)}+\|{\mathbf{G}}\|_{L^{p^{\prime}}(\Omega)}\right]$

for some constant $C=C(p,\delta,R_{0},\Omega)>0$.

###### Proof.

Let $f\in L^{p}(\Omega)$ and $\mathbf{F}\in L^{p}(\Omega;\mathbb{R}^{2})$.
Then $\operatorname{div}\mathbf{F}\in W^{-1,p}(\Omega)$. For each
$u\in{W}^{1,p}_{0}(\Omega)$, it follows from Proposition 3.1 that
$\mathbf{b}\cdot\nabla u\in W^{-1,p}(\Omega)$. Moreover, for each
$\varepsilon>0$, there exists a constant
$C_{\varepsilon}=C(\varepsilon,p,\mathbf{b},\Omega)>0$ such that

$\|{\mathbf{b}\cdot\nabla u}\|_{W^{-1,p}(\Omega)}\leq\varepsilon\|{\nabla
u}\|_{L^{p}(\Omega)}+C_{\varepsilon}\|{u}\|_{L^{p}(\Omega)}$ (3.7)

for all $u\in{W}^{1,p}_{0}(\Omega)$. By the remark of Theorem 3.2, there exist
$\gamma=\gamma(p,\delta)>0$, $\theta=\theta(p,\delta)>0$, and
$\lambda_{0}=\lambda_{0}(p,\delta,R_{0},\Omega)\geq 1$ such that under
Assumptions 2.2 $(\gamma)$ and 2.3 $(\theta)$, for $\lambda\geq\lambda_{0}$,
there exists a unique $\overline{u}=\mathcal{T}(u)\in{W}^{1,p}_{0}(\Omega)$
satisfying

$\int_{\Omega}A\nabla\overline{u}\cdot\nabla\phi\,d{x}+\lambda\int_{\Omega}\overline{u}\phi\,d{x}=\left<\operatorname{div}\mathbf{F}-\mathbf{b}\cdot\nabla
u,\phi\right>+\int_{\Omega}f\phi\,d{x}$

for all $\phi\in C_{c}^{\infty}(\Omega)$. Moreover, we have

$\lambda^{1/2}\|{\overline{u}}\|_{L^{p}(\Omega)}+\|{\nabla\overline{u}}\|_{L^{p}(\Omega)}\leq
C_{0}\left(\lambda^{-1/2}\|{f}\|_{L^{p}(\Omega)}+\|{\operatorname{div}\mathbf{F}-\mathbf{b}\cdot\nabla
u}\|_{W^{-1,p}(\Omega)}\right)$

for some constant $C_{0}=C_{0}(p,\delta,R_{0},\Omega)>0$. Choose
$\varepsilon>0$ so that $\varepsilon C_{0}=1/2$. Then we have

$\displaystyle\lambda^{1/2}\|{\overline{u}}\|_{L^{p}(\Omega)}+\|{\nabla\overline{u}}\|_{L^{p}(\Omega)}$
$\displaystyle\leq
C_{0}\lambda^{-1/2}\|{f}\|_{L^{p}(\Omega)}+C_{0}\|{\mathbf{F}}\|_{L^{p}(\Omega)}$
$\displaystyle\mathrel{\phantom{=}}+\frac{1}{2}\left(\|{\nabla
u}\|_{L^{p}(\Omega)}+C_{*}\|{u}\|_{L^{p}(\Omega)}\right),$

where $C_{*}=C_{*}(p,\mathbf{b},\delta,R_{0},\Omega)>0$. Moreover, we have

$\displaystyle\lambda^{1/2}\|{\mathcal{T}(u_{1})-\mathcal{T}(u_{2})}\|_{L^{p}(\Omega)}+\|{\nabla(\mathcal{T}(u_{1})-\mathcal{T}(u_{2}))}\|_{L^{p}(\Omega)}$
$\displaystyle\leq\frac{1}{2}\left(\|{\nabla(u_{1}-u_{2})}\|_{L^{p}(\Omega)}+C_{*}\|{u_{1}-u_{2}}\|_{L^{p}(\Omega)}\right)$

for $u_{1},u_{2}\in{W}^{1,p}_{0}(\Omega)$. Let
$\lambda_{1}:=\max(C_{*}^{2},\lambda_{0})+1$. Then for
$\lambda\geq\lambda_{1}$, $\mathcal{T}$ is a contraction on
${W}^{1,p}_{0}(\Omega)$ which is a Banach space equipped with the equivalent
norm $\|{\nabla\cdot}\|_{L^{p}(\Omega)}+\|{\cdot}\|_{L^{p}(\Omega)}$. Hence by
the Banach fixed point theorem, there exists a unique
$u\in{W}^{1,p}_{0}(\Omega)$ such that $\mathcal{T}(u)=u$, that is, $u$ is a
weak solution of the problem ($D$). Moreover, $u$ satisfies

$\lambda^{1/2}\|{u}\|_{L^{p}(\Omega)}+\|{u}\|_{L^{p}(\Omega)}\leq
2C_{0}\left(\lambda^{-1/2}\|{f}\|_{L^{p}(\Omega)}+\|{\mathbf{F}}\|_{L^{p}(\Omega)}\right).$

This completes the proof of (i). Following the exactly same argument, one can
also prove (ii) whose proof is omitted. This completes the proof of Theorem
3.3. ∎

##  4\. Uniqueness of weak solutions for the problem ($D$)

This section is devoted to a proof of the uniqueness part of Theorem 2.5 (i).
Below is the main theorem of this section.

######  Theorem 4.1.

Let $\Omega$ be a bounded domain in $\mathbb{R}^{2}$, $2<p<\infty$. Suppose
that $A$ satisfies (1.1), $\operatorname{div}A,\mathbf{b}\in
L^{2}(\Omega;\mathbb{R}^{2})$, and $\lambda\geq 0$. Then there exists a
constant
$\gamma=\gamma(p,\delta,\|{\operatorname{div}A}\|_{L^{2}(\Omega)},\|{\mathbf{b}}\|_{L^{2}(\Omega)})$
such that under Assumption 2.2 $(\gamma)$, if $u\in{W}^{1,p}_{0}(\Omega)$
satisfies

$\int_{\Omega}A\nabla u\cdot\nabla\phi+(\mathbf{b}\cdot\nabla u+\lambda
u)\phi\,d{x}=0$ (4.1)

for all $\phi\in C_{c}^{\infty}(\Omega)$, then $u$ is identically zero in
$\Omega$.

To prove Theorem 4.1, we use recent results due to Krylov [23, 24]. To state
these results, for $0<\delta<1$, let $\mathbb{S}_{\delta}$ be the set of
$n\times n$ real symmetric matrices which are measurable and whose eigenvalues
are in $[\delta,\delta^{-1}]$ and $\mathbf{b}:\Omega\rightarrow\mathbb{R}^{n}$
be a vector field. Write

$Lu=\sum_{i,j=1}^{n}a^{ij}D_{ij}u-\mathbf{b}\cdot\nabla u.$

The following theorem can be found in [23, Corollary 3.1], which generalizes
the classical theorem due to Alexsandrov [1].

######  Theorem 4.2.

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$ and $c$ be a nonnegative
measurable function on $\Omega$. Suppose that $0<\delta<1$ and $A$ is a
$\mathbb{S}_{\delta}$-valued function on $\mathbb{R}^{n}$ and $\mathbf{b}\in
L^{n}(\Omega;\mathbb{R}^{n})$. Then there exists a number $n/2<n_{0}<n$
depending on $n$, $\delta$, and $\|{\mathbf{b}}\|_{L^{n}(\Omega)}$ such that
if $n_{0}\leq p<\infty$, then there exists a constant $C$ depending on $n$,
$p$, $\delta$, $\|{\mathbf{b}}\|_{L^{n}(\Omega)}$, and the diameter of
$\Omega$ such that

$u(x)\leq C\|{(Lu-
cu)_{-}}\|_{L^{p}(\Omega)}+\sup_{\partial\Omega}u_{+}\quad\text{in }\Omega$
(4.2)

for all $u\in W^{2,p}_{\mathrm{loc}}(\Omega)\cap C(\overline{\Omega})$.

The following theorem is a special case of [24, Theorem 4.2].

######  Theorem 4.3.

Let $\Omega$ be a bounded $C^{1,1}$-domain in $\mathbb{R}^{n}$, $n\geq 2$,
$1<p<n$, and $\lambda\geq 0$. Assume that $A$ satisfies (1.1) and
$\mathbf{b}\in L^{n}(\Omega;\mathbb{R}^{n})$. Then there exists
$\gamma=\gamma(n,p,\delta)>0$ such that under Assumption 2.2 $(\gamma)$, for
every $g\in L^{p}(\Omega)$, there exists a unique strong solution
$u\in{W}^{1,p}_{0}(\Omega)\cap W^{2,p}(\Omega)$ satisfying

$-\sum_{i,j=1}^{n}a^{ij}D_{ij}u+\mathbf{b}\cdot\nabla u+\lambda
u=g\quad\text{in }\Omega.$

Now we are ready to prove the main theorem of this section.

###### Proof of Theorem 4.1.

By Theorem 4.2, there exists a number $1<n_{0}<2$ depending on $\delta$,
$\|{\operatorname{div}A}\|_{L^{2}(\Omega)}$, and
$\|{\mathbf{b}}\|_{L^{2}(\Omega)}$ such that for $n_{0}<q<2$, there exists a
constant $C$ depending on $q$, $\delta$,
$\|{\operatorname{div}A}\|_{L^{2}(\Omega)}$,
$\|{\mathbf{b}}\|_{L^{2}(\Omega)}$, and the diameter of $\Omega$ such that

$u(x)\leq C\|{(Lu-\lambda
u)_{-}}\|_{L^{q}(\Omega)}+\sup_{\partial\Omega}u_{+}\quad\text{in }\Omega$

for all $u\in W^{2,q}_{\mathrm{loc}}(\Omega)\cap C(\overline{\Omega})$, where

$Lu=\sum_{i,j=1}^{2}a^{ij}D_{ij}u-(\mathbf{b}-\operatorname{div}A)\cdot\nabla
u.$

Define $q=\frac{1}{2}\left(\max\\{2p/(p+2),n_{0}\\}+2\right)$. Then
$\max\\{n_{0},\frac{2p}{p+2}\\}<q<2$. Since $u\in{W}^{1,p}_{0}(\Omega)$ and
$p>2$, it follows from the Sobolev embedding theorem that $u\in
C(\overline{\Omega})$. Hence it suffices to show that $u\in
W^{2,q}_{\mathrm{loc}}(\Omega)$. To show this, we fix a bounded smooth
subdomain $\Omega^{\prime}\Subset\Omega$ and let $\zeta\in
C_{c}^{\infty}(\Omega^{\prime})$. Since $u$ satisfies (4.1), we have

$\int_{\Omega}A\nabla
u\cdot\nabla(\zeta\phi)\,d{x}+\int_{\Omega}(\mathbf{b}\cdot\nabla u+\lambda
u)(\zeta\phi)\,d{x}=0$ (4.3)

for all $\phi\in C_{c}^{\infty}(\Omega)$. Since $\operatorname{div}A\in
L^{2}(\Omega;\mathbb{R}^{2})$ and $u\in{W}^{1,p}_{0}(\Omega)$, $p>2$, it
follows that

$\displaystyle-\int_{\Omega}\operatorname{div}A\cdot(u\phi\nabla\zeta)\,d{x}$
$\displaystyle=\sum_{i,j=1}^{2}\int_{\Omega}a^{ij}D_{i}(u\phi D_{j}\zeta)dx$
(4.4)
$\displaystyle=\sum_{i,j=1}^{2}\int_{\Omega}a^{ij}(D_{i}u)(D_{j}\zeta)\phi\,d{x}+\sum_{i,j=1}^{2}\int_{\Omega}ua^{ij}(D_{i}\phi)(D_{j}\zeta)dx$
$\displaystyle\mathrel{\phantom{=}}+\sum_{i,j=1}^{2}\int_{\Omega}u\phi
a^{ij}D_{ij}\zeta\,d{x}+\int_{\Omega}u\phi\left(\sum_{i,j=1}^{2}a^{ij}D_{ij}\zeta\right)\,d{x}$

for all $\phi\in C_{c}^{\infty}(\Omega)$. Hence by (4.3) and (4.4), we get

$\displaystyle\int_{\Omega}A\nabla(u\zeta)\cdot\nabla\phi+(\mathbf{b}\cdot\nabla(\zeta
u)+\lambda_{1}(\zeta u))\phi\,d{x}=-\int_{\Omega}g\phi\,d{x},$ (4.5)

where

$g=u\operatorname{div}A\cdot\nabla\zeta+\sum_{i,j=1}^{2}(a^{ij}D_{ij}\zeta)u+\nabla
u\cdot A\nabla\zeta+A\nabla
u\cdot\nabla\zeta-u\mathbf{b}\cdot\nabla\zeta+(\lambda-\lambda_{1})(\zeta u).$

Note that $g\in L^{2}(\Omega)$ since $u\in{W}^{1,p}_{0}(\Omega)$, $p>2$, and
$\operatorname{div}A,\mathbf{b}\in L^{2}(\Omega;\mathbb{R}^{2})$.

By Theorem 3.3, there exist $\gamma_{0}=\gamma_{0}(p,\delta)>0$ and
$\lambda_{1}=\lambda_{1}(p,\delta,\mathbf{b},\Omega^{\prime})\geq 1$ such that
under Assumption 2.2 $(\gamma_{0})$, for $\lambda\geq\lambda_{1}$, $f\in
L^{p}(\Omega^{\prime})$, and $\mathbf{F}\in
L^{p}(\Omega^{\prime};\mathbb{R}^{2})$, there exists a unique weak solution
$v\in{W}^{1,p^{\prime}}_{0}(\Omega)$ such that

$-\operatorname{div}(A\nabla v)+\mathbf{b}\cdot\nabla v+\lambda
v=f+\operatorname{div}\mathbf{F}\quad\text{in }\Omega^{\prime},\quad
v=0\quad\text{on }\partial\Omega^{\prime}.$

For such $\lambda_{1}$ and
$q=\frac{1}{2}\left(\max\\{2p/(p+2),n_{0}\\}+2\right)$, it follows from
Theorem 4.3 that there exists $\gamma_{1}=\gamma_{1}(p,n_{0},\delta)>0$ such
that under Assumption 2.2 $(\gamma_{1})$, there exists a unique
$w\in{W}^{1,q}_{0}(\Omega^{\prime})\cap W^{2,q}(\Omega^{\prime})$ satisfying

$-\sum_{i,j=1}^{2}a^{ij}D_{ij}w+(\mathbf{b}-\operatorname{div}A)\cdot\nabla
w+\lambda_{1}w=-g\quad\text{in }\Omega^{\prime}.$

Set $\gamma=\min\\{\gamma_{0},\gamma_{1}\\}$ and choose $A$ so that $A$
satisfies (1.1), $\operatorname{div}A\in L^{2}(\Omega;\mathbb{R}^{2})$, and
Assumption 2.2 $(\gamma)$. Since $w\in{W}^{1,q}_{0}(\Omega^{\prime})\cap
W^{2,q}(\Omega^{\prime})$ and $\operatorname{div}A\in
L^{2}(\Omega;\mathbb{R}^{2})$, it follows from the Sobolev embedding theorem
that $w\in{W}^{1,q^{*}}_{0}(\Omega^{\prime})$ and

$\displaystyle\int_{\Omega^{\prime}}\operatorname{div}A\cdot(\phi\nabla
w)\,d{x}$
$\displaystyle=-\sum_{i,j=1}^{2}\int_{\Omega^{\prime}}a^{ij}D_{i}(\phi
D_{j}w)\,d{x}$
$\displaystyle=-\sum_{i,j=1}^{2}\int_{\Omega^{\prime}}(a^{ij}D_{ij}w)\phi\,d{x}-\int_{\Omega^{\prime}}A\nabla
w\cdot\nabla\phi\,d{x}$

for all $\phi\in C_{c}^{\infty}(\Omega^{\prime})$. Hence $w$ satisfies

$\int_{\Omega^{\prime}}A\nabla w\cdot\nabla\phi+(\mathbf{b}\cdot\nabla
w+\lambda_{1}w)\phi\,d{x}=-\int_{\Omega^{\prime}}g\phi\,d{x}$

for all $\phi\in C_{c}^{\infty}(\Omega^{\prime})$. Since $2p/(p+2)<q<2$ and
$\Omega^{\prime}$ is bounded, it follows that
${W}^{1,q^{*}}_{0}(\Omega^{\prime})\subset{W}^{1,p}_{0}(\Omega^{\prime})$.
Hence by (4.5) and Theorem 3.3, we have $w=\zeta u$ in $\Omega^{\prime}$,
which implies that $\zeta u\in W^{2,q}(\Omega^{\prime})$ for any $\zeta\in
C_{c}^{\infty}(\Omega^{\prime})$. Since there exists a sequence of smooth
bounded subdomains $\\{\Omega_{k}\\}$ satisfying
$\Omega_{k}\Subset\Omega_{k+1}\Subset\Omega$ and
$\bigcup_{k}\Omega_{k}=\Omega$ (see e.g. [5, Proposition 8.2.1]), we conclude
that $u\in W^{2,q}_{\mathrm{loc}}(\Omega)$.

Since $\operatorname{div}A\in L^{2}(\Omega;\mathbb{R}^{2})$, we have

$\sum_{i,j=1}^{2}\int_{\Omega}a^{ij}D_{i}\left[(D_{j}u)\phi\right]\,d{x}=-\int_{\Omega}\phi\operatorname{div}A\cdot\nabla
u\,d{x}$

for all $\phi\in C_{c}^{\infty}(\Omega)$. Hence $u$ satisfies

$-\sum_{i,j=1}^{2}a^{ij}D_{ij}u+(\mathbf{b}-\operatorname{div}A)\cdot\nabla
u+\lambda u=0\quad\text{in }\Omega.$

Therefore by Theorem 4.2, we conclude that $u$ is identically zero in
$\Omega$. This completes the proof of Theorem 4.1. ∎

##  5\. Proof of Theorem 2.5

This section is devoted to a proof of Theorem 2.5. For the case of
sufficiently large $\lambda$, this was done by Theorem 3.3. To treat the case
of small $\lambda$, we use a functional analytic argument as in Kang-Kim [19,
17] and Kim-Kwon [20] to deduce that it suffices to prove the uniqueness part
of (i). To explain this, let $2<p<\infty$ be fixed. Then

$(u,v)\mapsto\left<\mathcal{L}_{p}u,v\right>=\int_{\Omega}(A\nabla
u)\cdot\nabla v\,d{x}+\int_{\Omega}(\mathbf{b}\cdot\nabla u)v\,d{x}$

and

$(u,v)\mapsto\left<\mathcal{L}_{p^{\prime}}^{*}v,u\right>=\int_{\Omega}(A^{T}\nabla
v)\cdot\nabla u\,d{x}+\int_{\Omega}(v\mathbf{b})\cdot\nabla
u\,d{x}=\left<\mathcal{L}_{p}u,v\right>$

define bounded linear operators

$\mathcal{L}_{p}:{W}^{1,p}_{0}(\Omega)\rightarrow
W^{-1,p}(\Omega)\quad\text{and}\quad\mathcal{L}_{p^{\prime}}^{*}:{W}^{1,p^{\prime}}_{0}(\Omega)\rightarrow
W^{-1,p^{\prime}}(\Omega).$

Let
$\mathcal{I}_{p}:L^{p}(\Omega)\rightarrow(L^{p^{\prime}}(\Omega))^{\prime}$ be
the isomorphism defined by

$\left<\mathcal{I}_{p}u,v\right>=\int_{\Omega}uv\,d{x}\quad\text{for all
}(u,v)\in L^{p}(\Omega)\times L^{p^{\prime}}(\Omega).$

We also define $\operatorname{div}_{p}:L^{p}(\Omega;\mathbb{R}^{2})\rightarrow
W^{-1,p}(\Omega)$ by

$\left<\operatorname{div}_{p}\mathbf{F},\varphi\right>=-\int_{\Omega}\mathbf{F}\cdot\nabla\varphi\,d{x}\quad\text{for
all }\varphi\in{W}^{1,p^{\prime}}_{0}(\Omega).$

Then it is easy to show that $u\in{W}^{1,p}_{0}(\Omega)$ is a weak solution of
the problem ($D$) if and only if
$(\mathcal{L}_{p}+\lambda\mathcal{I}_{p})u=\mathcal{I}_{p}f+\operatorname{div}_{p}\mathbf{F}$,
while $v\in{W}^{1,p^{\prime}}_{0}(\Omega)$ is a weak solution of
($D^{\prime}$) if and only if
$(\mathcal{L}_{p^{\prime}}^{*}+\lambda\mathcal{I}_{p^{\prime}})v=\mathcal{I}_{p^{\prime}}g+\operatorname{div}_{p^{\prime}}\mathbf{G}$.

By Theorem 3.3, there exist $\gamma=\gamma(p,\delta)>0$,
$\theta=\theta(p,\delta)>0$, and $\lambda_{1}\geq 1$ such that under
Assumptions 2.2 $(\gamma)$ and 2.3 $(\theta)$, the operators
$\mathcal{L}_{p}+\lambda_{1}\mathcal{I}_{p}$ and
$\mathcal{L}_{p^{\prime}}^{*}+\lambda_{1}\mathcal{I}_{p^{\prime}}$ are
invertible. Then for $0\leq\lambda\neq\lambda_{1}$, the linear operators

$\mathcal{K}_{p,\lambda}=(\lambda_{1}-\lambda)(\mathcal{L}_{p}+\lambda_{1}\mathcal{I}_{p})^{-1}\circ\mathcal{I}_{p}:L^{p}(\Omega)\rightarrow
L^{p}(\Omega)$

and

$\mathcal{K}_{p^{\prime},\lambda}^{*}=(\lambda_{1}-\lambda)(\mathcal{L}_{p^{\prime}}^{*}+\lambda_{1}\mathcal{I}_{p^{\prime}})^{-1}\circ\mathcal{I}_{p^{\prime}}:L^{p^{\prime}}(\Omega)\rightarrow
L^{p^{\prime}}(\Omega)$

are bounded and even compact by Rellich-Kondrachov’s embedding theorem. Since

$\int_{\Omega}(\mathcal{K}_{p,\lambda}u)v\,d{x}=\int_{\Omega}(\mathcal{K}_{p^{\prime},\lambda}^{*}v)u\,d{x}$

for all $(u,v)\in L^{p}(\Omega)\times L^{p^{\prime}}(\Omega)$, it follows that

$\mathcal{K}_{p^{\prime},\lambda}^{*}=\mathcal{I}_{p^{\prime}}^{-1}\circ\mathcal{K}_{p,\lambda}^{\prime}\circ\mathcal{I}_{p^{\prime}},$
(5.1)

where
$\mathcal{K}_{p,\lambda}^{\prime}:(L^{p}(\Omega))^{\prime}\rightarrow(L^{p}(\Omega))^{\prime}$
is the adjoint operator of $\mathcal{K}_{p,\lambda}$. Note also that

$\ker(Id-\mathcal{K}_{p,\lambda})=\ker(\mathcal{L}_{p}+\lambda\mathcal{I}_{p})\quad\text{and}\quad\ker(Id-\mathcal{K}_{p^{\prime},\lambda}^{*})=\ker(\mathcal{L}_{p^{\prime}}^{*}+\lambda\mathcal{I}_{p^{\prime}}).$

By (5.1) and the Fredholm alternative theorem (see e.g. [3, Theorem 6.6]), we
have

$\mathrm{Im}(Id-\mathcal{K}_{p,\lambda})=\left\\{u\in
L^{p}(\Omega):\int_{\Omega}uv\,d{x}=0\quad\text{for all
}v\in\ker(\mathcal{L}_{p^{\prime}}^{*}+\lambda\mathcal{I}_{p^{\prime}})\right\\},$
(5.2) $\mathrm{Im}(Id-\mathcal{K}_{p^{\prime},\lambda}^{*})=\left\\{v\in
L^{p^{\prime}}(\Omega):\int_{\Omega}vu\,d{x}=0\quad\text{for all
}u\in\ker(\mathcal{L}_{p}+\lambda\mathcal{I}_{p})\right\\},$ (5.3)

and

$\dim\ker(\mathcal{L}_{p}+\lambda\mathcal{I}_{p})=\dim\ker(\mathcal{L}_{p^{\prime}}^{*}+\lambda\mathcal{I}_{p^{\prime}})<\infty$
(5.4)

for all $\lambda\geq 0$.

Now we are ready to prove Theorem 2.5.

###### Proof of Theorem 2.5.

Choose $(\gamma_{0},\theta_{0},\lambda_{1})$ from Theorem 3.3 and choose
$\gamma_{1}$ from Theorem 4.1, and let
$\gamma=\min\\{\gamma_{0},\gamma_{1}\\}$. Under Assumptions 2.2 $(\gamma)$ and
2.3 $(\theta)$, Theorem 2.5 holds for all $\lambda\geq\lambda_{1}$ by Theorem
3.3. Suppose that $0\leq\lambda<\lambda_{1}$. By (5.4) and Theorem 4.1, we
have

$\ker(\mathcal{L}_{p}+\lambda\mathcal{I}_{p})=\ker(\mathcal{L}_{p^{\prime}}^{*}+\lambda\mathcal{I}_{p^{\prime}})=\\{0\\}$
(5.5)

for any $\lambda\geq 0$. Hence by (5.2) and (5.3), we have

$\mathrm{Im}(Id-\mathcal{K}_{p,\lambda})=\mathrm{Im}(Id-\mathcal{K}_{p^{\prime},\lambda}^{*})=L^{p}(\Omega).$
(5.6)

Given $f\in L^{p}(\Omega)$ and $\mathbf{F}\in L^{p}(\Omega;\mathbb{R}^{2})$,
let

$w=(\mathcal{L}_{p}+\lambda_{1}\mathcal{I}_{p})^{-1}(\mathcal{I}_{p}f+\operatorname{div}_{p}\mathbf{F}).$

By (5.6), there exists $u\in L^{p}(\Omega)$ such that

$u-\mathcal{K}_{p,\lambda}u=w.$

Since $(\mathcal{L}_{p}+\lambda_{1}\mathcal{I}_{p})^{-1}$ maps
$W^{-1,p}(\Omega)$ to ${W}^{1,p}_{0}(\Omega)$, it follows from the definition
of $\mathcal{K}_{p,\lambda}$ that
$w,\mathcal{K}_{p,\lambda}u\in{W}^{1,p}_{0}(\Omega)$. Hence

$u\in{W}^{1,p}_{0}(\Omega)\quad\text{and}\quad(\mathcal{L}_{p}+\lambda\mathcal{I}_{p})u=\mathcal{I}_{p}f+\operatorname{div}_{p}\mathbf{F}.$

This proves that $u$ is a weak solution of the problem ($D$). Similarly, we
can also prove the existence of weak solution of the problem ($D^{\prime}$).
By (5.5), we also have uniqueness of weak solution of the problem
($D^{\prime}$). It remains to show the desired estimate.

Let $f\in L^{p}(\Omega)$ and $\mathbf{F}\in L^{p}(\Omega;\mathbb{R}^{2})$.
Define

$\left<\ell,\phi\right>=\int_{\Omega}f\phi\,d{x}-\int_{\Omega}\mathbf{F}\cdot\nabla\phi\,d{x}\quad\text{for
all }\phi\in C_{c}^{\infty}(\Omega).$

Then

$\ell\in
W^{-1,p}(\Omega)\quad\text{and}\quad\|{\ell}\|_{W^{-1,p}(\Omega)}\leq\|{f}\|_{L^{p}(\Omega)}+\|{\mathbf{F}}\|_{L^{p}(\Omega)}.$

Since the operator $\mathcal{L}_{p}+\lambda\mathcal{I}_{p}$ is bijective, it
follows that there exists $u_{\lambda}\in{W}^{1,p}_{0}(\Omega)$ such that

$(\mathcal{L}_{p}+\lambda\mathcal{I}_{p})u_{\lambda}=\ell.$

Moreover, it follows from the bounded inverse theorem (see e.g. [3, Corollary
2.7]) that there exists a constant $C_{\lambda}>0$ independent of
$u_{\lambda}$ and $\ell$ such that

$\|{u_{\lambda}}\|_{W^{1,p}(\Omega)}\leq
C_{\lambda}\|{\ell}\|_{W^{-1,p}(\Omega)}\leq
C_{\lambda}(\|{f}\|_{L^{p}(\Omega)}+\|{\mathbf{F}}\|_{L^{p}(\Omega)}).$

In fact, we can choose a constant independent of $\lambda$. Indeed, for
$0\leq\lambda\leq\lambda_{1}$, let

$E_{\lambda}=\left\\{\mu\in[0,\lambda_{1}]:C_{\lambda}|\lambda-\mu|<\frac{1}{2}\right\\}.$

For $\mu\in E_{\lambda}$, since the operator
$\mathcal{L}_{p}+\lambda\mathcal{I}_{p}$ is bijective, there exists
$u_{\mu}\in{W}^{1,p}_{0}(\Omega)$ such that

$(\mathcal{L}_{p}+\lambda\mathcal{I}_{p})u_{\mu}=\mathcal{I}_{p}f+\operatorname{div}_{p}\mathbf{F}+(\lambda-\mu)\mathcal{I}_{p}u_{\mu}.$

Moreover, we have

$\|{u_{\mu}}\|_{W^{1,p}(\Omega)}\leq
2C_{\lambda}(\|{f}\|_{L^{p}(\Omega)}+\|{\mathbf{F}}\|_{L^{p}(\Omega)})$

for all $\mu\in E_{\lambda}$. Hence by the compactness of $[0,\lambda_{1}]$,
there is a constant $C>0$ independent of $u_{\lambda}$, $f$, $\mathbf{F}$, and
$\lambda$ such that

$\|{u_{\lambda}}\|_{W^{1,p}(\Omega)}\leq
2C(\|{f}\|_{L^{p}(\Omega)}+\|{\mathbf{F}}\|_{L^{p}(\Omega)}).$

By the Poincaré inequality, we conclude that

$\|{\nabla
u_{\lambda}}\|_{L^{p}(\Omega)}+\lambda^{1/2}\|{u_{\lambda}}\|_{L^{p}(\Omega)}\leq
C(\|{f}\|_{L^{p}(\Omega)}+\|{\mathbf{F}}\|_{L^{p}(\Omega)})$

for all $\lambda\in[0,\lambda_{1}]$. This proves the desired estimate for a
solution of the problem ($D$). Following exactly the same argument, one can
derive a similar estimate for a weak solution of the problem ($D^{\prime}$).
This completes the proof of Theorem 2.5. ∎

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