# Elliptic equations with a singular drift
from a weak Morrey space

Misha Chernobai111Department of Mathematics, UBC, Canada<EMAIL_ADDRESS>Tim Shilkin222V.A. Steklov Mathematical Institute, St.-Petersburg, Russia, and
Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany,
<EMAIL_ADDRESS>
###### Abstract

In this paper we prove the existence, uniqueness and regularity of weak
solutions to the Dirichlet problem for an elliptic equation with a drift $b$
satisfying $\mathop{\rm div}b\leq 0$ in $\Omega$. We assume $b$ belongs to
some weak Morrey class which includes in the 3D case, in particular, drifts
having a singularity along the axis $x_{3}$ with the asymptotic $b(x)\sim
c/r$, where $r=\sqrt{x_{1}^{2}+x_{2}^{2}}$.

##  1 Introduction and Main Results

Assume $\Omega\subset\mathbb{R}^{n}$ is a bounded Lipschitz domain, $n\geq 3$.
We consider the following boundary value problem:

$\left\\{\quad\begin{gathered}-\Delta u+b\cdot\nabla u\,=\,-\mathop{\rm
div}f\qquad\mbox{in}\quad\Omega,\\\ u|_{\partial\Omega}\ =\
0.\qquad\quad\end{gathered}\right.$ (1.1)

Here $u:\Omega\to\mathbb{R}$ is unknown, $b:\Omega\to\mathbb{R}^{n}$ and
$f:\Omega\to\mathbb{R}^{n}$ are given functions. In particular, we are
interested in the drift of type

$n=3,\qquad b(x)=-\alpha\,\frac{x^{\prime}}{|x^{\prime}|^{2}},\qquad
x^{\prime}=(x_{1},x_{2},0),\qquad\alpha\in\mathbb{R}.$ (1.2)

Motivated by this example, we assume that the drift $b$ satisfies the
condition

$b\in L^{2,n-2}_{w}(\Omega),$ (1.3)

where $L^{p,\lambda}_{w}(\Omega)$ is the weak Morrey space equipped with the
quasi-norm

$\|b\|_{L^{p,\lambda}_{w}(\Omega)}\ :=\
\sup\limits_{x_{0}\in\Omega}\sup\limits_{R<\mathop{\rm
diam}\Omega}\,R^{-\frac{\lambda}{p}}\,\|b\|_{L_{p,w}(B_{R}(x_{0})\cap\Omega)},$

and $L_{p,w}(\Omega)$ is the weak Lebesgue space equipped with the quasi-norm

$\|b\|_{L_{p,w}(\Omega)}\ :=\
\sup\limits_{s>0}s~{}|\\{~{}x\in\Omega:~{}|b(x)|>s~{}\\}|^{\frac{1}{p}},$
(1.4)

where $|\Omega|$ denotes the $n$-dimensional Lebesgue measure of $\Omega$.
Note that the condition (1.3) implies in particular that $b$ belongs to the
usual Morrey space $L^{p,n-p}(\Omega)$ for any $p\in[1,2)$ (see Proposition
2.1 below and the definition of Morrey spaces at the end of this section).
Note also that both scales of spaces $L^{p,n-p}(\Omega)$ and
$L^{p,n-p}_{w}(\Omega)$ are critical ones, i.e. these spaces are invariant
under the scaling

$b^{\lambda}(x)=\lambda\,b(\lambda
x),\quad\lambda>0\quad\Longrightarrow\quad\quad\|b^{\lambda}\|_{L^{p,n-p}_{w}(\Omega_{\lambda})}=\|b\|_{L^{p,n-p}_{w}(\Omega)},$
(1.5)

where we assume $\Omega$ is star-shaped and denote
$\Omega_{\lambda}:=\\{\,x\in\mathbb{R}^{n}:\,x/\lambda\in\Omega\,\\}$.
Concerning further properties of weak Morrey spaces we refer to [12], [39].
Here we emphasise only that the scale of weak Morrey spaces
$L_{w}^{p,n-p}(\Omega)$ is convenient for the description of drifts $b$ which
have a certain asymptotic near some singular submanifold
$\Sigma\subset\Omega$. For example, for $n=3$ we have

$\begin{array}[]{ccl}b(x)\sim\frac{1}{x_{3}},&\Sigma\ =\mbox{ a plane
}&\quad\Longrightarrow\quad b\in L^{1,2}_{w}(\Omega),\phantom{\Big{|}}\\\
b(x)\sim\frac{1}{\sqrt{x_{1}^{2}+x_{2}^{2}}},&\Sigma\ =\mbox{ a line
}&\quad\Longrightarrow\quad b\in L^{2,1}_{w}(\Omega),\phantom{\Big{|}}\\\
b(x)\sim\frac{1}{\sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}},&\Sigma\ =\mbox{ a
point }&\quad\Longrightarrow\quad b\in
L^{3,0}_{w}(\Omega)=L_{3,w}(\Omega).\phantom{\Big{|}}\\\ \end{array}$

We define the bilinear form $\mathcal{B}[u,\eta]$ by

$\mathcal{B}[u,\eta]\ :=\ \int\limits_{\Omega}\eta\,b\cdot\nabla u~{}dx.$
(1.6)

Note that for $b$ satisfying (1.3) the bilinear form (1.6) generally speaking
is not well-defined for $u\in W^{1}_{2}(\Omega)$ and $\eta\in
C_{0}^{\infty}(\Omega)$ where

$W^{1}_{p}(\Omega)\,:=\,\\{\,u\in L_{p}(\Omega):\,\nabla u\in
L_{p}(\Omega)\,\\},\qquad p\in[1,+\infty],$

is the usual Sobolev space and $C_{0}^{\infty}(\Omega)$ is the linear space of
$C^{\infty}$–smooth functions compactly supported in $\Omega$.

Nevertheless, the bilinear form $\mathcal{B}[u,\eta]$ is well-defined at least
for $u\in W^{1}_{p}(\Omega)$ with $p>2$ and $\eta\in L_{q}(\Omega)$ with
$q>\frac{2p}{p-2}$. So, instead of the standard notion of weak solutions from
the energy class $W^{1}_{2}(\Omega)$ we introduce the definition of $p$-weak
solutions to the problem (1.1), see also [22], [24], [31]:

######  Definition 1.1.

Assume $p>2$, $b\in L_{p^{\prime}}(\Omega)$, $p^{\prime}=\frac{p}{p-1}$, and
$f\in L_{1}(\Omega)$. We say $u$ is a $p$-weak solution to the problem (1.1)
if $u\in\overset{\circ}{W}{{}^{1}_{p}}(\Omega)$ and $u$ satisfies the identity

$\begin{gathered}\int\limits_{\Omega}\nabla u\cdot\nabla\eta\,dx\ +\
\mathcal{B}[u,\eta]\ =\
\int\limits_{\Omega}f\cdot\nabla\eta~{}dx,\qquad\forall~{}\eta\in
C_{0}^{\infty}(\Omega).\end{gathered}$ (1.7)

If $b\in L_{2}(\Omega)$ and $u\in\overset{\circ}{W}{{}^{1}_{2}}(\Omega)$
satisfy (1.7) then we call $u$ a weak solution to the problem (1.1).
Obviously, in this case $p$-weak solutions are some subclass of weak
solutions.

Note that if $u$ is a weak or a $p$-weak solution to (1.1) and $f\in
L_{2}(\Omega)$ then by density arguments we can extend the class of test
functions in (1.7) from $\eta\in C_{0}^{\infty}(\Omega)$ to all functions
$\eta\in\overset{\circ}{W}{{}^{1}_{2}}(\Omega)\cap L_{\infty}(\Omega)$.

In this paper we always assume that the vector field $b$ has a sign-defined
divergence:

$\mathop{\rm div}b\,\leq\,0\quad\mbox{in}\quad\mathcal{D}^{\prime}(\Omega),$
(1.8)

where $\mathcal{D}^{\prime}(\Omega)$ is the space of distributions on
$\Omega$. In the case (1.2) the condition (1.8) corresponds to $\alpha\geq 0$.
The important impact of the condition (1.8) is due to the fact that in this
case the quadratic form $\mathcal{B}[u,u]$ provides a positive support to the
quadratic form of the elliptic operator in (1.1) (see Proposition 2.2 below),
while in the opposite case $\mathop{\rm div}b\geq 0$ in $\Omega$ the quadratic
form $\mathcal{B}[u,u]$ is (formally) non-positive and hence it “shifts” the
operator to the “spectral area”. It is well-known that in the case of a non-
regular drift $b$ the violation of the condition (1.8) potentially can ruin
the uniqueness for the problem (1.1) even in the class of smooth solutions.
For example, in the case $\Omega=B$ where
$B:=\\{\,x\in\mathbb{R}^{n}:\,|x|<1\,\\}$ the functions

$u(x)=c(|x|^{2}-1),\quad b(x)=n\,\tfrac{x}{|x|^{2}},\quad b\in
L_{n,w}(B),\quad\mathop{\rm div}b\geq 0\ \ \mbox{ in }\ \
\mathcal{D}^{\prime}(B)$ (1.9)

satisfy (1.1) with $f\equiv 0$. Note also that in the regular case $b\in
L_{n}(\Omega)$ the uniqueness for the problem (1.1) holds regardless of the
sign of $\mathop{\rm div}b$ due to the maximum principle, see, for example,
[16] and reference there.

The main results of the present paper are the following two theorems:

######  Theorem 1.1.

Assume $n\geq 3$, $\Omega\subset\mathbb{R}^{n}$ is a bounded Lipschitz domain
and $b$ satisfies (1.3), (1.8). Then there exist $p>2$ depending only on $n$,
$\Omega$ and $\|b\|_{L^{2,n-2}_{w}(\Omega)}$ such that for any $f\in
L_{p}(\Omega)$ there exists a unique $p$-weak solution $u\in
W^{1}_{p}(\Omega)$ to the problem (1.1). Moreover, this solution satisfies the
estimate

$\|u\|_{W^{1}_{p}(\Omega)}\ \leq\ c~{}\|f\|_{L_{p}(\Omega)},$ (1.10)

with a constant $c>0$ depending only on $n$, $\|b\|_{L^{2,n-2}_{w}(\Omega)}$
and the Lipschitz constant of $\partial\Omega$.

######  Theorem 1.2.

Assume $n\geq 3$, $\Omega\subset\mathbb{R}^{n}$ is a bounded Lipschitz domain
and $b$ satisfies (1.3), (1.8) and $p>2$. Then for any $q>n$ there exists
$\mu\in(0,1)$ depending only on $n$, $p$, $q$, $\Omega$ and
$\|b\|_{L^{2,n-2}_{w}(\Omega)}$ such that if $u$ is a $p$–weak solution to the
problem (1.1) corresponding to the right-hand side $f\in L_{q}(\Omega)$ then
$u$ is Hölder continuous on $\bar{\Omega}$ with the exponent $\mu$ and the
estimate

$\|u\|_{C^{\mu}(\bar{\Omega})}\ \leq\ c~{}\|f\|_{L_{q}(\Omega)},$ (1.11)

holds with the constant $c>0$ depending only on $q$, $p$, $n$,
$\|b\|_{L^{2,n-2}_{w}(\Omega)}$ and the Lipschitz constant of
$\partial\Omega$.

The study of the weak solvability and properties of weak solutions to drift-
diffusion equations has a long history. Below we focus on the brief overview
of results concerning the existence of weak solutions, their uniqueness,
boundedness and Hölder continuity.

Very often the study of the problem (1.1) is coupled with the study of the
dual problem

$\left\\{\quad\begin{gathered}-\Delta v-\mathop{\rm div}(bv)\,=\,\mathop{\rm
div}g\qquad\mbox{in}\quad\Omega,\\\ v|_{\partial\Omega}\ =\
0.\qquad\quad\end{gathered}\right.$ (1.12)

In the case of the divergence–free drift

$\mathop{\rm div}b=0\quad\mbox{in}\quad\mathcal{D}^{\prime}(\Omega)$ (1.13)

problems (1.1) and (1.12) are equivalent, but for the general drifts it is not
the case. The reason is that the dual problem (1.12) is presented in the
divergent form. Hence weak solutions, depending on the regularity of $b$, in
principle are well-defined in a wider class than the energy space
$W^{1}_{2}(\Omega)$ (one can consider classes of very weak solutions,
renormalized solutions etc.) For example, in a series of papers [4], [5], [6]
(see also [7] and reference there) the existence of weak (distributional)
solutions for the problem (1.12) in various functional classes under various
critical and supercritical assumptions on $b$ was studied.

Another approach to the solvability of the problem (1.1) with the rough
coefficients was developed in a series of papers [26], [27], [28], [29] (see
also references there). This approach is applicable to elliptic operators in
the non-divergent form (with the main part
$-a_{jk}(x)\frac{\partial^{2}u}{\partial x_{j}\partial x_{k}}$ with a non-
smooth uniformly elliptic matrix $a(x)=(a_{jk}(x))$ instead of the Laplacian)
and the drift $b$ belonging to the critical Morrey spaces. In this approach
the existence of solutions in Sobolev spaces $W^{2}_{p}(\Omega)$ with some
$p>1$ is investigated and the equation (1.1) is understood a.e. in $\Omega$.

For the problem (1.1) weak solutions from $W^{1}_{2}(\Omega)$ formally are
well-defined only for $b\in L_{2}(\Omega)$ which does not cover the case
(1.3). Nevertheless, weak solutions to elliptic and parabolic equations with
rough coefficients are commonly known to possess the higher integrability
property. This means that the gradient of a weak solution is integrable with
some exponent which is slightly greater than 2. One can ask what are the
optimal conditions on the drift $b$ which provide the higher integrability of
weak solutions to the problem (1.1)? The first results in this direction were
obtained earlier in [23], [24], [31] where the critical drift $b\in
L_{n,w}(\Omega)$ satisfying (1.8) was studied. So, our Theorem 1.1 can be
viewed as a far away extension of these results. Note also that the 2D case
was investigated separately in [8] and [32].

All results concerning weak solvability and properties of weak solutions to
the problems (1.1) and (1.12) formally can be split onto three groups: the
results related to the subcrtitical drifts, the results for critical drifts
and supercritical results. We remind that we call some functional class $X$
supercritical if for $b\in X$ and $b^{\lambda}$ defined in (1.5) the norm
$\|b^{\lambda}\|_{X(B)}$ computed over a unit ball $B$ blows up as $\lambda\to
0$.

The results concerning the drift $b$ belonging to subcritical Lesbegue spaces
are classical one, see [33]. The same is true for drifts belonging to the
critical Lebesgue space

$b\in L_{n}(\Omega).$ (1.14)

We call drifts satisfying (1.14) regular. In the regular case the drift term
provides a compact perturbation of the Laplace operator and hence the problem
(1.1) possesses the Fredholm property (i.e. the existence for (1.1) follows
from the uniqueness). Moreover, in the case (1.14) weak solutions are Hölder
continuous and the maximum principle holds (which provides the uniqueness of
weak solutions), see the survey of the results, for example, in [16]. So, we
can consider the regular case as a starting point of our investigation. Note
that the necessary and sufficient conditions for the boundedness of the
bilinear form (1.6) on the Sololev space $W^{1}_{2}(\mathbb{R}^{n})$ were
investigated in [36].

The counterexample (1.9) shows that even the minimal relaxation of the
condition (1.14) (still within the critical scale) can ruin the maximum
principle and the uniqueness for the problem (1.1). So, to recover the
uniqueness we need to impose some additional assumptions on $b$. For example,
one can consider the condition (1.8) or its particular case (1.13). (Motivated
by the properties of the quadratic form $\mathcal{B}[u,u]$ we call the
condition (1.8) “non-spectral”.) In the present paper we focus only on the
critical “non-spectral” case and we are going to investigate the critical
“spectral” case $\mathop{\rm div}b\geq 0$ in a separate forthcoming paper, see
[8] and [32] for the related 2D results.

For $b\in L_{2}(\Omega)$ satisfying (1.13) the uniqueness of weak solutions in
the energy class $W^{1}_{2}(\Omega)$ can be found in [46] (see also [45] for
the parabolic case). In [46] a counterexample to the uniqueness for the
problem (1.1) with a divergence-free drift was also constructed if the
condition $b\in L_{2}(\Omega)$ is violated (see [10] and references there for
the further developments in this direction). For $b\in L_{2}(\Omega)$
satisfying the “non-spectral” condition (1.8) the corresponding uniqueness
result in the energy class $W^{1}_{2}(\Omega)$ can be easily obtained by a
straightforward adaptation of the methods of [45], [46]. For $b\in
L_{2,w}(\Omega)$ satisfying (1.8) our Proposition 3.1 below provides a
uniqueness result for the problem (1.1) in a class of weak solutions which
possess the higher integrability property.

The next issue is the boundedness of weak solutions. It is well-known that
even in the divergence–free case (1.13) the local boundedness of weak
solutions requires some extra regularity of the drift, see [37], [16], [15],
[1], see also a recent survey [2] and reference there. Nevertheless, our
Proposition 3.1 shows that for a drift $b\in L_{2,w}(\Omega)$ satisfying (1.8)
$p$–weak solutions to the boundary value problem (1.1) are always bounded
without any extra assumptions on the regularity of $b$. We emphasize that our
result is global (i.e. it requires smooth boundary conditions) and can not be
localized. The phenomena of the dependence of local boundedness of weak
solutions on their behaviour on the boundary of the domain is discussed in
[16].

The last issue we are going to discuss is Hölder continuity of weak solutions.
Note that under the critical condition (1.14) Hölder continuity is the optimal
regularity. For the problems (1.1), (1.12) with a drift $b$ belonging to the
subcritical Morrey space $b\in L^{2,\lambda}(\Omega)$ with $\lambda\in(n-2,n)$
Hölder continuity of weak solutions is known for a long time, see [11] and
[38]. On the other hand, the counterexample in [14] shows that even in the
divergence–free case (1.13) the smallest violation of the critical scale can
lead to the loss of continuity by weak solutions. So, in this paper we focus
on results concerning drifts belonging to critical spaces.

For the divergence-free drifts from critical spaces the best known for today
results were obtained in [17], [40] (see also [41]) where Hölder continuity of
weak solutions was proved for the heat equation with a drift term satisfying
(1.13) (see also the related result in [35]). In its elliptic version, these
results assume that the drift $b$ satisfying (1.13) belongs to the space
$BMO^{-1}(\Omega)$, i.e. there exists a skew-symmetric matrix $A\in
BMO(\Omega)$ such that $b=\mathop{\rm div}A$ (see the definition of $BMO$
space at the end of this section). If we assume $\nabla A\in
L^{1,n-1}(\Omega)$ (which is in a sense the optimal regularity for the
critical Morrey scale $L^{p,n-p}(\Omega)$) then from Trudinger’s imbedding
[42] we obtain $A\in BMO(\Omega)$ and hence Hölder continuity of weak
solutions follows from [17], [40].

For the non–divergence free critical drifts satisfying the condition (1.8)
local a priori estimates of the Hölder norms for Lipschitz continuous
solutions to the problem (1.1) were obtained earlier in [37] under the
additional restriction $b\in L^{r,n-r}(\Omega)$ with $r\in(\tfrac{n}{2},n]$.
So, in our Theorem 1.2 we remove this restriction. Note also that our present
contribution can be viewed as a multidimensional analogue of the results
obtained earlier in [8] in the 2D case.

Here are some comments and possible extensions:

  * •

With minor changes in the proofs our technique allows us to obtain results
similar to our Theorems 1.1–1.2 if we replace the Laplace operator in (1.1) by
an elliptic operator in the divergence form $-\mathop{\rm div}(a(x)\nabla u)$
with a uniformly elliptic matrix $a(x)=(a_{jk}(x))$, $a_{jk}\in
L_{\infty}(\Omega)$. We consider this extension to be obvious.

  * •

In the case of the Laplace operator in (1.1) (or, more generally, in the case
of the operator $-\mathop{\rm div}(a(x)\nabla u)$ with a sufficiently smooth
matrix $a(x)$) our $p$-weak solutions in Theorem 1.1 possesses locally
integrable second derivatives $u\in W^{2}_{s,loc}(\Omega)$ with some $s>1$ (if
$f$ is sufficiently regular). Hence for $p$-weak solutions the equation (1.1)
is valid a.e. in $\Omega$. See the related results in [26], [27], [28], [29]
for the case of elliptic operators in the non-divergent form with non-smooth
coefficients.

  * •

In our investigation of Hölder continuity of weak solutions in Theorem 1.2 we
focus on the “optimal” conditions for $b$ but not for $f$. It is well-known
(see, for example, [20, Theorem 5.17]) that Hölder continuity of weak
solutions to the problem (1.1) is valid if the right hand side $f$ belongs to
some appropriate Morrey space. So, we believe that the condition $f\in
L_{q}(\Omega)$ with $q>n$ in Theorem 1.2 can be relaxed, see related results,
for example, in [11], [12].

Our paper is organized as follows. In Section 2 we introduce some auxiliary
results which are basic tools of our paper. In Section 3 we obtain the higher
integrability of weak solutions to the problem (1.1) and prove Theorems 1.1.
In Section 4 we introduce some De Giorgi–type classes which are convenient for
the study of the equations with the coefficients from Morrey spaces. These De
Giorgi classes are very similar to the classical ones (see [20], [21], [33])
and readers familiar with the De Giorgi technique can omit this section. In
Section 5 we show that $p$–weak solutions to the problem (1.1) belong to the
above De Giorgi classes and hence they are Hölder continuous. So, in Section 5
we prove Theorem 1.2. Finally, in Appendix we prove some property of the
mollification operator on the class of weak Morrey spaces.

In the paper we use the following notation. For any $a$, $b\in\mathbb{R}^{n}$
we denote by $a\cdot b=a_{k}b_{k}$ their scalar product in $\mathbb{R}^{n}$.
Repeated indexes assume the summation from 1 to $n$. An index after comma
means partial derivative with respect to $x_{k}$, i.e. $f_{,k}:=\frac{\partial
f}{\partial x_{k}}$. We denote by $L_{p}(\Omega)$ and $W^{k}_{p}(\Omega)$ the
usual Lebesgue and Sobolev spaces. We do not distinguish between functional
spaces of scalar and vector functions and omit the target space in notation.
$C_{0}^{\infty}(\Omega)$ is the space of smooth functions compactly supported
in $\Omega$. The space $\overset{\circ}{W}{{}^{1}_{p}}(\Omega)$ is the closure
of $C_{0}^{\infty}(\Omega)$ in $W^{1}_{p}(\Omega)$ norm. For $p>1$ we denote
$p^{\prime}=\frac{p}{p-1}$. The space of distributions on $\Omega$ is denoted
by $\mathcal{D}^{\prime}(\Omega)$. By $C^{\mu}(\bar{\Omega})$, $\mu\in(0,1)$
we denote the spaces of Hölder continuous functions on $\bar{\Omega}$ equipped
with the norm

$\|u\|_{C^{\mu}(\bar{\Omega})}\,:=\,\sup\limits_{x\in\Omega}|u(x)|\,+\,\sup\limits_{x,y\in\Omega:\,x\not=y}\frac{|u(x)-u(y)|}{|x-y|^{\mu}}.$

The symbols $\rightharpoonup$ and $\to$ stand for the weak and strong
convergence respectively. We denote by $B_{R}(x_{0})$ the ball in
$\mathbb{R}^{n}$ of radius $R$ centered at $x_{0}$ and write $B_{R}$ if
$x_{0}=0$. We write $B$ instead of $B_{1}$. For a domain
$\Omega\subset\mathbb{R}^{n}$ we also denote $\Omega_{R}(x_{0}):=\Omega\cap
B_{R}(x_{0})$. For an open set $\Omega_{0}\subset\mathbb{R}^{n}$ we write
$\Omega\Subset\Omega_{0}$ if $\bar{\Omega}$ is compact and
$\bar{\Omega}\subset\Omega_{0}$. For $u\in L_{\infty}(\Omega)$ we denote

$\mathop{\rm osc}\limits_{\Omega}u\,:=\,\mathop{\rm
esssup}\limits_{\Omega}u-\mathop{\rm essinf}\limits_{\Omega}u.$

We denote by $L^{p,\lambda}(\Omega)$ the Morrey space equipped with the norm

$\|u\|_{L^{p,\lambda}(\Omega)}\ :=\
\sup\limits_{x_{0}\in\Omega}\sup\limits_{R<\mathop{\rm
diam}\Omega}R^{-\frac{\lambda}{p}}\|u\|_{L_{p}(\Omega_{R}(x_{0}))}.$

We denote by $\chi_{\omega}$ the indicator function of the set
$\omega\subset\mathbb{R}^{n}$. For $u:\Omega\to\mathbb{R}$ and
$k\in\mathbb{R}$ we denote $(u-k)_{+}:=\max\\{u-k,0\\}$. We denote by
$BMO(\Omega)$ the space of functions with a bounded mean oscillation equipped
with the semi-norm

$[u]_{BMO(\Omega)}\ :=\ \sup\limits_{x_{0}\in\Omega}\sup\limits_{R<\mathop{\rm
diam}\Omega}R^{-n}\int\limits_{\Omega_{R}(x_{0})}|u(x)-(u)_{\Omega_{R}(x_{0})}|\,dx,$

where $(f)_{\omega}$ stands for the average of $f$ over the domain
$\omega\subset\mathbb{R}^{n}$. Also we denote by $f*g$ the convolution of
functions $f$, $g:\mathbb{R}^{n}\to\mathbb{R}$:

$(f)_{\omega}\ :=\ -\\!\\!\\!\\!\\!\\!\int\limits_{\omega}f~{}dx\ =\
\frac{1}{|\omega|}~{}\int\limits_{\omega}f~{}dx,\qquad(f*g)(x)\ :=\
\int\limits_{\mathbb{R}^{n}}f(x-y)g(y)~{}dy.$

Acknowledgement. The research of Tim Shilkin was supported by the project
20-11027X financed by Czech science foundation (GAČR). The research of Misha
Chernobai was partially supported by Natural Sciences and Engineering Research
Council of Canada (NSERC) grants RGPIN-2018-04137 and RGPIN-2023-04534.

##  2 Auxiliary results

In this section we present several auxiliary results. For reader’s convenience
we formulate these results not in their full generality but only in the form
they are used in our paper.

The first result shows that by relaxing an exponent of the integrability of a
function we can always switch from a weak Morrey norm to a regular one.

######  Proposition 2.1.

For any $p\in(1,n)$ and $1\leq q<p\leq r\leq n$ there are positive constants
$c_{1}$ and $c_{2}$ depending only on $n$, $p$ , $q$ and $r$ such that

$c_{1}\,\|b\|_{L^{q,n-q}(\Omega)}\,\leq\,\|b\|_{L^{p,n-p}_{w}(\Omega)}\,\leq\,c_{2}\,\|b\|_{L^{r,n-r}(\Omega)}$

###### Proof.

The result follows from the Hölder inequality for Lorentz norms, see [18,
Section 4.1]. ∎

The next result shows that under the condition (1.8) the quadratic form
corresponding the drift term is non-negative.

######  Proposition 2.2.

Assume $p\geq 2$, $p^{\prime}=\frac{p}{p-1}$, and $b\in
L_{p^{\prime}}(\Omega)$ satisfies (1.8). Let $\mathcal{B}[u,\eta]$ be the
bilinear form defined in (1.6). Then

$\forall\,u\in
L_{\infty}(\Omega)\cap\overset{\circ}{W}{{}^{1}_{p}}(\Omega)\qquad\mathcal{B}[u,u]\
\geq\ 0.$ (2.1)

###### Proof.

The result follows from the inclusions
$|u|^{2}\in\overset{\circ}{W}{{}^{1}_{p}}(\Omega)$, $b\in
L_{p^{\prime}}(\Omega)$, $p^{\prime}=\frac{p}{p-1}$, and the inequality

$\mathcal{B}[u,u]=\ \frac{1}{2}\,\int\limits_{\Omega}b\cdot\nabla|u|^{2}\,dx\
\geq\ 0.$

∎

The next result is the general Marcinkiewicz interpolation theorem in Lorentz
spaces, see [3, Theorem 5.3.2].

######  Proposition 2.3.

Assume $\Omega$, $\Omega^{\prime}\subset\mathbb{R}^{n}$ are bounded domains
and assume for some $1<p_{0}<p_{1}<\infty$ a linear operator
$T:L_{p_{j}}(\Omega)\to L_{p_{j}}(\Omega^{\prime})$ is bounded for $j=0,1$ and

$\|T\|_{L_{p_{0}}(\Omega)\to
L_{p_{0}}(\Omega^{\prime})}\,\leq\,M_{0},\qquad\|T\|_{L_{p_{1}}(\Omega)\to
L_{p_{1}}(\Omega^{\prime})}\,\leq\,M_{1}$

Then for any $p\in(p_{0},p_{1})$ the operator $T$ is bounded from
$L_{p,w}(\Omega)$ to $L_{p,w}(\Omega^{\prime})$ and

$\|T\|_{L_{p,w}(\Omega)\to
L_{p,w}(\Omega^{\prime})}\,\leq\,M_{0}^{1-\theta}M_{1}^{\theta}$

where $\theta\in(0,1)$ satisfies
$\frac{1}{p}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}}$.

The next proposition proved by Chiarenza and Frasca in [9] is the well-known
extension of the result of C. Fefferman [13] for $p=2$. This theorem is one of
basic tools in our proofs of both Theorems 1.1 and 1.2.

######  Proposition 2.4.

Assume $p\in(1,n)$, $r\in(1,\frac{n}{p}]$ and $V\in L^{r,n-pr}(\Omega)$. Then

$\int\limits_{\Omega}|V|\,|u|^{p}\,dx\ \leq\
c_{n,r,p}\,\|V\|_{L^{r,n-rp}(\Omega)}\|\nabla
u\|_{L_{p}(\Omega)}^{p},\qquad\forall\,u\in C_{0}^{\infty}(\Omega).$

with the constant $c_{n,r,p}>0$ depending only on $n$, $r$ and $p$.

##  3 Higher integrability of weak solutions

In this section we present the proof of Theorem 1.1. First we show that in the
case of (1.8) any $p$–weak solution corresponding to a sufficiently regular
right-hand side is bounded, see the related result in [34].

######  Proposition 3.1.

Assume $p\geq 2$, $p^{\prime}=\frac{p}{p-1}$, $b\in L_{p^{\prime}}(\Omega)$
satisfies (1.8), $q>n$ and $f\in L_{q}(\Omega)$. Then any weak solution $u\in
W^{1}_{p}(\Omega)$ to the problem (1.1) (in the sense of the integral identity
(1.7)) is essentially bounded and satisfies the estimate

$\|u\|_{L_{\infty}(\Omega)}\ \leq\ c~{}\|f\|_{L_{q}(\Omega)},$ (3.1)

with a constant $c>0$ depending only on $\Omega$, $n$ and $q$.

###### Proof.

For $m>0$ we define a truncation $T_{m}:\mathbb{R}\to\mathbb{R}$ by
$T_{m}(s):=m$ for $s\geq m$, $T_{m}(s)=s$ for $s<m$. For any $s\in\mathbb{R}$
we also denote $(s)_{+}:=\max\\{s,0\\}$. Now we fix some $m>0$ and denote
$\bar{u}:=T_{m}(u)$. Then for any $k\geq 0$ we have

$(\bar{u}-k)_{+}\in
L_{\infty}(\Omega)\cap\overset{\circ}{W}{{}^{1}_{p}}(\Omega),\qquad\nabla(\bar{u}-k)_{+}=\chi_{\Omega[k<u<m]}\nabla
u$

where $\Omega[k<u<m]=\\{\,x\in\Omega:\,k<u(x)<m\,\\}$. Approximating
$\eta:=(\bar{u}-k)_{+}$ by smooth functions we can take $\eta$ as a test
function in (1.7). For $k\geq m$ we have $\eta\equiv 0$ and hence
$\mathcal{B}[u,\eta]=0$. For $k<m$ the condition (1.8) implies

$\begin{gathered}\mathcal{B}[u,\eta]\ =\
\tfrac{1}{2}\,\int\limits_{\Omega[k<u<m]}b\cdot\nabla|\eta|^{2}\,dx\ +\
(m-k)\,\int\limits_{\Omega[u\geq m]}b\cdot\nabla u\,dx\\\
=\mathcal{B}[\eta,\eta]\ +\
(m-k)\,\int\limits_{\Omega}b\cdot\nabla(u-m)_{+}\,dx\ \geq\
\mathcal{B}[\eta,\eta]\ \geq\ 0\end{gathered}$

Hence for $A_{k}:=\\{\,x\in\Omega:\,\bar{u}(x)>k\,\\}$ from (1.7) we obtain

$\int\limits_{\Omega}|\nabla(\bar{u}-k)_{+}|^{2}\,dx\ \leq\
\|f\|_{L_{q}(\Omega)}^{2}|A_{k}|^{1-\frac{2}{q}},\qquad\forall\,k\geq 0,$

which implies (see [33, Chapter II, Lemma 5.3])

$\mathop{\rm esssup}\limits_{\Omega}\bar{u}\ \leq\
c\,|\Omega|^{\delta}\,\|f\|_{L_{q}(\Omega)},\qquad\delta:=\tfrac{1}{n}-\tfrac{1}{q},$

with some constant $c>0$ depending only on $n$ and $q$. As this estimate is
uniform with respect to $m>0$ we conclude $u$ is essentially bounded from
above and

$\mathop{\rm esssup}\limits_{\Omega}u\ \leq\
c\,|\Omega|^{\delta}\,\|f\|_{L_{q}(\Omega)}.$

Applying the same procedure to $\bar{u}:=T_{m}(-u)$ instead of
$\bar{u}:=T_{m}(u)$ we obtain $u\in L_{\infty}(\Omega)$ as well as (3.1). ∎

Proposition 3.1 implies the uniqueness theorem for the problem (1.1) in the
class of $p$-weak solutions.

######  Proposition 3.2.

Assume $p\geq 2$, $p^{\prime}=\frac{p}{p-1}$, and $b\in
L_{p^{\prime}}(\Omega)$ satisfies (1.8). Then for any $f\in L_{p}(\Omega)$ the
problem (1.1) can have at most one weak solution in the class
$W^{1}_{p}(\Omega)$.

###### Proof.

Assume $u_{1}\in W^{1}_{p}(\Omega)$ and $u_{2}\in W^{1}_{p}(\Omega)$ are two
weak solutions of (1.1) corresponding to the same $f\in L_{p}(\Omega)$. Then
$u:=u_{1}-u_{2}$ is a weak solution of (1.1) corresponding to $f\equiv 0$.
From Proposition 3.1 we obtain $u\equiv 0$ in $\Omega$. ∎

The next proposition is the energy estimate. Note that the constant $c>0$ in
this estimate does not depend on the drift $b$ satisfying the condition (1.8).

######  Proposition 3.3.

Assume $p\geq 2$, $p^{\prime}=\frac{p}{p-1}$, $b\in L_{p^{\prime}}(\Omega)$
satisfies (1.8), and $f\in L_{q}(\Omega)$ with $q>n$. Then for any weak
solution $u\in W^{1}_{p}(\Omega)$ to the problem (1.1) the following estimate
hold:

$\|u\|_{W^{1}_{2}(\Omega)}\ \leq\ c~{}\|f\|_{L_{2}(\Omega)}$

with some constant $c>0$ depending only on $n$ and $\Omega$.

###### Proof.

From Proposition 3.1 we obtain $u\in L_{\infty}(\Omega)$ and hence the test
function $\eta:=u\in
L_{\infty}(\Omega)\cap\overset{\circ}{W}{{}^{1}_{p}}(\Omega)$ is admissible
for (1.7). From Proposition 2.2 we obtain $\mathcal{B}[u,u]\geq 0$ and hence
from (1.7) we obtain the inequality

$\|\nabla u\|_{L_{2}(\Omega)}^{2}\ \leq\ \int\limits_{\Omega}f\cdot\nabla
u\,dx$

which the result follows from. ∎

Our next result is the estimate of the bilinear form corresponding to the
drift term. This result is a direct consequence of the
Feffermann–Chiarenza–Frasca estimate, see Proposition 2.4.

######  Proposition 3.4.

Assume $r\in\left(\frac{2n}{n+2},2\right)$ and $b\in L^{r,n-r}(\Omega)$. Then
there exists $c>0$ depending only on $n$ and $r$ such that for any $u\in
W^{1}_{2}(\Omega)$ and $\zeta\in C_{0}^{\infty}(\Omega)$ the following
estimate holds:

$\int\limits_{\Omega}\zeta^{3}\,|b|\,|u|^{2}\,dx\ \leq\
c\,\|b\|_{L^{r,n-r}(\Omega)}\,\|\nabla(\zeta^{2}u)\|_{L_{2}(\Omega)}\,\|\nabla(\zeta
u)\|_{L_{\frac{2n}{n+2}}(\Omega)}.$ (3.2)

Moreover, for any $\theta\in\left(\frac{n}{r}-\frac{n}{2},1\right)$ there
exists $c>0$ depending only on $n$, $r$ and $\theta$ such that for any $u\in
W^{1}_{2}(\Omega)$ and any $\zeta\in C_{0}^{\infty}(\Omega)$ satisfying
$0\leq\zeta\leq 1$ we have

$\int\limits_{\Omega}\zeta^{1+\theta}\,|b|\,|u|^{2}\,dx\ \leq\
c\,\|b\|_{L^{r,n-r}(\Omega)}\|u\|_{L_{2}(\Omega)}^{1-\theta}\|\nabla(\zeta
u)\|_{L_{2}(\Omega)}^{1+\theta}\,|\Omega|^{\theta/n}.$ (3.3)

If we additionally assume $u\in\overset{\circ}{W}{{}^{1}_{2}}(\Omega)$ then
the estimates (3.2) and (3.3) remains true for an arbitrary $\zeta\in
C_{0}^{\infty}(\mathbb{R}^{n})$.

###### Proof.

To prove (3.2) we apply the Hölder inequality

$\int\limits_{\Omega}\zeta^{3}\,|b|\,|u|^{2}\,dx\ \leq\
c\,\|\zeta^{2}u\|_{L_{\frac{2n}{n-2}}(\Omega)}\|\zeta
bu\|_{L_{\frac{2n}{n+2}}(\Omega)}.$

Denote $q:=\frac{2n}{n+2}$, $V:=|b|^{q}$, $r_{1}:=\frac{r}{q}$, $r_{1}>1$.
Then

$V\in
L^{r_{1},n-qr_{1}}(\Omega),\qquad\|V\|_{L^{r_{1},n-qr_{1}}(\Omega)}\,=\,\|b\|^{q}_{L^{r,n-r}(\Omega)}$

and from Proposition 2.4 we obtain

$\|\zeta bu\|_{L_{q}(\Omega)}\ \leq\
c\,\|V\|_{L^{r_{1},n-qr_{1}}(\Omega)}^{1/q}\|\nabla(\zeta
u)\|_{L_{q}(\Omega)}\ \leq\ c\,\|b\|_{L^{r,n-r}(\Omega)}\,\|\nabla(\zeta
u)\|_{L_{q}(\Omega)}.$

On the other hand, by the Gagliardo-Nirenberg inequality we obtain

$\|\zeta^{2}u\|_{L_{\frac{2n}{n-2}}(\Omega)}\,\leq\,c\,\|\nabla(\zeta^{2}u)\|_{L_{2}(\Omega)}$

and we arrive at (3.2).

Now we turn to the proof of (3.3). Take $q\in\left(\frac{2n}{n+2},r\right)$
such that

$\tfrac{1}{q^{\prime}}=\tfrac{1-\theta}{2}+\theta\,\tfrac{n-2}{2n},\qquad
q^{\prime}=\tfrac{q}{q-1}.$

Then

$\int\limits_{\Omega}\zeta^{1+\theta}\,|b|\,|u|^{2}\,dx\ \leq\
c\,\|\zeta^{\theta}u\|_{L_{q^{\prime}}(\Omega)}\|\zeta bu\|_{L_{q}(\Omega)}.$

Denote $V:=|b|^{q}$, $r_{1}:=\frac{r}{q}$, $r_{1}>1$. Then $V\in
L^{r_{1},n-qr_{1}}(\Omega)$ and

$\|\zeta bu\|_{L_{q}(\Omega)}\ \leq\
c\,\|V\|_{L^{r_{1},n-qr_{1}}(\Omega)}^{1/q}\|\nabla(\zeta
u)\|_{L_{q}(\Omega)}\ \leq\ c\,\|b\|_{L^{r,n-r}(\Omega)}\,\|\nabla(\zeta
u)\|_{L_{q}(\Omega)}$

On the other hand, we can interpolate

$\begin{gathered}\|\zeta^{\theta}u\|_{L_{q^{\prime}}(\Omega)}\,=\|(\zeta|u|)^{\theta}|u|^{1-\theta}\|_{L_{q^{\prime}}(\Omega)}\,\leq\\\
\leq\,\|(\zeta|u|)^{\theta}\|_{L^{\frac{2n}{(n-2)\theta}}(\Omega)}\||u|^{1-\theta}\|_{L_{\frac{2}{1-\theta}}(\Omega)}\leq\,c\,\|\nabla(\zeta
u)\|_{L_{2}(\Omega)}^{\theta}\|u\|_{L_{2}(\Omega)}^{1-\theta}\end{gathered}$

with some constant $c>0$ depending only on $n$. So, we obtain

$\int\limits_{\Omega}\zeta^{1+\theta}\,|b|\,|u|^{2}\,dx\ \leq\
c\,\|u\|_{L_{2}(\Omega)}^{1-\theta}\|\nabla(\zeta
u)\|_{L_{2}(\Omega)}^{\theta}\,\|b\|_{L^{r,n-r}(\Omega)}\,\|\nabla(\zeta
u)\|_{L_{q}(\Omega)}.$

Using the Hölder inequality we obtain

$\|\nabla(\zeta u)\|_{L_{q}(\Omega)}\ \leq\ \|\nabla(\zeta
u)\|_{L_{2}(\Omega)}|\Omega|^{\frac{1}{q}-\frac{1}{2}}\ =\ \|\nabla(\zeta
u)\|_{L_{2}(\Omega)}|\Omega|^{\frac{\theta}{n}}$

which completes the proof. ∎

The next proposition is the higher integrability estimate which is the main
result of the present section. In this proposition we assume that $b$ is
smooth and we derive (1.10) as an a priori estimate for a sufficiently regular
drift.

######  Proposition 3.5.

Assume $b\in C^{\infty}(\bar{\Omega})$ satisfies (1.8) and assume $f\in
L_{q}(\Omega)$ with $q>n$. Let $u\in\overset{\circ}{W^{1}_{2}}(\Omega)\cap
L_{\infty}(\Omega)$ satisfy (1.1) in terms of distributions (i.e. the identity
(1.7) holds). Then there exists $p>2$ depending only on $n$, $q$ , $\Omega$
and $\|b\|_{L^{2,n-2}_{w}(\Omega)}$ such that
$u\in\overset{\circ}{W^{1}_{p}}(\Omega)$ and the estimate (1.10) holds with
some constant $c>0$ depending only on $n$, $q$,
$\|b\|_{L^{2,n-2}_{w}(\Omega)}$ and the Lipschitz constant of
$\partial\Omega$.

###### Proof.

Let us fix some $r\in(\frac{2n}{n+2},2)$. First we consider an internal point
$x_{0}\in\Omega$. Take $R>0$ such that $B_{2R}(x_{0})\subset\Omega$ and choose
a cut-off function $\zeta\in C_{0}^{\infty}(B_{2R}(x_{0}))$ so that
$\zeta\equiv 1$ on $B_{R}(x_{0})$ and $|\nabla\zeta|\leq c/R$. Denote
$\bar{u}=u-(u)_{B_{2R}(x_{0})}$. Then the function $\eta=\zeta^{4}\bar{u}$ is
admissible for (1.7). It is easy to see that

$\mathcal{B}[u,\zeta^{4}\bar{u}]\ =\
\mathcal{B}[\zeta^{2}\bar{u},\zeta^{2}\bar{u}]\ -\
2\,\int\limits_{B_{2R}(x_{0})}\zeta^{3}|\bar{u}|^{2}\,b\cdot\nabla\zeta\,dx.$

Taking into account (2.1) from (1.7) we obtain

$\|\zeta^{2}\nabla u\|_{L_{2}(B_{2R}(x_{0}))}^{2}\ \leq\
c\,\Big{(}\|\bar{u}\nabla\zeta\|_{L_{2}(B_{2R}(x_{0}))}^{2}+\|f\|_{L_{2}(B_{2R}(x_{0}))}^{2}\Big{)}\
+\
c\,\int\limits_{B_{2R}(x_{0})}\zeta^{3}|\bar{u}|^{2}\,b\cdot\nabla\zeta~{}dx.$

We estimate the last term with the help of (3.2):

$\begin{gathered}\int\limits_{B_{2R}(x_{0})}\zeta^{3}|\bar{u}|^{2}\,b\cdot\nabla\zeta~{}dx\
\leq\\\ \leq\
\|\nabla\zeta\|_{L_{\infty}(B_{2R}(x_{0}))}\,\|b\|_{L^{r,n-r}(\Omega)}\,\|\nabla(\zeta^{2}\bar{u})\|_{L_{2}(B_{2R}(x_{0}))}\,\|\nabla(\zeta\bar{u})\|_{L_{\frac{2n}{n+2}}(B_{2R}(x_{0}))}.\end{gathered}$

Using the Young inequality we get

$\begin{gathered}\|\zeta^{2}\nabla u\|_{L_{2}(B_{2R}(x_{0}))}^{2}\ \leq\
c\,\Big{(}\|\bar{u}\nabla\zeta\|_{L_{2}(B_{2R}(x_{0}))}^{2}+\|f\|_{L_{2}(B_{2R}(x_{0}))}^{2}\Big{)}\
+\\\ +\
c\,\|\nabla\zeta\|_{L_{\infty}(B_{2R}(x_{0}))}^{2}\,\|b\|_{L^{r,n-r}(\Omega)}^{2}\,\|\nabla(\zeta\bar{u})\|_{L_{\frac{2n}{n+2}}(B_{2R}(x_{0}))}^{2}.\end{gathered}$
(3.4)

Taking into account $\zeta\equiv 1$ on $B_{R}(x_{0})$ and
$\|\nabla\zeta\|_{L_{\infty}(B_{2R}(x_{0}))}\leq\frac{c}{R}$ and the
inequalities

$\|\bar{u}\|_{L_{2}(B_{2R}(x_{0}))}\,\leq\,c\,\|\nabla
u\|_{L_{\frac{2n}{n+2}}(B_{2R}(x_{0}))},$ (3.5)
$\|\nabla(\zeta\bar{u})\|_{L_{\frac{2n}{n+2}}(B_{2R}(x_{0}))}\,\leq\,c\,\|\nabla
u\|_{L_{\frac{2n}{n+2}}(B_{2R}(x_{0}))},$

we arrive at

$\|\nabla u\|_{L_{2}(B_{R}(x_{0}))}\ \leq\ c\,\|f\|_{L_{2}(B_{2R}(x_{0}))}\ +\
\frac{c}{R}\,\Big{(}1+\|b\|_{L^{r,n-r}(\Omega)}\Big{)}\,\|\nabla
u\|_{L_{\frac{2n}{n+2}}(B_{2R}(x_{0}))}.$

Here $c$ is a global constant that depends only on $r$. Next we derive similar
estimates near the boundary. Extend $u$ and $f$ by zero outside $\Omega$ and
denote this extension by $\tilde{u}$ and $\tilde{f}$. Note that as
$\partial\Omega$ is Lipschitz there exist $R_{0}>0$ and $\delta_{0}>0$
depending on $\Omega$ such that

$\forall\,R<R_{0},\qquad\forall\,x_{0}\in\partial\Omega\qquad|B_{R}(x_{0})\setminus\Omega|\,\geq\,\delta_{0}|B_{R}|.$
(3.6)

Take $x_{0}\in\partial\Omega$, $R<R_{0}/2$ and denote
$\Omega_{R}(x_{0}):=B_{R}(x_{0})\cap\Omega$. Choose a cut-off function
$\zeta\in C_{0}^{\infty}(B_{2R}(x_{0}))$ so that $\zeta\equiv 1$ on
$B_{R}(x_{0})$. Then the function

$\eta\ :=\ \zeta^{4}u\ \in\
L_{\infty}(\Omega_{2R}(x_{0}))\cap\overset{\circ}{W}{{}^{1}_{2}}(\Omega_{2R}(x_{0}))$

is admissible for the identity (1.7). Proceeding similar to the internal case
we arrive at the inequality (3.4) with $\bar{u}$ replaces by $\tilde{u}$. Then
instead of (3.5) we use the inequality

$\|\tilde{u}\|_{L_{2}(B_{2R}(x_{0}))}\,\leq\,c\,\|\nabla\tilde{u}\|_{L_{\frac{2n}{n+2}}(B_{2R}(x_{0}))},$

which holds with some constant $c>0$ depending only on $n$ and $\delta_{0}$ in
(3.6) (and independent on $x_{0}$ and $R$) as $\tilde{u}$ satisfies
$|\\{\,x\in B_{2R}(x_{0}):\,\tilde{u}(x)=0\,\\}|\geq\delta_{0}|B_{2R}|$. Then
after routine computations we obtain for any $x_{0}\in\partial\Omega$ and
$R<R_{0}$

$\|\nabla u\|_{L_{2}(\Omega_{R}(x_{0}))}\ \leq\
c\,\|f\|_{L_{2}(\Omega_{2R}(x_{0}))}\ +\
\frac{c}{R}\,\Big{(}1+\|b\|_{L^{r,n-r}(\Omega)}\Big{)}\,\|\nabla
u\|_{L_{\frac{2n}{n+2}}(\Omega_{2R}(x_{0}))}.$

Combining internal and boundary estimates in the standard way and dividing the
result by $R^{n/2}$ for any $x_{0}\in\bar{\Omega}$ and $R<R_{0}$ we obtain the
reverse Hölder inequality for $\nabla u$:

$\begin{gathered}\left(\ \ \ \
-\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{\Omega_{R}(x_{0})}|\nabla
u|^{2}~{}dx\right)^{\frac{1}{2}}\ \leq\ c_{1}\,\left(\ \ \ \
-\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{\Omega_{2R}(x_{0})}|f|^{2}~{}dx\right)^{\frac{1}{2}}\
+\\\ +\ c_{2}\,\Big{(}1+\|b\|_{L^{r,n-r}(\Omega)}\Big{)}\left(\ \ \ \
-\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\int\limits_{\Omega_{2R}(x_{0})}|\nabla
u|^{\frac{2n}{n+2}}~{}dx\right)^{\frac{n+2}{2n}}.\end{gathered}$

From Proposition 2.1 we conclude

$\|b\|_{L^{r,n-r}(\Omega)}\,\leq\,c\,\|b\|_{L^{2,n-2}_{w}(\Omega)}.$

Now the higher integrability of $u$ follows via Gehring’s lemma, see, for
example, [19, Chapter V]. Namely, there exists $p>2$ and $c_{*}>0$ depending
only on $c_{1}$, $c_{2}$ and $\|b\|_{L^{2,n-2}_{w}(\Omega)}$ such that $\nabla
u\in L_{p}(\Omega)$ and the following estimate holds:

$\|\nabla u\|_{L_{p}(\Omega)}\ \leq\ c_{*}~{}(\|\nabla
u\|_{L_{2}(\Omega)}+\|f\|_{L_{p}(\Omega)}).$

Combining this estimate with Proposition 3.3 we obtain (1.10). ∎

Finally we can prove Theorem 1.1.

###### Proof.

We follow the method similar to [31, Theorem 2.1], and [43],[44]. Let us fix
some $q>n$. Let $p>2$ be the exponent defined in Proposition 3.5 and denote
$p^{\prime}=\frac{p}{p-1}$. As $\Omega$ is Lipschitz we can find the sequence
of $C^{2}$-smooth domains $\\{\Omega_{k}\\}_{k=1}^{\infty}$ such that

$\Omega_{k+1}\Subset\Omega_{k},\qquad\bigcup\limits_{k}\Omega_{k}=\Omega.$

Moreover, it is possible to construct domains $\Omega_{k}$ so that the
Lipschitz constants of $\partial\Omega_{k}$ are controlled uniformly by the
Lipschitz constant of $\partial\Omega$. In particular, we can assume there are
exist positive constants $\hat{\delta}_{0}$ and $\hat{R}_{0}$ independent on
$k\in\mathbb{N}$ such that

$\forall\,k\in\mathbb{N},\qquad\forall\,R<\hat{R}_{0},\qquad\forall\,x_{0}\in\partial\Omega_{k}\qquad|B_{R}(x_{0})\setminus\Omega_{k}|\,\geq\,\hat{\delta}_{0}\,|B_{R}|.$
(3.7)

For a canonical domain given as a subgraph of a Lipschitz function existence
of such approximation can be obtained by mollification and shift of the graph,
and for general bounded Lipschitz domain the standard localization works.

Now we take a sequence of positive numbers $\varepsilon_{k}\to 0$ such that
$\varepsilon_{k}<\mathop{\rm dist}\\{\bar{\Omega}_{k},\partial\Omega\\}$, and
define the mollification of the drift $b$:

$b_{k}(x)\,:=\,\int\limits_{\Omega}\omega_{\varepsilon_{k}}(x-y)b(y)\,dy,\qquad
x\in\Omega,$

where $\omega_{\varepsilon}(x):=\varepsilon^{-n}\omega(x/\varepsilon)$ and
$\omega\in C_{0}^{\infty}(\mathbb{R}^{n})$ is the standard Sobolev kernel,
i.e.

$\omega\,\geq\,0,\qquad\mathop{\rm
supp}\omega\in\bar{B},\qquad\int\limits_{\mathbb{R}^{n}}\omega(x)\,dx\,=\,1,\qquad\omega(x)=\omega_{0}(|x|).$
(3.8)

Then $b_{k}\in C^{\infty}(\bar{\Omega})$ and as $\varepsilon_{k}<\mathop{\rm
dist}\\{\bar{\Omega}_{k},\partial\Omega\\}$ from (1.8) for any $k$ we obtain

$\mathop{\rm div}b_{k}\,\leq\,0\qquad\mbox{in}\quad\Omega_{k}.$ (3.9)

Moreover, there is a constant $c>0$ independent on $k$ such that

$\|b_{k}\|_{L^{2,n-2}_{w}(\Omega_{k})}\,\leq\,c\,\|b\|_{L^{2,n-2}_{w}(\Omega)},\qquad\|b_{k}-b\|_{L_{p^{\prime}}(\Omega)}\to
0\quad\mbox{as}\quad k\to 0.$ (3.10)

For the reader’s convenience we prove the inequality (3.10) in Appendix.

For $f\in L_{p}(\Omega)$ we can find $f_{k}\in C^{\infty}_{0}(\Omega_{k})$
such that $\|f_{k}-f\|_{L_{p}(\Omega)}\to 0$. Let $u_{k}\in
W^{1}_{2}(\Omega)\cap L_{\infty}(\Omega)$ be smooth solutions to the
regularized problem

$\left\\{\quad\begin{gathered}-\Delta u_{k}+b_{k}\cdot\nabla u_{k}\ =\
-\mathop{\rm div}f_{k}\qquad\mbox{in}\quad\Omega_{k},\\\
u_{k}|_{\partial\Omega_{k}}\ =\ 0,\end{gathered}\right.$ (3.11)

and extend functions $u_{k}$ by zero from $\Omega_{k}$ onto $\Omega$. As the
drift $b_{k}$ of in (3.11) satisfies the properties (3.9) and (3.10) from
Proposition 3.5 we obtain the estimate

$\|u_{k}\|_{W^{1}_{p}(\Omega)}\ \leq\ c\,\|f_{k}\|_{L_{p}(\Omega)}$

with a constant $c>0$ depending only on $n$, $q$,
$\|b\|_{L^{2,n-2}_{w}(\Omega)}$ and the constant $\hat{\delta}_{0}$ in (3.7)
which is independent on $k$. Hence we can take a subsequence $u_{k}$ such that

$u_{k}\rightharpoonup u\quad\mbox{in}\quad W^{1}_{p}(\Omega).$

Take any $\eta\in C_{0}^{\infty}(\Omega)$, due to our construction of the
sequence $\Omega_{k}$ we can find $N$ such that for all $k\geq
N,~{}\mathop{\rm supp}\eta\subset\Omega_{k}$, therefore $\eta$ is a suitable
test function in (3.11). As $b_{k}\to b$ in $L_{p^{\prime}}(\Omega)$ we can
pass to the limit in the identity

$\begin{gathered}\int\limits_{\Omega}\nabla
u_{k}\cdot\Big{(}\nabla\eta+b_{k}\eta\Big{)}\,dx\ =\
\int\limits_{\Omega}f_{k}\cdot\nabla\eta~{}dx,\end{gathered}$

and obtain (1.7). Hence $u\in\overset{\circ}{W}{{}^{1}_{p}}(\Omega)$ is a
$p$-weak solution to the problem (1.1). Its uniqueness follows from
Proposition 3.2. ∎

##  4 Modified De Giorgi classes

In this section we introduce the modified De Giorgi classes which are
convenient for the study of solutions to the elliptic equations with
coefficients from Morrey spaces. The only difference of our classes from the
standard De Giorgi classes (see, for example [20, §8.3], [33, §II.6]) is the
additional multiplier $1+\frac{R^{\alpha}}{(R-\rho)^{\alpha}}$ in the
inequalities (4.1) and (4.2).

######  Definition 4.1.

For $u\in W^{1}_{2}(\Omega)$ we define $A_{k}:=\\{\,x\in\Omega:\,u(x)>k\,\\}$.
We say $u\in DG(\Omega,k_{0})$ if there exist constants $\gamma>0$, $F>0$,
$\alpha\geq 0$, $q>n$ such that for any $B_{R}(x_{0})\subset\Omega$, any
$0<\rho<R$ and any $k\geq k_{0}$ the following inequality holds

$\begin{gathered}\int\limits_{A_{k}\cap B_{\rho}(x_{0})}|\nabla u|^{2}\,dx\
\leq\
\frac{\gamma^{2}}{(R-\rho)^{2}}\left(1+\frac{R^{\alpha}}{(R-\rho)^{\alpha}}\right)\,\int\limits_{A_{k}\cap
B_{R}(x_{0})}|u-k|^{2}\,dx\ +\\\ +\ F^{2}\,|A_{k}\cap
B_{R}(x_{0})|^{1-\frac{2}{q}}.\end{gathered}$ (4.1)

We write $u\in DG(\Omega)$ if $u\in DG(\Omega,k_{0})$ for any
$k_{0}\in\mathbb{R}$. To avoid overloaded notation, when we need to specify
constants in Definition 4.1 we allow some terminological license and say that
the class $D(\Omega,k_{0})$ corresponds to the constants $\gamma$, $F$,
$\alpha$, $q$ instead of including these constants in the notation of the
functional class.

The second modified De Giorgi class characterizes the behavior of Sobolev
functions near the boundary of the domain:

######  Definition 4.2.

We say $u\in DG(\partial\Omega)$ if
$u\in\overset{\circ}{W}{{}^{1}_{2}}(\Omega)$ and there exist constants
$\gamma>0$, $F>0$, $\alpha\geq 0$, $q>n$, $R_{0}>0$ such that for any
$x_{0}\in\partial\Omega$, any $0<\rho<R\leq R_{0}$ and any $k\geq 0$ the
following inequality holds

$\begin{gathered}\int\limits_{A_{k}\cap\Omega_{\rho}(x_{0})}|\nabla
u|^{2}\,dx\ \leq\
\frac{\gamma^{2}}{(R-\rho)^{2}}\left(1+\frac{R^{\alpha}}{(R-\rho)^{\alpha}}\right)\,\int\limits_{A_{k}\cap\Omega_{R}(x_{0})}|u-k|^{2}\,dx\
+\\\ +\ F^{2}\,|A_{k}\cap\Omega_{R}(x_{0})|^{1-\frac{2}{q}}\end{gathered}$
(4.2)

where we denote $\Omega_{R}(x_{0}):=\Omega\cap B_{R}(x_{0})$.

We emphasize that while $u\in DG(\Omega)$ implies that (4.1) holds for any
$k\in\mathbb{R}$ the inclusion $u\in DG(\partial\Omega)$ implies that (4.2)
holds only for non-negative $k\in\mathbb{R}$. Such choice of the boundary De
Giorgi class is motivated by the homogeneous Dirichlet boundary conditions in
(1.1).

The main result of this section is the following theorem:

######  Theorem 4.1.

Assume $\Omega\subset\mathbb{R}^{n}$ is a bounded Lipschitz domain and denote
by $DG(\Omega)$ and $DG(\partial\Omega)$ the modified De Giorgi classes
corresponding to some positive constants $F$, $\gamma$, $\alpha$ and $q>n$.

  * (i)

Assume $\pm u\in DG(\Omega)$. Then there exists $\mu\in(0,1)$ depending only
on $n$, $q$, $\gamma$ such that $u\in C^{\mu}_{loc}(\Omega)$ and for any
$\Omega^{\prime}\Subset\Omega$

$\|u\|_{C^{\mu}(\bar{\Omega}^{\prime})}\ \leq\
c(\Omega,\Omega^{\prime})\,\Big{(}\|u\|_{L_{2}(\Omega)}+F\Big{)}$

with some constant $c(\Omega,\Omega^{\prime})>0$ depending only on $\Omega$,
$\Omega^{\prime}$, $n$, $q$, $\gamma$ and $\alpha$.

  * (ii)

Assume $\pm u\in DG(\Omega)\cap DG(\partial\Omega)$. Then there exists
$\mu\in(0,1)$ depending only on $n$, $q$, $\gamma$ such that $u\in
C^{\mu}(\bar{\Omega})$ and

$\|u\|_{C^{\mu}(\bar{\Omega})}\ \leq\
c(\Omega)\,\Big{(}\|u\|_{L_{2}(\Omega)}+F\Big{)}$

with some constant $c(\Omega)>0$ depending only on $\Omega$, $R_{0}$, $n$,
$q$, $\gamma$ and $\alpha$.

Our modified De Giorgi classes in Definitions 4.1 and 4.2 are very similar to
ones introduced earlier in [33]. Our proof of Theorem 4.1 is standard and
follows to the general line in [33]. But due to the minor difference in the
proof and for the reader convenience we present the proof of Theorem 4.1 in
this section.

First we present some technical lemma.

######  Lemma 4.1.

Assume $\\{Y_{m}\\}_{m=1}^{\infty}\subset\mathbb{R}$ are nonnegative and there
exist $\varepsilon>0$, $\varepsilon_{1}\geq 0$, $b\geq 1$ and $c_{0}\geq 1$
such that the following inequality holds

$Y_{m+1}\ \leq\
c_{0}~{}b^{m}~{}Y_{m}^{1+\varepsilon}(1+Y_{m}^{\varepsilon_{1}}),\qquad\forall~{}m\in\mathbb{N}.$
(4.3)

Assume $Y_{0}\leq\theta$ where $\theta\ :=\
(2c_{0})^{-1/\varepsilon}~{}b^{-1/\varepsilon^{2}}$. Then

$Y_{m}\ \leq\ \theta~{}b^{-m/\varepsilon},\qquad\forall~{}m\in\mathbb{N}.$

###### Proof.

We use the induction with respect to $m$. For $m=0$ the statement holds by
assumption. Now assuming $Y_{m}\ \leq\ \theta~{}b^{-m/\varepsilon}$ from (4.3)
we estimate

$\begin{gathered}Y_{m+1}\ \leq\
c_{0}~{}b^{m}~{}Y_{m}^{1+\varepsilon}(1+Y_{m}^{\varepsilon_{1}})\ \leq\
c_{0}~{}b^{m}~{}(\theta~{}b^{-m/\varepsilon})^{1+\varepsilon}\Big{(}1+(\theta~{}b^{-m/\varepsilon})^{\varepsilon_{1}}\Big{)}\
\leq\\\ \leq\ 2\,c_{0}\,\theta^{1+\varepsilon}\,b^{-\frac{m}{\varepsilon}}\ =\
2\,c_{0}\,\theta^{1+\varepsilon}\,b^{1/\varepsilon}\,b^{-\frac{m+1}{\varepsilon}}\end{gathered}$

To finish the proof we note that
$\theta=(2c_{0})^{-1/\varepsilon}~{}b^{-1/\varepsilon^{2}}$ implies
$2c_{0}\,\theta^{1+\varepsilon}\,b^{1/\varepsilon}=\theta$. ∎

Now we prove the local maximum estimate:

######  Lemma 4.2.

Assume $u\in DG(B_{R},k_{0})$. Then $u$ is essentially bounded from above in
$B_{R/2}$ and

$\sup\limits_{B_{R/2}}~{}(u-k_{0})_{+}\ \leq\ c_{*}\,\left[\Big{(}\
-\\!\\!\\!\\!\\!\\!\\!\int\limits_{B_{R}}|(u-k_{0})_{+}|^{2}~{}dx\Big{)}^{1/2}+F\,R^{1-\frac{n}{q}}\right]$
(4.4)

where $(u-k_{0})_{+}:=\max\\{u-k_{0},0\\}$ and $c_{*}>0$ depends only on $n$,
$\gamma$ and $\alpha$.

###### Proof.

Without loss of generality we can assume $R=1$ and $k_{0}=0$. Denote

$H\ :=\ c_{*}\left[\Big{(}\ -\\!\\!\\!\\!\\!\\!\int\limits_{B}u_{+}^{2}~{}dx\
\Big{)}^{1/2}+F~{}\right],$

where the precise value of the constant $c_{*}=c_{*}(n,\gamma,\alpha)>0$ will
be fixed later. Define

$\begin{gathered}\rho_{m}:=\tfrac{1}{2}+\tfrac{1}{2^{m+1}},\quad
k_{m}:=H-\tfrac{H}{2^{m}},\quad
J_{m}:=\int\limits_{B_{\rho_{m}}}(u-k_{m})^{2}_{+}~{}dx,\\\
\mu_{m}\,:=\,|\\{\,x\in B_{\rho_{m}}:\,u(x)>k_{m+1}\,\\}|.\end{gathered}$

From $J_{m}\geq(k_{m+1}-k_{m})^{2}\mu_{m}$ we obtain

$\mu_{m}\,\leq\,\frac{2^{2m+2}}{H^{2}}~{}J_{m}.$ (4.5)

Define $\rho_{m}^{\prime}:=\frac{1}{2}(\rho_{m+1}+\rho_{m})$ and define cut-
off functions

$\zeta_{m}\in C^{\infty}_{0}(B_{\rho_{m}^{\prime}}),\qquad\zeta_{m}\equiv
1\quad\mbox{on}\quad
B_{\rho_{m+1}},\qquad|\nabla\zeta_{m}|\leq\frac{c}{\rho_{m}-\rho_{m+1}}.$

From the Hölder inequality and the imbedding
$W^{1}_{2}(B_{\rho_{m}^{\prime}})\hookrightarrow
L_{\frac{2n}{n-2}(B_{\rho_{m}^{\prime}})}$ we obtain

$J_{m+1}\ \leq\
c(n)~{}\mu_{m}^{\frac{2}{n}}~{}\Big{(}~{}\|\nabla(u-k_{m+1})_{+}\|_{L_{2}(B_{\rho_{m}^{\prime}})}^{2}\
+\ \frac{c(n)}{(\rho_{m}-\rho_{m+1})^{2}}~{}J_{m}~{}\Big{)}.$ (4.6)

Using $u\in DG(B)$ we estimate

$\|\nabla(u-k_{m+1})_{+}\|_{L_{2}(B_{\rho_{m}}^{\prime})}^{2}\ \leq\
\frac{\gamma^{2}}{(\rho_{m}-\rho_{m+1})^{2}}\left(1+\frac{\rho_{m}^{\alpha}}{(\rho_{m}-\rho_{m+1})^{\alpha}}\right)\,J_{m}\
+\ F^{2}\mu_{m}^{1-\frac{2}{q}}$ (4.7)

and hence

$J_{m+1}\ \leq\
\frac{c(n,\gamma)}{(\rho_{m}-\rho_{m+1})^{2}}~{}\mu_{m}^{\frac{2}{n}}~{}\left(1+\frac{\rho_{m}^{\alpha}}{(\rho_{m}-\rho_{m+1})^{\alpha}}\right)~{}J_{m}\
+\ F^{2}\mu_{m}^{1+\frac{2}{n}-\frac{2}{q}}.$

Denote

$Y_{m}\,:=\,\frac{J_{m}}{H^{2}}.$

Taking into account $\rho_{m}-\rho_{m+1}=\frac{1}{2^{m+2}}$, $\rho_{m}\leq 1$
from (4.5), (4.6), (4.7) we obtain

$Y_{m+1}\ \leq\ c\,2^{2m(1+\frac{2}{n})}\left(1+2^{\alpha
m}\right)\,Y_{m}^{1+\frac{2}{n}}\ +\
c\,\tfrac{F^{2}}{H^{2}}\,2^{m\left(1+\frac{2}{n}-\frac{2}{q}\right)}Y_{m}^{1+\frac{2}{n}-\frac{2}{q}}.$

Using and $\frac{F}{H}\leq 1$ we arrive at (4.3)

$Y_{m+1}\ \leq\
c_{0}~{}b^{m}~{}Y_{m}^{1+\varepsilon}(1+Y_{m}^{\varepsilon_{1}}),$ (4.8)

where $b:=2^{2+\alpha+\frac{4}{n}}$, $\varepsilon=\frac{2}{n}-\frac{2}{q}$,
$\varepsilon_{1}=\frac{2}{q}$ and $c_{0}$ depends only on $n$, $\gamma$,
$\alpha$. So, the sequence $\\{Y_{m}\\}$ satisfies the assumptions of Lemma
4.1. Now fix $c_{*}>\theta^{-1/2}$ where $\theta$ is defined in Lemma 4.1.
Then $Y_{0}<\theta$ holds and hence by Lemma 4.1 we obtain $Y_{m}\to 0$ which
gives (4.4). ∎

Now we prove so-called “Density lemma” (or “Thin set lemma”):

######  Lemma 4.3.

Assume $u\in DG(\Omega,k_{0})$. Then there exists $\theta_{0}\in(0,1)$
depending only on the constants $n$, $\gamma$, $\alpha$, $q$ in Definition 4.1
of De Giorgi class such that for any
$B_{4\rho}:=B_{4\rho}(x_{0})\Subset\Omega$ if

$|B_{2\rho}\cap A_{k_{0}}|\,\leq\,\theta_{0}\,|B_{2\rho}|$

then either

$\sup\limits_{B_{\rho}}(u-k_{0})_{+}\ \leq\
\frac{1}{2}~{}\sup\limits_{B_{4\rho}}(u-k_{0})_{+}$ (4.9)

or

$\sup\limits_{B_{4\rho}}(u-k_{0})_{+}\ \leq\
4c_{*}\,F\,(2\rho)^{1-\frac{n}{q}}$ (4.10)

where $c_{*}>0$ is a constant from (4.4).

###### Proof.

Take $\theta_{0}\in(0,1)$ such that $c_{*}~{}\theta_{0}^{1/2}=\frac{1}{4}$.
Assume (4.10) does not hold. Then from (4.4) we obtain (4.9). ∎

Now we prove a result called “How to find a thin set”:

######  Lemma 4.4.

Assume $u\in DG(\Omega,k_{0})$. Then for any $\delta\in(0,1)$ and any
$\theta\in(0,1)$ there exists $s\in\mathbb{N}$ depending only on $\theta$,
$\delta$ and the constants $n$, $\gamma$, $\alpha$, $q$ in Definition 4.1 of
De Giorgi class such that if for some
$B_{4\rho}:=B_{4\rho}(x_{0})\Subset\Omega$ the following estimate is valid

$|B_{2\rho}\setminus A_{k_{0}}|\ \geq\ \delta~{}|B_{2\rho}|$

then either

$|B_{2\rho}\cap A_{\bar{k}}|\ \leq\ \theta~{}|B_{2\rho}|,$ (4.11)

or

$M(4\rho)-k_{0}\,\leq\,2^{s}\,F\,\rho^{1-\frac{n}{q}}.$ (4.12)

Here we denote $\bar{k}=M(4\rho)-\frac{1}{2^{s}}(M(4\rho)-k_{0})$ and
$M(4\rho)\,:=\,\sup\limits_{B_{4\rho}}u$.

###### Proof.

Denote $M:=M(4\rho)$ and assume $s\in\mathbb{N}$ is some number whose value
will be fixed later. From $u\in DG(B_{4\rho})$ for any $k_{0}\leq
k\leq\bar{k}$ we obtain

$\int\limits_{B_{2\rho}\cap A_{k}}~{}|\nabla u|^{2}~{}dxdt\ \leq\
c(n,\gamma,\alpha)~{}\rho^{n-2}~{}\left((M-k)^{2}+F^{2}\rho^{2(1-\frac{n}{q})}\right).$
(4.13)

Assume (4.12) is not valid. Then for any $k_{0}\leq k\leq\bar{k}$

$F\,\rho^{1-\frac{n}{q}}\,\leq\,\frac{1}{2^{s}}\,(M-k_{0})\,=\,M-\bar{k}\,\leq\,M-k$

and from (4.13) we obtain

$\int\limits_{B_{2\rho}\cap A_{k}}~{}|\nabla u|^{2}~{}dxdt\ \leq\
2\,c(n,\gamma,\alpha)~{}\rho^{n-2}~{}(M-k)^{2}.$ (4.14)

Define $k_{m}:=M-\frac{1}{2^{m}}(M-k_{0})$ and denote

$D_{m}:=\\{\ x\in B_{2\rho}~{}|~{}k_{m}\leq u(x)<k_{m+1}\ \\}.$

For simplicity in the rest of the proof of this lemma we will abuse the
notation of $A_{m}$ in Definition 4.1 by denoting

$A_{m}:=\\{\ x\in B_{2\rho}~{}|~{}u(x)>k_{m}\ \\}.$

From the De Giorgi inequality (see [33, Lemma 3.5]) we derive

$(k_{m+1}-k_{m})|A_{m+1}|\ \leq\
C(n)\frac{|\,B_{2\rho}\,|^{1+\frac{1}{n}}}{|\,B_{2\rho}\setminus A_{m}\,|}\
\int\limits_{D_{m}}|\nabla u(x)|~{}dx.$

Note that

$|\,B_{2\rho}\setminus A_{m}\,|\ \geq\ \delta~{}|B_{2\rho}|$

and hence

$(k_{m+1}-k_{m})~{}|A_{m+1}|\ \leq\ \frac{C(n)}{\delta}~{}\rho\
\int\limits_{D_{m}}|\nabla u|~{}dx.$

Using the Hölder inequality and (4.14) we obtain

$(k_{m+1}-k_{m})|A_{m+1}|\ \leq\
\frac{C(n,\gamma,\alpha)}{\delta}~{}\rho^{\frac{n}{2}}\
|\,D_{m}\,|^{1/2}(M-k_{m}).$

Note that

$k_{m+1}-k_{m}=\frac{1}{2^{m+1}}(M-k_{0}),\qquad
M-k_{m}=\frac{1}{2^{m}}(M-k_{0}).$

Hence

$|A_{m+1}|^{2}\ \leq\ C(n,\gamma,\alpha,\delta)~{}\rho^{n}\ |\,D_{m}\,|,\qquad
m=1,2,3,\ldots$

Taking a summation over $m$ from 1 to $s-1$ we gat

$\sum\limits_{m=1}^{s-1}|A_{m+1}|^{2}\ \leq\
C(n,\gamma,\alpha,\delta)~{}\rho^{n}\ \sum\limits_{m=1}^{s-1}|\,D_{m}\,|.$

Note that

$\sum\limits_{m=1}^{s-1}|\,D_{m}\,|\ =\ |A_{1}\setminus
A_{2}|+\ldots+|A_{s-1}\setminus A_{s}|\ =\ |A_{1}\setminus A_{s}|\ \leq\
|B_{2\rho}|,$ $A_{s}\subset A_{m},\quad
m=1,\ldots,s-1\qquad\Longrightarrow\qquad\sum\limits_{m=1}^{s-1}|A_{m+1}|^{2}\
\geq\ (s-1)|A_{s}|^{2}.$

So, for any $s\in\mathbb{N}$ we obtain

$|A_{s}|\ \leq\
\frac{C(n,\gamma,\alpha,\delta)}{\sqrt{s-1}}~{}\rho^{\frac{n}{2}}~{}|B_{2\rho}|^{\frac{1}{2}}.$

Let $\theta\in(0,1)$ be arbitrary. Find $s\in\mathbb{N}$,
$s=s(\delta,\theta,n,\gamma,\alpha)$, so that

$\frac{C(n,\gamma,\alpha,\delta)}{\sqrt{s-1}}~{}\rho^{\frac{n}{2}}~{}|B_{2\rho}|^{\frac{1}{2}}\
\leq\ \theta~{}|B_{2\rho}|.$

Then

$|A_{s}|\ \leq\ \theta~{}|B_{2\rho}|,$

which implies (4.11). ∎

Our next result is a local estimate of oscillations:

######  Lemma 4.5.

Assume $\pm u\in DG(\Omega)$. Then there exist constants $\sigma\in(0,1)$ and
$c_{1}>0$ depending only on the constants $n$, $\gamma$, $\alpha$, $q$ in the
Definition 4.1 of the De Giorgi class such that

$\forall\,B_{4\rho}(x_{0})\Subset\Omega\qquad\mathop{\rm
osc}\limits_{B_{\rho}(x_{0})}u\ \leq\ \sigma~{}\mathop{\rm
osc}\limits_{B_{4\rho}(x_{0})}u\ +\ c_{1}\,F\,\rho^{1-\frac{n}{q}}.$ (4.15)

###### Proof.

Take any $B_{4\rho}:=B_{4\rho}(x_{0})\Subset\Omega$ and denote

$m(\rho)=\inf\limits_{B_{\rho}}u,\qquad M(\rho)=\sup\limits_{B_{\rho}}u,\qquad
k_{0}:=\frac{m(4\rho)+M(4\rho)}{2}.$

Let $\theta_{0}\in(0,1)$ be the constant from Lemma 4.3 and denote by
$s\in\mathbb{N}$ the constant from Lemma 4.4 which corresponds to
$\delta=\frac{1}{2}$, $\theta=\theta_{0}$ and the constants $n$, $\gamma$,
$\alpha$, $q$ in Definition 4.1 of the De Giorgi class. Let $c_{*}>0$ be the
constant from (4.4) and take

$c_{1}\ =\ \max\\{16c_{*},2^{s+1}\\}$ (4.16)

and assume

$\mathop{\rm osc}\limits_{B_{\rho}(x_{0})}u\ >\
c_{1}\,F\,\rho^{1-\frac{n}{q}}.$

does not hold. Then both (4.10) and (4.12) are not valid. Consider two cases:

$\begin{gathered}\mbox{(1)}\qquad\big{|}~{}\\{\ u\leq k_{0}\ \\}\cap
B_{2\rho}~{}\big{|}\ \geq\ \tfrac{1}{2}~{}|B_{2\rho}|,\\\
\mbox{(2)}\qquad\big{|}~{}\\{\ u\geq k_{0}\ \\}\cap B_{2\rho}~{}\big{|}\ \geq\
\tfrac{1}{2}~{}|B_{2\rho}|.\end{gathered}$

Case (1). Take

$\bar{k}:=M(4\rho)-\frac{1}{2^{s}}(M(4\rho)-k_{0}).$

Applying Lemma 4.4 with $\delta=\frac{1}{2}$ and $\theta=\theta_{0}$ we
conclude

$|B_{2\rho}\cap A_{\bar{k}}|\ \leq\ \theta_{0}~{}|B_{2\rho}|.$

Then from Lemma 4.3 we obtain

$\sup\limits_{B_{\rho}}(u-\bar{k})_{+}\ \leq\
\frac{1}{2}~{}\sup\limits_{B_{4\rho}}(u-\bar{k})_{+},$

and hence

$M(\rho)\ \leq\ \frac{1}{2}~{}\big{(}M(4\rho)+\bar{k}\big{)},$

which means

$M(\rho)\ \leq\ M(4\rho)-\frac{1}{2^{s+2}}\Big{(}M(4\rho)-m(4\rho)\Big{)}.$

Adding this with $-m(\rho)\leq-m(4\rho)$ we arrive at

$\mathop{\rm osc}\limits_{B_{\rho}}u\ \leq\ \sigma~{}\mathop{\rm
osc}\limits_{B_{4\rho}}u,\qquad\mbox{where}\qquad\sigma:=1-\frac{1}{2^{s+2}}.$

Case (2). We denote

$v:=-u,\qquad l_{0}:=-k_{0},\qquad M_{v}(\rho):=\sup\limits_{B_{\rho}}v,\qquad
m_{v}(\rho):=\inf\limits_{B_{\rho}}v.$

Hence we have $v\in DG(\Omega)$. From $(2)$ we get

$|~{}\\{v\leq l_{0}\\}\cap B_{2\rho}\\}~{}|\ \geq\
\tfrac{1}{2}~{}|B_{2\rho}|.$

Hence similar to above

$M_{v}(\rho)\ \leq\
M_{v}(4\rho)-\frac{1}{2^{s+2}}\Big{(}M_{v}(4\rho)-m_{v}(4\rho)\Big{)},$

which gives

$\mathop{\rm osc}\limits_{B_{\rho}}v\ \leq\
\left(1-\frac{1}{2^{s+2}}\right)~{}\mathop{\rm
osc}\limits_{B_{4\rho}}v,\qquad\mbox{where}\qquad\sigma:=1-\frac{1}{2^{s+2}}.$

As

$\mathop{\rm osc}\limits_{B_{\rho}}v\ =\ \mathop{\rm
osc}\limits_{B_{\rho}}u,\qquad\mathop{\rm osc}\limits_{B_{4\rho}}v\ =\
\mathop{\rm osc}\limits_{B_{4\rho}}u,$

the result follows. ∎

The next result follows easily by iterations of the estimate of oscillations:

######  Lemma 4.6.

Assume $u\in L_{\infty}(\Omega)$ satisfies (4.15) with some $\sigma\in(0,1)$.
Then

$\forall~{}B_{R}(x_{0})\subset\Omega,\quad\forall\,\rho<R\qquad\mathop{\rm
osc}\limits_{B_{\rho}(x_{0})}u\,\leq\,c_{2}\,\left(\Big{(}\frac{\rho}{R}\Big{)}^{\mu}\,\|u\|_{L_{\infty}(\Omega)}+F\,\rho^{\mu}\right)$

where $\mu\in(0,1)$ depends only on $\sigma$, $n$, $q$ and $c_{2}$ depends on
$\sigma$, $n$, $q$ and $c_{1}$.

Now part (i) of Theorem 4.1 follows from Lemma 4.6.

Now we turn to the investigation of the boundary regularity.

######  Lemma 4.7.

Assume $\pm u\in\pm u\in DG(\Omega)\cap DG(\partial\Omega)$. Then there exist
constants $\sigma\in(0,1)$, $c_{1}>0$, $R_{1}>0$ depending only on the
constants $n$, $\gamma$, $\alpha$, $q$, $R_{0}$, in the Definitions 4.1 and
4.2 of the De Giorgi classes and the Lipschitz constant of $\partial\Omega$
such that

$\forall\,x_{0}\in\partial\Omega,\quad\forall\,\rho\leq R_{1}\qquad\mathop{\rm
osc}\limits_{\Omega_{\rho}(x_{0})}u\ \leq\ \sigma~{}\mathop{\rm
osc}\limits_{\Omega_{4\rho}(x_{0})}u\ +\ c_{1}\,F\,\rho^{1-\frac{n}{q}}.$
(4.17)

Here we denote $\Omega_{\rho}(x_{0}):=\Omega\cap B_{\rho}(x_{0})$.

###### Proof.

Take arbitrary $\Omega_{0}\Supset\Omega$ and denote by $\tilde{u}$ the zero
extension of $u$ from $\Omega$ onto $\Omega_{0}$. Obviously, if $\pm u\in
DG(\Omega)\cap DG(\partial\Omega)$ then $\pm\tilde{u}\in DG(\Omega_{0},0)$ and
hence, in particular, $u\in L_{\infty}(\Omega)$. As $\partial\Omega$ is
Lipschitz there exists $\delta_{0}\in(0,1)$ and
$0<R_{1}<\min\left(R_{0},\mathop{\rm
dist}\\{\Omega,\partial\Omega_{0}\\}\right)$ such that for any
$x_{0}\in\partial\Omega$ and any $R\in(0,R_{1})$

$|\\{\,x\in\Omega_{0}:\,\tilde{u}(x)=0\,\\}\cap
B_{R}(x_{0})|\,\geq\,\delta_{0}\,|B_{R}|.$

Denote $\tilde{v}:=-\tilde{u}$. Then $\tilde{u}$, $\tilde{v}\in
DG(\Omega_{0},0)$. Take any $x_{0}\in\partial\Omega$ and assume
$B_{4\rho}(x_{0})\Subset\Omega_{0}$. Denote

$m(\rho)=\inf\limits_{B_{\rho}(x_{0})}\tilde{u},\qquad
M(\rho)=\sup\limits_{B_{\rho}(x_{0})}\tilde{u},$
$m_{\tilde{v}}(\rho)=\inf\limits_{B_{\rho}(x_{0})}\tilde{v},\qquad
M_{\tilde{v}}(\rho)=\sup\limits_{B_{\rho}(x_{0})}\tilde{v},$

Note that

$m(\rho)\leq 0\leq M(\rho)$

and

$|\\{\,x\in\Omega_{0}:\,\tilde{u}(x)\leq 0\,\\}\cap
B_{2\rho}(x_{0})|\,\geq\,\delta_{0}\,|B_{2\rho}|,$
$|\\{\,x\in\Omega_{0}:\,\tilde{v}(x)\leq 0\,\\}\cap
B_{2\rho}(x_{0})|\,\geq\,\delta_{0}\,|B_{2\rho}|.$

Let $\theta_{0}\in(0,1)$ be the constant from Lemma 4.3 and denote by
$s\in\mathbb{N}$ the constant from Lemma 4.4 which corresponds to
$\delta=\frac{1}{2}$, $\theta=\theta_{0}$ and the constants $n$, $\gamma$,
$\alpha$, $q$ in the definition of the De Giorgi class $DG(\Omega_{0},0)$.
Define $c_{1}>0$ by (4.16) and assume

$\mathop{\rm osc}\limits_{B_{\rho}(x_{0})}\tilde{u}\ >\
c_{1}\,F\,\rho^{1-\frac{n}{q}}.$

Then (4.10) and (4.12) are not valid for both $\tilde{u}$ and $\tilde{v}$.
Using Lemma 4.4 with $\delta=\delta_{0}$, $\theta=\theta_{0}$ and $k_{0}=0$ we
obtain

$M(\rho)\ \leq\ \sigma~{}M(4\rho),\qquad M_{\tilde{v}}(\rho)\ \leq\
\sigma~{}M_{\tilde{v}}(4\rho),\qquad\mbox{where}\qquad\sigma:=1-\frac{1}{2^{s+2}}.$

Taking the sum of these two inequalities and taking into account

$M_{\tilde{v}}(\rho)=-m(\rho),\qquad M_{\tilde{v}}(4\rho)=-m(4\rho)$

we obtain

$\mathop{\rm osc}\limits_{\Omega_{\rho}(x_{0})}u\ \leq\ \sigma~{}\mathop{\rm
osc}\limits_{\Omega_{4\rho}(x_{0})}u.$

∎

The next result is the boundary analogue of Lemma 4.6.

######  Lemma 4.8.

Assume $u\in L_{\infty}(\Omega)$ satisfies (4.17) with some $\sigma\in(0,1)$.
Then

$\forall~{}x_{0}\in\partial\Omega,\quad\forall\,\rho<R\leq
R_{1}\qquad\mathop{\rm
osc}\limits_{\Omega_{\rho}(x_{0})}u\,\leq\,c_{2}\,\left(\Big{(}\frac{\rho}{R}\Big{)}^{\mu}\,\|u\|_{L_{\infty}(\Omega)}+F\,\rho^{\mu}\right)$

where $\mu\in(0,1)$ depends only on $\sigma$, $n$, $q$ and $c_{2}$ depends on
$\sigma$, $n$, $q$ and $c_{1}$.

Now part (ii) of Theorem 4.1 follows by the standard combination of estimates
in Lemmas 4.6 and 4.8.

##  5 Hölder continuity of weak solutions

In this section we give the proof of Theorem 1.2. Assume $p>2$ and
$u\in\overset{\circ}{W}{{}^{1}_{p}}(\Omega)$ is a $p$-weak solution to the
problem (1.1) corresponding to the right-hand side $f\in L_{q}(\Omega)$ with
$q>n$. From Proposition 3.1 we conclude $u\in L_{\infty}(\Omega)$.

Let us fix some $r\in\left(\frac{2n}{n+2},2\right)$ and
$\theta\in\left(\frac{n}{r}-\frac{n}{2},1\right)$. Then from (1.3) and
Proposition 2.1 we obtain

$b\in
L^{r,n-r}(\Omega),\qquad\|b\|_{L^{r,n-r}(\Omega)}\,\leq\,c\,\|b\|_{L^{2,n-2}_{w}(\Omega)}.$

First we consider the case $B_{R}(x_{0})\subset\Omega$. Take some radius
$\rho<R$ and a cut-off function $\zeta\in C_{0}^{\infty}(B_{R}(x_{0}))$ such
that

$0\leq\zeta\leq 1,\qquad\zeta\equiv 1\quad\mbox{on}\quad
B_{\rho}(x_{0}),\qquad|\nabla\zeta|\,\leq\,\frac{c}{R-\rho}.$ (5.1)

Assume $k\in\mathbb{R}$ is arbitrary and denote

$\tilde{u}\,:=\,(u-k)_{+}\,\equiv\,\max\\{u-k,0\\},\qquad\tilde{u}\in
L_{\infty}(\Omega)\cap W^{1}_{p}(\Omega).$

Fix $m:=\frac{1}{1-\theta}$ and note that $2m-1=m(1+\theta)$. Take
$\eta=\zeta^{2m}\tilde{u}$ in (1.7). It is easy to see that

$\mathcal{B}[u,\eta]\ =\ \mathcal{B}[\zeta^{m}\tilde{u},\zeta^{m}\tilde{u}]\
-\
m\,\int\limits_{\Omega}\zeta^{2m-1}\,b\cdot\nabla\zeta\,|\tilde{u}|^{2}\,dx.$

Taking into account (2.1) and $2m-1=m(1+\theta)$ we obtain

$\begin{gathered}\|\zeta^{m}\nabla\tilde{u}\|_{L_{2}(B_{R}(x_{0}))}^{2}\ \leq\
c\,\|\tilde{u}\nabla\zeta\|_{L_{2}(B_{R}(x_{0}))}^{2}+m\,\int\limits_{\Omega}\zeta^{m(1+\theta)}\,b\cdot\nabla\zeta\,|\tilde{u}|^{2}\,dx\
+\\\ +\ \|f\|_{L_{q}(\Omega)}^{2}\,|A_{k}\cap
B_{R}(x_{0})|^{1-\frac{2}{q}}\end{gathered}$ (5.2)

where $A_{k}:=\\{\,x\in\Omega:\,u(x)>k\,\\}$. Taking into account (5.1) and
applying the estimate (3.3) we obtain

$\int\limits_{\Omega}\zeta^{m(1+\theta)}\,b\cdot\nabla\zeta\,|\tilde{u}|^{2}\,dx\
\leq\
\frac{c\,R^{\theta}}{R-\rho}\,\|b\|_{L^{r,n-r}(\Omega)}\,\left\|\nabla(\zeta^{m}\tilde{u})\right\|_{L_{2}(B_{R}(x_{0}))}^{1+\theta}\|\tilde{u}\|_{L_{2}(B_{R}(x_{0}))}^{1-\theta}.$

Taking arbitrary $\varepsilon>0$ and applying the Young inequality we obtain

$\begin{gathered}\int\limits_{\Omega}\zeta^{m}b\cdot\nabla\zeta\,|\tilde{u}|^{2}\,dx\
\leq\
\varepsilon\,\left\|\nabla\left(\zeta^{m}\tilde{u}\right)\right\|_{L_{2}(B_{R}(x_{0}))}^{2}\
+\\\ +\
\frac{c_{\varepsilon}}{(R-\rho)^{2}}\Big{(}\frac{R}{R-\rho}\Big{)}^{\frac{2\theta}{1-\theta}}\,\|b\|_{L^{r,n-r}(B_{R}(x_{0}))}^{\frac{2}{1-\theta}}\|\tilde{u}\|_{L_{2}(B_{R}(x_{0}))}^{2}.\end{gathered}$

So, if we fix sufficiently small $\varepsilon>0$ from (5.2) for any
$k\in\mathbb{R}$ and $0<\rho<R$ we obtain

$\begin{gathered}\tfrac{1}{2}\,\|\nabla(u-k)_{+}\|_{L_{2}(B_{\rho}(x_{0}))}^{2}\
\leq\\\ \leq\
\frac{c}{(R-\rho)^{2}}\,\left(1+\Big{(}\frac{R}{R-\rho}\Big{)}^{\frac{2\theta}{1-\theta}}\,\|b\|_{L^{r,n-r}(\Omega)}^{\frac{2}{1-\theta}}\right)\,\|(u-k)_{+}\|_{L_{2}(B_{R}(x_{0}))}^{2}+\\\
+\ \|f\|_{L_{q}(\Omega)}^{2}\,|A_{k}\cap
B_{R}(x_{0})|^{1-\frac{2}{q}}.\end{gathered}$

Hence we obtain that $u\in DG(\Omega)$. Applying the same arguments to $-u$
instead of $u$ we also obtain $-u\in DG(\Omega)$.

Now we turn to the estimates near the boundary. Assume
$x_{0}\in\partial\Omega$ and denote $\Omega_{R}(x_{0}):=\Omega\cap
B_{R}(x_{0})$. Take a cut-off function $\zeta\in C_{0}^{\infty}(B_{R}(x_{0}))$
satisfying (5.1). Then for any $k\geq 0$ the function
$\eta=\zeta^{2m}(u-k)_{+}$ vanishes on $\partial\Omega$ and hence it is
admissible for the identify (1.7). Repeating all previous arguments we arrive
at

$\begin{gathered}\tfrac{1}{2}\,\|\nabla(u-k)_{+}\|_{L_{2}(\Omega_{\rho}(x_{0}))}^{2}\
\leq\\\ \leq\
\frac{c}{(R-\rho)^{2}}\,\left(1+\Big{(}\frac{R}{R-\rho}\Big{)}^{\frac{2\theta}{1-\theta}}\,\|b\|_{L^{r,n-r}(\Omega)}^{\frac{2}{1-\theta}}\right)\,\|(u-k)_{+}\|_{L_{2}(\Omega_{R}(x_{0}))}^{2}+\\\
+\
\|f\|_{L_{q}(\Omega)}^{2}\,|A_{k}\cap\Omega_{R}(x_{0})|^{1-\frac{2}{q}},\end{gathered}$

which holds for any $k\geq 0$. Hence $u\in DG(\partial\Omega)$. Applying the
same arguments to $-u$ instead of $u$ we also obtain $-u\in
DG(\partial\Omega)$. Now Theorem 1.2 follows directly from Theorem 4.1.

##  6 Appendix

In this section we prove the estimate (3.10).

######  Proposition 6.1.

Assume $\Omega\subset\mathbb{R}^{n}$ is a bounded domain and let $b$ satisfies
(1.3). Assume $\Omega^{\prime}\Subset\Omega$, $0<\varepsilon<\mathop{\rm
dist}\\{\bar{\Omega}^{\prime},\partial\Omega\\}$ and denote by
$b_{\varepsilon}\in C^{\infty}(\bar{\Omega})$ the convolution

$b_{\varepsilon}(x)\,:=\,\int\limits_{\Omega}\omega_{\varepsilon}(x-y)b(y)\,dy,\qquad
x\in\Omega,$

where $\omega_{\varepsilon}(x):=\varepsilon^{-n}\omega(x/\varepsilon)$ and
$\omega\in C_{0}^{\infty}(\mathbb{R}^{n})$ is the standard Sobolev kernel
satisfying properties (3.8). Then $b_{\varepsilon}$ satisfies the estimate

$\begin{gathered}\|b_{\varepsilon}\|_{L^{2,n-2}_{w}(\Omega^{\prime})}\,\leq\,c\,\|b\|_{L^{2,n-2}_{w}(\Omega)}.\end{gathered}$
(6.1)

with some constant $c>0$ depending only on $n$.

###### Proof.

Take arbitrary $x_{0}\in\Omega^{\prime}$. Note that
$B_{\varepsilon}(x)\subset\Omega$ for any $x\in\Omega^{\prime}$. First we
consider the case $R\in(0,\varepsilon)$. From the definition of the
convolution we have a pointwise estimate

$|b_{\varepsilon}(x)|\,\leq\,\frac{1}{\varepsilon^{n}}\,\int\limits_{B_{\varepsilon}(x)}|b(y)|\,dy,\qquad\forall\,x\in
B_{R}(x_{0})\cap\Omega^{\prime}$

and hence from the definition of the Morrey norm $L^{1,n-1}(\Omega)$ we obtain

$|b_{\varepsilon}(x)|\,\leq\,\frac{1}{\varepsilon}\,\|b\|_{L^{1,n-1}(\Omega)}\,\qquad\forall\,x\in
B_{R}(x_{0})\cap\Omega^{\prime}.$

Taking into account the estimate

$\|b\|_{L^{1,n-1}(\Omega)}\,\leq\,c\,\|b\|_{L^{2,n-2}_{w}(\Omega)}$

with a constant $c>0$ depending only on $\Omega$ we arrive at

$|b_{\varepsilon}(x)|\,\leq\,\frac{1}{\varepsilon}\,\|b\|_{L^{1,n-1}(\Omega)}\,\leq\,\frac{c}{\varepsilon}\,\|b\|_{L^{2,n-2}_{w}(\Omega)},\qquad\forall\,x\in
B_{R}(x_{0})\cap\Omega^{\prime}.$

Hence we obtain

$\|b_{\varepsilon}\|_{L_{2}(B_{R}(x_{0})\cap\Omega^{\prime})}^{2}\,\leq\,c\,\left(\frac{R}{\varepsilon}\right)^{2}\,\|b\|_{L^{2,n-2}_{w}(\Omega)}^{2}\,R^{n-2}.$

Applying the trivial estimate

$\|b_{\varepsilon}\|_{L_{2,w}(B_{R}(x_{0})\cap\Omega^{\prime})}\leq\|b_{\varepsilon}\|_{L_{2}(B_{R}(x_{0})\cap\Omega)}$

for $R\in(0,\varepsilon)$ we arrive at

$\|b_{\varepsilon}\|_{L_{2,w}(B_{R}(x_{0})\cap\Omega^{\prime})}^{2}\,\leq\,c\,\|b\|_{L^{2,n-2}_{w}(\Omega)}^{2}\,R^{n-2}.$
(6.2)

Now consider the case $R\in(\varepsilon,\mathop{\rm diam}\Omega)$. For any
$f\in L_{1}(B_{2R}(x_{0})\cap\Omega)$ we define a convolution operator

$(T_{\varepsilon}f)(x)\,:=\,\int\limits_{B_{\varepsilon}(x)}\omega_{\varepsilon}(x-y)f(y)\,dy,\qquad
x\in B_{R}(x_{0})\cap\Omega^{\prime}.$

From the Hölder inequality for $p\in[1,+\infty)$ we obtain

$|(T_{\varepsilon}f)(x)|^{p}\,\leq\,\int\limits_{B_{\varepsilon}(x)}\omega_{\varepsilon}(x-y)|f(y)|^{p}\,dy,\qquad
x\in B_{R}(x_{0})\cap\Omega^{\prime}.$

As for $\varepsilon<\min\left(R,\mathop{\rm
dist}\\{\bar{\Omega}^{\prime},\partial\Omega\\}\right)$ we have the inclusion

$\forall\,x\in B_{R}(x_{0})\cap\Omega^{\prime}\qquad B_{\varepsilon}(x)\subset
B_{2R}(x_{0})\cap\Omega,$

we conclude

$\int\limits_{B_{R}(x_{0})\cap\Omega^{\prime}}|(T_{\varepsilon}f)(x)|^{p}\,dx\,\leq\,\int\limits_{B_{2R}(x_{0})\cap\Omega}|f(y)|^{p}\,dy,\qquad
x\in B_{R}(x_{0})\cap\Omega^{\prime}.$

The last inequality implies that for $\varepsilon<\min\left(R,\mathop{\rm
dist}\\{\bar{\Omega}^{\prime},\partial\Omega\\}\right)$ the linear operator
$T_{\varepsilon}$ is bounded from $L_{p}(B_{2R}(x_{0})\cap\Omega)$ to
$L_{p}(B_{R}(x_{0})\cap\Omega^{\prime})$ for any $p\in[1,+\infty)$ and the
following estimate holds:

$M_{p}\,:=\,\|T_{\varepsilon}\|_{L_{p}(B_{2R}(x_{0})\cap\Omega)\to
L_{p}(B_{R}(x_{0})\cap\Omega^{\prime})}\,\leq\,1.$

Using Proposition 2.3 we conclude that the following interpolation estimate
holds:

$\|T_{\varepsilon}\|_{L_{2,w}(B_{2R}(x_{0})\cap\Omega)\to
L_{2,w}(B_{R}(x_{0})\cap\Omega^{\prime})}\,\leq\,c\,M_{3}^{1/2}\,M_{3/2}^{1/2}\,\leq\,c$

with some constant $c>0$ depending only on $n$. Hence we obtain

$\|b_{\varepsilon}\|_{L_{2,w}(B_{R}(x_{0})\cap\Omega^{\prime})}\,\leq\,c\,\|b\|_{L_{2,w}(B_{2R}(x_{0})\cap\Omega)}.$

From the definition of the Morrey quasinorm $L^{2,n-2}_{w}(\Omega)$ we obtain

$\|b_{\varepsilon}\|_{L_{2,w}(B_{R}(x_{0})\cap\Omega^{\prime})}^{2}\,\leq\,c\,\|b\|_{L_{w}^{2,n-2}(\Omega)}^{2}\,(2R)^{n-2}.$

Hence for any $R\in(\varepsilon,\mathop{\rm diam}\Omega)$ we arrive at the
estimate (6.2). So, the estimate (6.2) is valid for all $R\in(0,\mathop{\rm
diam}\Omega)$. Hence for any $\Omega^{\prime}\Subset\Omega$ and any
$\varepsilon\in(0,\mathop{\rm
dist}\\{\bar{\Omega}^{\prime},\partial\Omega\\})$ we obtain (6.1). ∎

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