\documentclass[reqno]{amsart}
\usepackage[notref,notcite]{showkeys}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 120, pp. 1--6.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/120\hfil Spatial regularity ]
{Remark on the spatial regularity for the Navier-Stokes equations}

\author[C. He\hfil EJDE-2008/120\hfilneg]
{Cheng He}

\address{Cheng He \newline
Institute of Applied Mathematics,
Academy of Mathematics and Systems Science, Academia Sinica,
Beijing, 100080,  China}
\email{chenghe@amss.ac.cn}

\thanks{Submitted November 24, 2006. Published August 28, 2008.}
\subjclass[2000]{35Q76D}
\keywords{Partial regularity; Navier-Stokes equations;
  Leray-Hopf weak solution}

\begin{abstract}
  Let $u$ be a Leray-Hopf weak solution to the Navier-Stokes
  equations. We will show that the  set of possible singular
  points of the vector field resulting from
  integrating the velocity $u$ with respect to time
  has Hausdorff dimension zero.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}

\section{Introduction}

 Let us consider the viscous incompressible fluid flow moving
within a region $\Omega$ of the three dimensional space $\mathbb{R}^3$, which
can be described by the  Navier-Stokes equations
\begin{equation}\label{e1.1}
\begin{gathered}
 \partial_tu -\nu\Delta u + (u\cdot\nabla)u =
-\nabla \pi, \quad \text{in } \Omega\times (0, \infty),\\
\nabla\cdot u = 0, \quad \text{in } \Omega\times(0,\infty)
\end{gathered}
\end{equation}
with the homogeneous boundary condition
\begin{equation}\label{e1.2}
u = 0 \quad\text{on }\partial\Omega\times(0, \infty)
\end{equation}
 when the boundary is not empty,
 and the initial condition
\begin{equation}\label{e1.3}
u(x, 0) = u_0(x) \quad\text{in }\Omega.
\end{equation}
Here $u = u(x,t)  = (u_1(x,t), u_2(x,t), u_3(x,t))$
denotes the unknown velocity vector field,
and $\pi = \pi(x,t)$  denotes the scalar pressure;
$\nu$ is the viscosity; and $u_0(x)$ is the initial velocity vector field.
For simplicity, the viscosity $\nu$ is normalized to 1.


For the initial value problem and the initial boundary value problem
to the Navier-Stokes equations, the existence of a class of global
weak solutions was shown by Leray and Hopf in their pioneering works
\cite{leray} and \cite{hopf} a long time ago.
Since then, much effort has been made to try to establish uniqueness
and regularity of weak solutions. However, these two
remarkable questions remain open. It is still not known
whether or not a weak solution can develop singularities at finite
time, even for sufficiently smooth initial data.
A lot of attention has been turned to the study of partial regularity
of  weak solutions to the Navier-Stokes equations.
The first analysis about the possible singular set was done by  Leray.

Following Caffarelli, Kohn and Nirenberg \cite{ckn},
a point $(x, t)$ (or $t$) is called a singular point of a weak solution
$u$ to the Navier-Stokes equations if and only if $u$ is not essentially
bounded in any neighborhood of $(x, t)$ (or $t$).
Leray \cite{leray} showed that the singularities, if exist, can occur
 at most on a set of $t$ with  Lebesgue measure zero.

Scheffer \cite{scheffer76}-\cite{scheffer80}  began the development
of the analysis about the set of possible singular points, and
established various partial regularity results for a class of weak
solutions. Scheffer's results showed that the set of possible time
singular points of the weak solution has $1/2$-dimensional Hausdorff
measure zero,
 and that the set of possible space-time singular points of
the weak solution has $5/3$-dimensional Hausdorff measure  zero.
See also \cite{foias,sohr85}.
Later, Caffarelli, Kohn and Nirenberg \cite{ckn} improved Scheffer's
results and showed that the set of possible
space-time singular points of a class of special weak solutions,
named as suitable weak solutions, has one-dimensional Hausdorff measure zero.
Note that suitable weak solutions differ essentially from the usual
weak solutions in the sense that they should satisfy a generalized
energy inequality.
A simplified proof of the main results of
\cite{ckn} was presented in \cite{lin}; see also
\cite{struwe,grunau,maremonti}.

It is well-known that there exists a large time $T$ such that after
the time $T$ the weak solution is smooth and the interval $(0, T)$
can be expressed as $ \cup_{i\in I}(a_i, b_i)\cup \mathcal{T}$,
where the set $I$ is at most a countable set, $(a_i, b_i)$ with $i
\in I$ are disjoint open intervals in $(0, T)$, the set
$\mathcal{T}$ has $1/2$-dimensional Hausdorff measure zero,
 and $u$ belongs to $C^\infty$ for $(x,t) \in \Omega\times (a, b)$
for each interval $(a, b)$ whose closure is contained in some
of the intervals $(a_i, b_i)$
(cf. Fioas and Temam \cite{foias}, Heywood \cite{heywood80}, Leray
\cite{leray}, Scheffer \cite{scheffer76}, sohr and W. von Wahl
\cite{sohr85}, Miyakawa and Sohr \cite{miyakawa}).

Applying their own local regularity theory,
Caffarelli, Kohn and Nirenberg \cite{ckn}
showed that the suitable weak solution is regular in the region
$\{(x, t):|x|^2t > k\}$ if the initial velocity
$u_0 \in L^2(\mathbb{R}^3)$ and
$|x|^{1/2}u_0 \in L^2(\mathbb{R}^3)$, or in the region
$\{(x, t):t > 0,|x| > R'\}$ for some $R' > R$
if the initial velocity $u_0 \in L^2(R^3)$ and
$\int_{|x|>R}|\nabla u_0|^2dx < \infty$.
Similar results has been obtained by Maremonti \cite{maremonti} in
the case of an exterior domain.
To the best of our knowledge, up to now, the set
of possible singular points is still not fully understood.

In this paper, we try to estimate the Hausdorff dimension of the set
of possible spatial singular points of weak solutions in some sense.
As is well-known, the set of possible time singular points of
a class of weak solutions has $1/2$-dimensional Hausdorff measure zero
( see \cite{foias,giga,leray,scheffer76,serrin63}).
 As for the set of possible
spatial-time singular points, it is known that the 1-dimensional
Hausdorff measure vanishes (cf. \cite{ckn,grunau,lin,maremonti,tian}).
However, as far as we know, no such results are available for the
set of possible spatial singular points.

It is difficult to study directly the set of
possible spatial singular points of the weak solutions. So we will
study the partial regularity of the weak solution by integrating the
solution in time. For this purpose, let $u$ be a weak solution to
the Navier-Stokes equations \eqref{e1.1} and define
$U(x) =\int^T_0u(x,s)ds$ for some $T> 0$.
By the definition of weak solutions,
$U(x) =\int^T_0u(x,s)ds$ is well-defined in the sense of Bochner
(See below). Then following ideas in \cite{foias}, we will show that
the set of points at any neighborhood of which $U$ is essentially
unbounded has Hausdorff dimension zero. This implies the
corresponding estimate of the Hausdorff dimension of the set of
possible spatial singular points of the weak solution $u$ in some
sense.

We conclude this introduction by introducing some function spaces
used in this paper. Let $L^p(\Omega)$, $1 \leq p \leq \infty$, represent
the usual Lesbegue space of scalar functions as well as that of
vector-valued functions with norm denoted by $\|\cdot\|_p$. Let
$C_{0,\sigma}^\infty(\Omega)$ denote the set of all $C^\infty$ vector
functions with compact support in $\Omega$ such that
$\mathop{\rm div}\phi=0$.
Let $L^p(0, T; X)$, $1 \leq p \leq \infty$, be the set of function
$f(t)$ defined on $(0, T)$ with values in X such that
$\int^T_0\|f(t)\|^p_X dt < \infty$ for a given Banach space $X$ with
norm $\|\cdot \|_X$.

\section{Main Results}

 In this article, we only consider four types of the domains:
(1) $\mathbb{R}^3$, (2) a bounded domain in $\mathbb{R}^3$,
(3) a half-space in $\mathbb{R}^3_+$, and (4) an exterior domain in
$\mathbb{R}^3$.

 We will consider the Leray-Hopf weak solutions defined
as follows:

\textbf{Definition.}
A \emph{Leray-Hopf weak solution} of the
system \eqref{e1.1}-\eqref{e1.3} in
$Q_\infty \equiv \Omega\times (0, \infty)$ is a vector field
$u: Q_\infty \to \mathbb{R}^3$ such that
\begin{gather}\label{e2.1}
u \in L^\infty(0, \infty; L^2_\sigma(\Omega)) \quad\text{and }\nabla u
\in L^2(0, \infty; L^2(\Omega)), \\
\label{e2.2}
\int_{Q_\infty}\Big(u\cdot\partial_tw + u\otimes u:\nabla w - \nabla
u:\nabla w \Big)\,dx\,dt = 0
\end{gather}
for any $w \in C^\infty_{0,\sigma}(Q_\infty)$ and  any
$t \in [0, \infty)$, $u$ satisfies the energy inequality
\begin{equation}\label{e2.3}
\|u(t)\|_2^2 + 2\int^t_0\|\nabla u(\tau)\|_2^2d\tau \leq
\|u_0\|_2^2,
\end{equation}
and $u$ takes the initial value in the sense that
\begin{equation}\label{e2.4}
\|u(\cdot,t) - u_0(\cdot)\|_2 \to
0\quad\text{as }t\to 0.
\end{equation}

 It is well-known now that  Leray \cite{leray} and Hopf \cite{hopf}
constructed a global Leray-Hopf weak solution .
Here we intend to study the spatial partial regularity of the
Leray-Hopf weak solution, in some sense. In fact, we are interested
in the estimate of the Hausdorff dimension of the set of possible
singular points of the vector resulting from integrating the
velocity $u$ with respect to time.
For simplicity, assume that $u_0 \in C^\infty_{0,\sigma}(\Omega)$.
The argument can be applied to general initial data $u_0$.

First we introduce the following result which was obtained by Giga
and Sohr \cite{giga91}.

 \begin{lemma} \label{lem2.1}
Let $u_0 \in C^\infty_{0,\sigma}(\Omega)$.
Then there exists a weak solution $(u, \pi)$ such that
 \begin{gather}\label{e2.5}
u \in L^\infty(0, \infty; L^2(\Omega)),\quad
\nabla u \in L^2(0, \infty; L^2(\Omega)), \\
\label{e2.6}
\partial_tu,\; \partial^2_xu, \;\nabla \pi \in L^p(0, \infty;L^q(\Omega))
\end{gather}
for any $1 < p, q < \infty$ with $1/p + 3/2q= 2$.
Also $u$ satisfies the energy inequality
\begin{equation}\label{e2.7}
\|u(t)\|_2^2 + 2\int^t_0\|\nabla u(\tau)\|_2^2d\tau \leq
\|u_0\|_2^2.
\end{equation}
So $\pi$ can be chosen such that
\begin{equation}\label{e2.8}
\nabla u,\;\pi \in L^p(0, \infty;L^{q^*}(\Omega)), \quad
\frac{1}{q^*} = \frac{1}{q}- \frac{1}{3}.
\end{equation}
\end{lemma}

As stated in the introduction, there is a time $T$ such that, after
 $T$, a Leray-Hopf weak solution $u$ is smooth. For this
$T > 0$, it is easy to see that
$$
U(x) := \int^T_0u(x, t)dt, \quad
\Pi(x) :=\int^T_0\pi(x,t)dt
$$
are well-defined in $L^{q^*}(\Omega)$. Then $(U,\Pi)$ satisfies the equations
\begin{equation}\label{e2.9}
- \Delta U  = f,\quad
f = u_0 - u(T) -\int^T_0(u\cdot\nabla u)(t)dt -\nabla \Pi.
\end{equation}

Define
$$
\Omega_0 =: \big\{x \in \Omega: U(x) \quad
\text{ is essentially unbounded in any neighborhood of }x\big\}.
$$

Now our main result can be stated as follows:


\begin{theorem} \label{thmA}
Let $u_0 \in C^\infty_{0,\sigma}(\Omega)$. Then $\Omega_0$
has Hausdorff dimension zero.
\end{theorem}

To prove our main theorem,  we need the following lemma
established by Foias and Temam \cite{foias}.

\begin{lemma}[{\cite[Lemma 4.2]{foias}}] \label{lem3.1}
For $a > 0$ and $f \in L^1(\mathbb{R}^n)$, let $\Lambda_a(f)$ be the set of
$x \in \mathbb{R}^n$ such that there exists $m_x$ with
$$
\int_{|y-x|\leq 2^{-m}}|f(y)|dy \leq 2^{-am} \quad
\text{for all }m \geq m_x.
$$
Then $\mathbb{R}^n\setminus\Lambda_a(f)$ has  Hausdorff
dimension less than or equal to  $a$.
\end{lemma}

\marginpar{Plase see addendum}

\begin{proof}[Proof of Theorem \ref{thmA}]
We will follow the ideas in \cite{foias}.
 Since $U \in L^{q^*}(\Omega)$,
\begin{equation}\label{e3.1}
U(x) = \frac{1}{4\pi}\int_{\mathbb{R}^3}\frac{1}{|x-y|}f(y)dy =: U_0
\quad \text{when } \partial\Omega = \emptyset,
\end{equation}
and
\begin{equation}\label{e3.2}
U(x) =  W (x) + \frac{1}{4\pi}\int_{\Omega}\frac{1}{|x-y|}f(y)dy
\quad\text{when $\partial\Omega \neq \emptyset$ with
a harmonic function $W$.}
\end{equation}
It is well-known that the function $W$ is smooth in the
interior of $\Omega$ and is bounded in the subdomain with positive
distance away from the boundary.
Extend $f$ to the outside of $\Omega$ by zero. So we only consider
the partial regularity of $U_0$.
For any $x_0 \in \mathbb{R}^3$, we have
\begin{equation}\label{e3.3}
\begin{aligned}
\frac{1}{r^3}\int_{|x-x_0|\leq r}|U_0(y)|dy
&\leq \frac{1}{4\pi}\frac{1}{r^3}\int_{|x-x_0|\leq r}
  \int_{\mathbb{R}^3}\frac{1}{|x-y|}|f(y)|dydx\\
&\leq \frac{1}{4\pi}\int_{\mathbb{R}^3}\frac{1}{|x_0-x|}
  \cdot\frac{1}{r^3}\int_{|x-y|\leq r}|f(y)|dydx \\
&\leq \frac{1}{4\pi}\int_{\mathbb{R}^3}\frac{1}{|x_0-x|}f^*(x)dx
\end{aligned}
\end{equation}
with
$$
f^*(x) = \sup_r \frac{1}{r^3}\int_{|x-y|\leq r}|f(y)|dy.
$$
By \eqref{e2.6}, we know that $f \in L^q(\mathbb{R}^3)$ for any
$1 < q < 3/2$. So it follows from the inequality on maximal
functions that $f^* \in L^q(\mathbb{R}^3)$. Let
$$
M_j = \big\{x\in \mathbb{R}^3:~|x-x_0| \leq 2^{-j}\big\}.
$$
For any $x_0 \in \mathbb{R}^3$, we have
\begin{equation}\label{e3.4}
\begin{aligned}
&\frac{1}{r^3}\int_{|x-x_0|\leq r}|U_0(x)|dx \\
&\leq \frac{1}{4\pi}\int_{\mathbb{R}^3}\frac{1}{|x_0-x|}f^*(x)dx\\
&\leq \frac{1}{4\pi}\int_{\mathbb{R}^3\setminus M_1}\frac{1}{|x_0-x|}f^*(x)dx
 + \sum^\infty_1\frac{1}{4\pi}\int_{M_j\setminus
 M_{j+1}}\frac{1}{|x_0-x|}f^*(x)dx\\
&\leq C\|f^*\|_q\Big(\int_{\mathbb{R}^3\setminus
 M_1}\frac{1}{|x-x_0|^{q\over q-1}}dx\Big)^{1-{1\over q}} \\
&\quad + C\sum^\infty_1\Big(\int_{M_j\setminus
 M_{j+1}}\frac{1}{|x-x_0|^{q\over q-1}}dx\Big)^{1-{1\over q}}
 \Big(\int_{M_j}|f^*(x)|^qdx\Big)^{1\over q}\\
&\leq C + C\sum^\infty_1 2^{j -3j(1-{1\over
 q})}\cdot\Big(\int_{M_j}|f^*(x)|^qdx\Big)^{1\over q}.
\end{aligned}
\end{equation}
It is obvious that $|f^*|^q \in L^1(\mathbb{R}^3)$.
Let $\Lambda_a(f^*) $ be the set of these $x_0 \in R^3$ such that
there exists $j_{x_0}$ with
$$
\int_{|x_0-x|\leq 2^{-j}}|f^*(x)|^qdx \leq 2^{-aj}
$$
for all $j \geq j_{x_0}$. Thus, for any $x_0 \in \Lambda_a(f^*)$,
\eqref{e3.4} gives us
\begin{equation}\label{e3.5}
\frac{1}{r^3}\int_{|x-x_0|\leq
r}|U_0(x)|dx \leq C + C \sum^\infty_1 2^{j -3j(1-{1\over q})}\cdot 2
^{-{aj\over q}} \leq C_1
\end{equation}
provided that $a > 3 - 2q$.
Note that the constant $C_1$ is independent of $x_0$.
 Let
$$
\overline{U}_0 = \frac{3}{4\pi r^3}\int_{|x-x_0| \leq r}U_0(x)dx
$$
denote the average of $U_0$ in the ball centered at $x_0$ with
radius $r$. Then, by \eqref{e3.5}, we have
$$
\frac{1}{r^3}\int_{|x-x_0|\leq r}\big|U_0(x) - \overline{U}_0\big|dx
\leq \frac{2}{r^3}\int_{|x-x_0|\leq r}\big|U_0(x)\big|dx \leq
2C_1
$$
provided that $a > 3 - 2q$. This implies
\begin{equation}\label{e3.6}
\sup_{x_0\in \Lambda_a(f^*),~r > 0}\frac{1}{r^3}\int_{|x-x_0|\leq
r}\big|U_0(x) - \overline{U}_0\big|dx \leq 2C_1
\end{equation}
provided that $a > 3-2q$.

Then, for any $x_0 \in \Lambda_a(f^*)$ with $a > 3 - 2q$,
\eqref{e3.6} tells us that
$$
|U_0(x_0) | < \infty.
$$
Therefore,
$$
\Omega_0 = \big\{x\in \mathbb{R}^3:~|U_0(x)| = \infty\big\} \subset \mathbb{R}^3\setminus
\Lambda_a(f^*).
$$
Applying Lemma \ref{lem3.1}, we deduce that the Hausdorff
dimension of $\Omega_0$ is less or equal to $a$.
Letting $a \to 3-2q$, we deduce that the Hausdorff dimension
of $\Omega_0$ does not exceed $3-2q$.
Since $q \in (1, 3/2)$ is arbitrary, we deduce that
the Hausdorff dimension of $\Omega_0$ is zero.
This completes the proof
\end{proof}


 \subsection*{Acknowledgment} The author is supported in part by The
Key Project 10431060 from the National Natural Science Foundation,
The 973 key Program 2006CB805902, and Knowledge Innovation Funds
from  CAS (KJCX3-SYW-S03), of China.
The author  would like to thank the anonymous referee for making
helpful comments on improving presentation of this paper.


\begin{thebibliography}{00}

\bibitem{ckn} L. Caffarelli, R. Kohn \& L. Nirenberg;
Partial regularity of suitable weak solution of the
Navier-Stokes equations,
\textit{Comm. Pure Appl. Math.}, \textbf{35}(1982), 771-837.

\bibitem{foias} C. Foias \& R. Temam;
 Some analytic and geometric properties of
the solutions of the evolution Navier-Stokes equations, \emph{J.
Math. Pures Appl.}, \textbf{9}(1979), 339-368.

\bibitem{giga} Y. Giga; Solutions for semilinear parabolic
equations in $L^p$ and regularity of weak solutions of the
Navier-Stokes system, \emph{J. Diff. Equas.}, \textbf{61}(1986), 186-212.

\bibitem{giga91} Y. Giga and H. Sohr;
Abstract $L^p$ estimates for the Cauchy problem
with applications to the Navier-Stokes equations in exterior
domains. \emph{J. Funct. Anal}. \textbf{102} (1991), 72--94.

\bibitem{grunau} H. C. Grunau;
Boundeness for large $|x|$ of suitable weak
solutions of the Navier-Stokes equations with prescribed velocity
at infinity,  \emph{Comm. Math.Phys.}, \textbf{151}(1993), 577-587.

\bibitem{heywood80} J. G. Heywood;
 The Navier-Stokes equations: On the existence,
regularity and decay of solutions, \emph{Indiana Univ. Math. J.},
\textbf{29}(1980), 641-681.

\bibitem{hopf} E. Hopf;
 \"{U}ber die Anfangswertaufgabe f\"{u}r die hydrodynamischen
Grundgleichungen, Math. Nachr., \textbf{4}(1951), 213--231.

\bibitem{leray} J. Leray;
 Sur le mouvement d'un liquide visqueux emplissant
l'espace. \emph{Acta Math.}, \textbf{63}(1934), 193-248.

\bibitem{lin} F. Lin; A new proof of the Caffarelli-Kohn-Nirenberg Theorem,
\emph{Comm. Pure Appl. Math.}, \textbf{51}(1998), 241-257.

\bibitem{maremonti} P. Maremonti;
 Partial regularity of a generalized solution to
the Navier-Stokes equations in exterior domains, \emph{Comm. Math.
Phys.}, \textbf{110}(1087), 75-87.

\bibitem{miyakawa} T. Miyakawa and H. Sohr;
 On the energy inequality, smoothness and large time behavior
in $L^2$ for weak solutions of the Navier- Stokes equations
in exterior domains, \emph{Math.Z.} \textbf{199}(1988), 455-478.

\bibitem{scheffer76} V. Scheffer;
 Turbulence and Hausdorff dimension, in
\emph{Turbulence and Navier-Stokes equations}, Lect. Notes in Math.
No. 565, 94-112, 1976.

\bibitem{scheffer762} V. Scheffer;
 Partial regularity of solutions to the
Navier-Stokes equations, \emph{Pacific J. Math.}, \textbf{66}(1976),
535-552.

\bibitem{scheffer77} V. Scheffer;
 Hausdorff measure and the Navier-Stokes
equations, \emph{Comm. Math. Phys.}, \textbf{55}(1977), 97-112.

\bibitem{scheffer78} V. Scheffer;
 The Navier-Stokes equations in space dimension
four, \emph{Comm. Math. Phy.}, \textbf{61}(1978), 41-68.

\bibitem{scheffer80} V. Scheffer;
 The Navier-Stokes equations on a bounded
domain, \emph{Comm. Math. Phys.}, \textbf{73}(1980), 1-42.

\bibitem{serrin63} J. Serrin;
 The initial value problem for the Navier-Stokes
equations, in \emph{Nonlinear Problem}, (R.E.Langer, Ed), 69-98,
Univ. of Wisconsin Press. Madison, 1963.

\bibitem{sohr85} H. Sohr and W. von Wahl;
 A new proof of Leray's structure
theorem and the smoothness of weak solutions of the Navier-Stokes
equations for large $|x|$. \emph{Bayreuth Math. Schr.}, \textbf{
20}(1985), 153-204.

\bibitem{struwe} M. Struwe;
 On partial regularity results for the Navier-Stokes
equations, \emph{Comm. Pure Appl. Math.}, \textbf{41}(1988), 437-458.

\bibitem{tian} G. Tian \& Z. Xin;
Gradient estimation on Navier-Stokes equations,
\emph{Comm. Anal. Geom.}, \textbf{7}(1999), No. 2, 221-257.

\end{thebibliography}



\section*{Addendum posted on September 30, 2008}

Following a suggestion from the anonymous refeee (to whom the
author wants to express his gratitude), the proof of the main
theorem is rewritten as follows:

\begin{proof}[Proof of Theorem \ref{thmA}]

We will follow the ideas in \cite{foias}.
 Since $U \in L^{q^*}(\Omega)$,
\begin{equation} \label{e4.10}
U(x) =
\frac{1}{4\pi}\int_{\mathbb{R}^3}\frac{1}{|x-y|}f(y)dy =: U_0 \quad
\text{when } \partial\Omega = \emptyset, 
\end{equation}
and
\begin{equation} \label{e4.11}
U(x) =  W (x) + \frac{1}{4\pi}\int_{\Omega}\frac{1}{|x-y|}f(y)dy
\quad\text{when $\partial\Omega \neq \emptyset$ with a harmonic
function $W$.} 
\end{equation}
It is well-known that the function $W$
is smooth in the interior of $\Omega$ and is bounded in the
subdomain with positive distance away from the boundary. Extend $f$
to the outside of $\Omega$ by zero. So we only consider the partial
regularity of $U_0$. For any $x_0 \in \mathbb{R}^3$, we have
\begin{equation} \label{e4.12}
\begin{aligned}
\frac{1}{r^3}\int_{|x-x_0|\leq r}|U_0(y)|dy 
&\leq \frac{1}{4\pi}\frac{1}{r^3}\int_{|x-x_0|\leq r}
  \int_{\mathbb{R}^3}\frac{1}{|x-y|}|f(y)|dy\,dx\\
&\leq \frac{1}{4\pi}\int_{\mathbb{R}^3}\frac{1}{|x_0-x|}
  \cdot\frac{1}{r^3}\int_{|x-y|\leq r}|f(y)|dy\,dx \\
&\leq \frac{1}{4\pi}\int_{\mathbb{R}^3}\frac{1}{|x_0-x|}f^*(x)dx :=
F(x_0)
\end{aligned}
\end{equation}
with
$$
f^*(x) = \sup_r \frac{1}{r^3}\int_{|x-y|\leq r}|f(y)|dy.
$$
By \eqref{e2.6}, we know that $f \in L^q(\mathbb{R}^3)$ for any 
$1 < q < 3/2$. So it follows from the inequality on maximal functions that
$f^* \in L^q(\mathbb{R}^3)$. Since $U_0(x)$ is continuous in $x$,
from \eqref{e4.12}, we deduce that 
$$
|U(x_0)| \leq F(x_0) \quad \forall x_0\in  \mathbb{R}^3.
$$
Let
$$
M_j = \big\{x\in \mathbb{R}^3:|x-x_0| \leq 2^{-j}\big\}.
$$
For each $x_0 \in \mathbb{R}^3$,
\begin{equation} \label{e4.13}
\begin{aligned}
F(x_0)
&= \frac{1}{4\pi}\int_{\mathbb{R}^3}\frac{1}{|x_0-x|}f^*(x)dx\\
&\leq \frac{1}{4\pi}\int_{\mathbb{R}^3\setminus
M_1}\frac{1}{|x_0-x|}f^*(x)dx
 + \sum^\infty_1\frac{1}{4\pi}\int_{M_j\setminus
 M_{j+1}}\frac{1}{|x_0-x|}f^*(x)dx\\
&\leq C\|f^*\|_q\Big(\int_{\mathbb{R}^3\setminus
 M_1}\frac{1}{|x-x_0|^{q\over q-1}}dx\Big)^{1-{1\over q}} \\
&\quad + C\sum^\infty_1\Big(\int_{M_j\setminus
 M_{j+1}}\frac{1}{|x-x_0|^{q\over q-1}}dx\Big)^{1-{1\over q}}
 \Big(\int_{M_j}|f^*(x)|^qdx\Big)^{1/q}\\
&\leq C + C\sum^\infty_1 2^{j -3j(1-{1\over  q})}
\Big(\int_{M_j}|f^*(x)|^qdx\Big)^{1/q}.
\end{aligned}
\end{equation}
 It is obvious that $|f^*|^q \in L^1(\mathbb{R}^3)$.
Let $\Lambda_a(f^*) $ be the set of these $x_0 \in \mathbb{R}^3$ such that
there exists $j_{x_0}$ with
$$
\int_{|x_0-x|\leq 2^{-j}}|f^*(x)|^qdx \leq 2^{-aj}
$$
for all $j \geq j_{x_0}$. Thus, for each $x_0 \in \Lambda_a(f^*)$,
\eqref{e4.13} gives us
\begin{equation} \label{e4.14}
\begin{aligned}
|F(x_0)| &\leq C + C \sum_1^{j_{x_0}-1}2^{j -3j(1-{1\over  q})}
\Big(\int_{M_j}|f^*(x)|^qdx\Big)^{1\over q}+ C\sum^\infty_{j_{x_0}}
2^{j -3j(1-{1\over q})} 2 ^{-{aj\over q}} \\
&\leq C_1(x_0)
\end{aligned}
\end{equation}
provided that $a > 3 - 2q$.
Then, for each $x_0 \in \Lambda_a(f^*)$ with $a > 3 - 2q$, 
\eqref{e4.14} tells us that
$$
|U_0(x_0) | \leq F(x_0) \leq C_1(x_0) < \infty.
$$
It is obvious that
$$
\Omega_0 = \big\{x\in \mathbb{R}^3:~|U_0(x)| = \infty\big\} \subset
\mathbb{R}^3\setminus \Lambda_a(f^*).
$$
Applying Lemma \ref{lem3.1}, we deduce that the Hausdorff dimension of
$\Omega_0$ is less or equal to $a$. Letting $a \to 3-2q$, we deduce
that the Hausdorff dimension of $\Omega_0$ does not exceed $3-2q$.
Since $q \in (1, 3/2)$ is arbitrary, we deduce that the Hausdorff
dimension of $\Omega_0$ is zero. This completes the proof
\end{proof}




\end{document}