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{\tenrm\ifodd\pageno\rightheadline \else
\leftheadline\fi}\fi}
\def\rightheadline{EJDE--1999/08\hfil Partial regularity for flows \hfil\folio}
\def\leftheadline{\folio\hfil Changyou Wang
\hfil EJDE--1999/08}
\voffset=2\baselineskip
\vbox {\eightrm\noindent\baselineskip 9pt %
 Electronic Journal of Differential Equations,
Vol. {\eightbf 1999}(1999) No.~08, pp. 1--8.\hfill\break
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\hfil\break ftp  ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)}
\footnote{}{\vbox{\hsize=10cm\eightrm\noindent\baselineskip 9pt %
1991 {\eighti Subject Classification:} 35B65, 35K65. 
\hfil\break
{\eighti Key words and phrases:} H-surfaces, Lorentz space, Hardy space. 
\hfil\break
\copyright 1999 Southwest Texas State University  and
University of North Texas.\hfil\break
Submitted August 29, 1998. Published March 4, 1999.
} }

\bigskip\bigskip

\centerline{PARTIAL REGULARITY FOR FLOWS OF $H$-SURFACES, II}
\medskip
\centerline{Changyou Wang} 
\bigskip\bigskip

{\eightrm\baselineskip=10pt \narrower
\centerline{\eightbf Abstract}
We study the regularity of weak solutions to the heat 
equation for $H$-surfaces. Under the assumption that the 
function $H:{\Bbb R}^3 \to {\Bbb R}$ is bounded and Lipschitz, 
we show that the solution is $C^{2,\alpha}$ on its domain, 
except for a set of measure zero.
\bigskip}
\bigbreak

\centerline{\S 1. Introduction} \medskip\nobreak

Let $\Omega\subset {\Bbb R}^2$ be a bounded domain with smooth boundary, and 
let $H$ be a bounded Lipschitz function on ${\Bbb R}^3$. 
A map $u\in C^2(\Omega,{\Bbb R}^3)$ is called an $H$-surface
(parametrized over $\Omega$) if $u$ satisfies
$$
-\Delta u=2H(u)u_{x_1}\wedge u_{x_2}\,.\eqno(1.1)
$$
It is well known that if
$u=(u^1,u^2,u^3)$ is a conformal representation of a surface $S\subset {\Bbb R}^3$,
then the mean curvature of $S$, at the point $u$, is $H(u)$ (see [S3]).  
The existence of surfaces with constant mean curvature under various boundary
conditions has been studied by Hildebrandt [Hs], Wente [W], Struwe [S1]
[S2] [S3], and Brezis-Coron [BC].  
The regularity of weak solutions to (1.1) has also been studied; see for example
Wente [W], Heinz [He], Tomi [T], and Bethuel-Ghidaglia [BG] for earlier results.
Moreover, Bethuel [B] proved that weak solutions to (1.1) are $C^{2,\alpha}$
for any bounded Lipschitz function $H$.

The heat equation of $H$-surfaces is defined by
$$
\partial_{t} u-\Delta u =2H(u) u_{x_1}\wedge u_{x_2},\quad  \hbox{in }
\Omega\times {\Bbb R}^+. \eqno(1.2) 
$$
This equation describes an evolution process of (1.1), which
models the deformation of a surface into another surface
with mean curvature $H$ at time infinity.
The existence of global smooth solutions to (1.2), under
special conditions on the $H$-function, has been studied in [R] and [S2].
In particular, Struwe [S2] considered free boundary conditions of 
(1.2), with constant $H$,
and obtained a global weak solution to (1.2), which is
smooth except for finitely many singular points.
Rey [R] has established  the existence of a global smooth 
solutions to (1.1) with the Dirichlet boundary conditions $u=\phi$, 
provided that $\phi\in H^1\cap L^{\infty}(\Omega,{\Bbb R}^3)$
and
$$\|\phi\|_{L^\infty (\Omega)}\|H\|_{L^\infty ({\Bbb R}^3)} <1\,. \eqno(1.3)$$

Motivated by the notion of weak solution
to (1.1), we say that $u:\Omega\times {\Bbb R}^+\to {\Bbb R}^3$ is a weak solution
to (1.2),
if $\partial_t u, Du \in L^2_{\hbox{loc}}(\Omega\times {\Bbb R}^+)$ and
$u$ satisfies (1.2) in the sense of distributions.  

In this note, we consider the partial regularity of weak solutions to
(1.2).  The motivation is two folds.  First, (1.2) is a parabolic counterpart of
the elliptic system (1.1) which exhibits full regularity, and in the case of a
single equation we know that the parabolic equation roughly has the same
regularity as its elliptic counterpart.  
Second, the nonlinear term in (1.2) is of the same order as that in 
the flows of harmonic maps from surfaces (see [S3]),
and the best regularity for heat equations of harmonic maps from surfaces is
that there are finitely many singular points (see Freire [F] or Wang [Wa]).
This suggests that weak solutions to (1.2) may have regularity similar to that
of heat equations of harmonic maps from surfaces.  However, the heat equation of
harmonic maps is the negative gradient flow of the Dirichlet energy functional,
which satisfies the energy inequality property, but it is not clear whether 
smooth solutions to (1.2) satisfy
$$
\int_\Omega |Du|^2(\cdot, t)\le \int_\Omega |Du|^2(\cdot, s), 
\quad 0\le s \le t<\infty\,. \eqno(1.4)
$$      
This makes the study of the size and the dimension of singular sets
of weak solutions to (1.2) much more difficult.
In fact, we are only able
to show in Theorem~1 that the singular set
has zero Lebesgue measure, which is far from the
conjecture that the singular set has (parabolic) Hausdorff
dimension at most $1$.

In [Wa1], we studied the partial regularity of weak
solutions to (1.2) under the condition that
$H$ is a bounded Lipschitz function depending
only on two variables. A uniqueness result can be found in Chen [Ch]. 

\proclaim Theorem 1.
Let  $H(p): {\Bbb R}^3\to {\Bbb R}$ be bounded and Lipschitz continuous,
and let $u\in H^1(\Omega\times {\Bbb R}^+, {\Bbb R}^3)$ be a weak  
solution of (1.2).  
Then there exists a closed subset $\Sigma=\cup_{t>0}\Sigma_t
\subset \Omega\times {\Bbb R}^+$, with $\Sigma_t\subset\Omega\times\{t\}$
 finite for almost all $t>0$, such that  $u\in C^{2,\alpha}
(\Omega\times {\Bbb R}^+\setminus\Sigma, {\Bbb R}^3)$. In particular, $
\Sigma $ has Lebesgue measure zero. 

\bigbreak
\centerline{\bf \S 2. Proof of the main theorem } \medskip\nobreak
The goal of this section is to prove Theorem 1. 
First we show that the solution $u: B_1\times (0,1]\to {\Bbb R}^3$
has spatial H\"older continuity in $B_{1/2}$
uniformly with respect to $t\in [1/2,1]$, under the
assumption that $\int_{B_1}|Du|^2$ is small
and $\int_{B_1}|\partial_t u|^2$ is bounded,
uniformly with respect to $t\in [0,1]$. Then based
on the spatial continuity of $u$, and a simple observation, we obtain
the continuity of $u$ in the time direction.
 Finally, by elementary covering and suitable rescaling arguments, 
 we show that $u$ has regularity almost everywhere.

To make the proof clear, we review a few concepts. First, we
recall the definition of Lorentz spaces [Z]. 
For an open set $W\subset {\Bbb R}^2$ and  $1\le q\le\infty$, let
$$L^{2,q}(W)=\{f: W\to {\Bbb R} \hbox{ measurable },
\|f\|_{L^{2,q}(W)}<\infty\}\,.$$
The norm in this space is defined by 
$$
\|f\|_{L^{2,q}(W)}=
\cases{ (\int_0^\infty [t^{1/2}f^*(t)]^q{1\over t}\,dt)^{1/q}, 
& if $1\le q<\infty$ ; \cr
\sup_{t>0}t^{1/2}f^*(t), & if $q=\infty$,\cr}
$$
where $f^*(t):=\inf\{s>0: |\{x\in W: |f(x)|>s\}|\le t\}$ is the 
the rearrangement of $f$.
Notice that 
$L^{2,1}\subset L^{2,2}(\equiv L^2)\subset L^{2,\infty}$,
and that $L^{2,1}$ and $L^{2,\infty}$ are dual of each other.
For $x_0\in {\Bbb R}^2$, $0<r<\infty$, let $B(x_0,r)=\{y\in {\Bbb R}^2: |y-x_0|\le r\}$. 


\proclaim Lemma 2.1.  For  $f, g\in H^1(B(x_0,r))$,
let $v\in H^1_0(B(x_0,r))$ be the solution to 
$$\displaylines{\hfill
-\Delta v = f_x g_y-f_y g_x,\quad \hbox{in }B(x_0,r), \hfill\rlap (2.1)\cr
 v = 0,\quad \hbox{on }\partial B(x_0,r).\cr}$$
Then $Dv\in L^{2,1}(B(x_0,r))$ and
$$\|Dv\|_{L^{2,1}(B(x_0,r))}\le C\|Df\|_{L^2(B(x_0,r))}\|Dg\|_{L^2(B(x_0,r))}.
\eqno(2.2)$$
$$\|Dv\|_{L^{2,\infty}(B(x_0,r))}\le C\|Df\|_{L^2(B(x_0,r))}\|Dg\|_{L^{2,\infty}
(B(x_0,r))}.\eqno(2.3)$$

The proof of the Lemma above can be found in H\'elein [Hf1],
Theorems 3.33--3.38, page 146-155.


\proclaim  Lemma 2.2.  For $f\in L^1(B(x_0,r))$, let $v\in H^1(B(x_0,r))$
be the solution to
$$-\Delta v =f, \quad \hbox{in } B(x_0,r)\,.$$
Then there exists a $C>0$ such that, for any $\theta\in (0,1/4)$,
$$\|Dv\|_{L^{2,\infty}(B(x_0,\theta r))}\le C\theta 
\|Dv\|_{L^{2,\infty}(B(x_0, r))}+C\|f\|_{L^1(B(x_0,r))}\,.\eqno(2.4)$$

\noindent{\bf Proof}. Let $\bar f:{\Bbb R}^2\to {\Bbb R}$
be an extension of $f$ such that $\bar f=0$
outside $B(x_0,r)$. Let 
$\bar v\in W^{1,1}({\Bbb R}^2)$ be a solution to
$$-\Delta \bar v =\bar f,\quad \hbox{ in }{\Bbb R}^2.$$
Then 
$$D(\bar v)(z)=\int_{{\Bbb R}^2}DK(z-x)\bar f(x)\,dx\,,$$
where $K(z)={1\over 2\pi}\log(|z|^{-1})$. It is well known
(cf. [Z]) that $DK\in L^{2,\infty}({\Bbb R}^2)$. Hence, it follows
from the convolution property that
$D\bar v\in L^{2,\infty}({\Bbb R}^2)$, and that
$$\|D\bar v\|_{L^{2,\infty}({\Bbb R}^2)}\le C\|DK\|_{L^{2,\infty}({\Bbb R}^2)}\|f\|_{L^1({\Bbb R}^2)}
\le C\|f\|_{L^1({\Bbb R}^2)}.\eqno(2.5)$$
Since $v-\bar v$ is a harmonic function on $B(x_0,r)$,
an estimate of harmonic functions in [Hf1] implies
that
$$\|D(v-\bar v)\|_{L^{2,\infty}(B(x_0,\theta r))}
\le C\theta\|D(v-\bar v)\|_{L^{2,\infty}(B(x_0, r))},\eqno(2.6)$$
for any $\theta\in (0,1/4)$.
Hence 
$$\|Dv\|_{L^{2,\infty}(B(x_0,\theta r))}\le C\theta
\|Dv\|_{L^{2,\infty}(B(x_0, r))}+C\|D\bar v\|_{L^{2,\infty}(B(x_0, r))},$$
this, combined with (2.5), implies (2.4). \quad$\diamondsuit$  

The key part of the proof of Theorem 1 is the following
decay property.

\proclaim Lemma 2.3. There exist $\epsilon_0>0$
and $\theta_0\in (0, 1/4)$ such that if $u\in H^1(B_1\times (0,1],{\Bbb R}^3)$
is a weak solution of (1.2) and
$\sup_{t\in (0,1]}\int_{B_1}|Du|^2\le\epsilon_0^2$
then, for $x_0\in B_{1/2}$, $0<r<1/4$, $\theta\in (0,\theta_0)$,
we have 
$$\|Du\|_{L^{2,\infty}({B(x_0,\theta r)})}\le
{1\over 2}\|Du\|_{L^{2,\infty}(B(x_0,r))}+C\|\partial_t u\|_{L^2(B(x_0,r))}
 r\,,  \eqno(2.7)$$
for almost all $t\in [1/2, 1]$.

\noindent{\bf Proof}. The argument here is inspired by
that of Bethuel [Bf]. For
$x_0\in B_{1/2}$, $r\in(0,1/4)$, and
$t\in [1/2,1]$. We apply the $L^2$-Hodge decomposition
theorem to get that
there exist $A\in H^1(B(x_0,r), {\Bbb R}^3)$ and $B\in H^1_0(B(x_0,r), {\Bbb R}^3)$
such that
$$(2H(u)u_x^i, 2H(u)u_y^i)=(A_x^i, A_y^i)+(B_y^i, -B_x^i),
\ i=1, 2, 3, \eqno(2.8)$$
and
$$\|DA\|_{L^2(B(x_0,r))}+\|DB\|_{L^2(B(x_0,r))}\le C\|Du\|_{L^2(B(x_0,r))}.
\eqno(2.9)$$
Then we have
$$\displaylines{\hfill
 \Delta B =2(H(u)_y u_x -H(u)_x u_y),\quad \hbox {in }B(x_0,r),\hfill
\rlap(2.10)\cr
B =0,\quad \hbox{on }\partial B(x_0,r)\,.\cr}$$
Hence Lemma 2.1 implies 
$$\|DB\|_{L^{2,1}(B(x_0,r))}\le C\|Du\|_{L^2(B(x_0,r))}^2\,.\eqno(2.11)$$
Using (2.8), we can write (1.2) as
$$\Delta u=\partial_t u -A_x\wedge u_y -B_y\wedge u_y, \hbox{ in } B(x_0,r).
\eqno(2.12)$$
Let $w\in H^1_0(B(x_0,r), {\Bbb R}^3)$ be  the solution to
$$\displaylines{\hfill
\Delta w =-A_x\wedge u_y ,\quad \hbox {in }B(x_0,r), \hfill\rlap(2.13)\cr
w =0,\quad \hbox{ on }\partial B(x_0,r).\cr}$$
Hence, by Lemma 2.1,   
$$\eqalignno{\|Dw\|_{L^{2,\infty}(B(x_0,r))}&\le
C\|DA\|_{L^2(B(x_0,r))}\|Du\|_{L^{2,\infty}(B(x_0,r))}\cr
&\le
C\|Du\|_{L^2(B(x_0,r))}\|Du\|_{L^{2,\infty}(B(x_0,r))}. &(2.14)\cr}$$
Now we can apply Lemma 2.2, to
$u-w$ on $B(x_0,r)$, to conclude that,
for $\theta\in(0,1/4)$, 
$$\eqalignno{\|D(u-w)\|_{L^{2,\infty}(B(x_0,\theta r))}
\le& C\theta \|D(u-w)\|_{L^{2,\infty}(B(x_0,r))}\cr
&+\|\partial_t u\|_{L^1(B(x_0,r))}+\|B_y\wedge u_y\|_{L^1(B(x_0,r))}\cr
\le& C\theta \|D(u-w)\|_{L^{2,\infty}(B(x_0,r))}\cr
&+\|\partial_t u\|_{L^2(B(x_0,r))}r+\|DB\|_{L^{2,1}(B(x_0,r))}\|Du\|_{L^{2,\infty}
(B(x_0,r))}\cr
\le& C\theta \|D(u-w)\|_{L^{2,\infty}(B(x_0,r))}\cr
&+\|\partial_t u\|_{L^2(B(x_0,r))}r
+C\|Du\|_{L^2(B(x_0,r))}^2 \|Du\|_{L^{2,\infty}
(B(x_0,r))}\cr}$$
This, combined with (2.14), imply that
$$\eqalignno{\|Du\|_{L^{2,\infty}(B(x_0,\theta r))}
\le& (C\theta +C\|Du\|_{L^2(B(x_0,r))}^2)\|Du\|_{L^{2,\infty}(B(x_0,r))}\cr
&+C\|Dw\|_{L^{2,\infty}(B(x_0,r))}+\|\partial_t u\|_{L^2(B(x_0,r))} r\cr
\le& (C\theta+C \epsilon_0)\|Du\|_{L^{2,\infty}(B(x_0,r))}+
\|\partial_t u\|_{L^2(B(x_0,r))} r.\cr}$$
Therefore, if we choose $\theta_0\in(0,1/4)$ and
$\epsilon_0>0$ sufficiently small then (2.7) follows. \quad$\diamondsuit$ 

A direct consequence of Lemma 2.3 is the following.

\proclaim Corollary 2.4. There exist $\epsilon_0>0$
and $\alpha_0\in (0,1)$ such that if $u\in H^1(B_1\times (0,1],{\Bbb R}^3)$
is a weak solution to (1.2) with 
$\sup_{t\in (0,1]}\int_{B_1}|Du|^2\le\epsilon_0^2$ and \hfil\break
 $\Lambda =\sup_{t\in(0,1]}\int_{B_1}|\partial_t u|^2<\infty$, then
$u(t,\cdot)\in C^{\alpha_0}(B_{1/2},{\Bbb R}^3)$ 
for $t\in [1/2,1]$, and
$$\sup_{t\in [1/2,1]}\|u(t,\cdot)\|_{C^{\alpha_0}(B_{1/2})}
\le C(\epsilon_0,\Lambda )\,.\eqno(2.15)$$

\noindent{\bf Proof.} Notice that the $L^{2,\infty}$-norm
is conformally invariant. Hence we can iterate (2.7) of Lemma 2.3
to conclude that there exists $\theta_0\in (0,1/4)$
such that for any $x_0\in B_{1/2}$, $r\in (0,1/4)$,
and $t\in [1/2,1]$,
$$\|Du\|_{L^{2,\infty}(B(x_0,\theta_0^kr))}
\le 2^{-k}\|Du\|_{L^{2,\infty}(B(x_0,r))}+C(1-\theta_0)^{-1}\Lambda  r, 
\eqno(2.16)$$ 
for all $k\ge 1$.
This certainly implies (see for example [GT] Lemma 8.23) that there
exists $\alpha_0\in (0,1)$ such that for all $t\in [1/2,1]$
$$\|Du\|_{L^{2,\infty}(B(x,r))}\le Cr^{\alpha_0}
\|Du\|_{L^{2,\infty}(B(x,1/4)}+C\Lambda  r^{1\over 2},\eqno(2.17)$$
for any $x\in B_{1/2}$ and $0<r\le 1/4$.
On the other hand, we know that
$L^{2,\infty}\subset L^{1,p}$ for any $p\in (1,2)$. In particular,
$$\eqalignno{r^{2-p}\int_{B(x,r)}|Du|^p 
&\le C\|Du\|_{L^{2,\infty}(B(x,r))}^p\cr
&\le Cr^{p\alpha_0}
\|Du\|_{L^{2,\infty}(B(x,1/4)}^p+C\Lambda  r^{p/2}\,, &(2.18)\cr}$$
for any $x\in B_{1/2}$, $r\in (0,1/4)$,
and $t\in [1/2,1]$.
This, combined with Morrey Lemma (see [Mc]),  imply 
$u(t,\cdot)\in C^{\alpha_0}(B_{1/2},{\Bbb R}^3)$ 
for $t\in [1/2,1]$, and (2.15). \quad$\diamondsuit$ 

Based on Corollary 2.3 and a simple observation, we actually
get the H\"older continuity of $u$ in the time direction as follows.

\proclaim Corollary 2.5. There exist
$\epsilon_0>0$ and $\alpha_1\in (0,1)$ such that
if $u\in H^1(B_1\times [0,1],{\Bbb R}^3)$ is a weak solution
to (1.2) with
$$\sup_{t\in (0,1]}\int_{B_1}|Du|^2\le\epsilon_0^2,$$
and $\Lambda =\sup_{t\in(0,1]}\int_{B_1}|\partial_t u|^2<\infty$, then
$u\in C^{\alpha_1}(B_{1/2}\times [1/2,1], {\Bbb R}^3)$.

\noindent{\bf Proof}. For any $x\in B_{1/2}$, $r\in (0, 1/4)$,
and ${1\over 2}\le t_1<t_2\le 1$. We have 
$$\eqalignno{|u(x,t_1)-u(x,t_2)| \le& |u(x,t_1)-{1\over |B(x,r)|}
\int_{B(x,r)}u(y,t_1)|\cr
&+|u(x,t_2)-{1\over |B(x,r)|}
\int_{B(x,r)}u(y,t_2)|\cr
&+{1\over |B(x,r)|}\int_{B(x,r)}|u(y,t_1)-u(y,t_2)|\cr
\le& \hbox{osc}_{B(x,r)}u(\cdot,t_1)+\hbox{osc}_{B(x,r)}u(\cdot,t_2)\cr
&+{1\over |B(x,r)|}\int_{B(x,r)}\,dy\int_{t_1}^{t_2}|\partial_t u(y,t)|\,dt\cr
\le& C(\epsilon_0,\Lambda )r^{\alpha_0}+\|u_t\|_{L^2(B_1\times (0,1])}
{\sqrt{t_2-t_1}\over r}.\cr}$$
Here osc denotes the oscillation and  we have used H\"older inequality and
$\alpha_0\in (0,1)$ is given by Corollary 2.4. 
Now if we choose $t_1, t_2$ such that
$|t_1-t_2|\le 4^{1/(2(1+\alpha_0))}$, and let
$r=|t_1-t_2|^{1/( 2(1+\alpha_0))}(\le {1\over 4})$,
then
$$|u(x,t_1)-u(x,t_2)|\le (C(\epsilon_0,\Lambda )+
\|\partial_t u\|_{L^2(B_1\times (0,1])})|t_1-t_2|
^{\alpha_0/(2(1+\alpha_0))},
\ \forall x\in B_{1/2}.\eqno(2.19)$$
Let $\alpha_1=\alpha_0/( 2(1+\alpha_0))$. Then (2.15) and (2.19)
imply that $u\in C^{\alpha_1}(B_{1/2}\times [1/2,1],{\Bbb R}^3)$.
\quad$\diamondsuit$ 

\medskip
\noindent{\bf Completion of the proof of Theorem 1}.

\noindent Define the parabolic metric:  $\delta((x,t),(y,s))
=\max\{|x-y|,\sqrt{|t-s|}\}$. 
For $(x,t)\in\Omega\times {\Bbb R}^+$ and 
$R\in (0,\delta((x,t),\partial (\Omega\times
{\Bbb R}^+)))$. Define 
$$\eqalignno{M_{R}^1(x,t)&=\limsup_{s\uparrow t}\int_{B_R(x)}|Du|^2(x,s)\cr
             M_{R}^2(x,t)&=\limsup_{s\uparrow t}\int_{B_R(x)}|\partial_tu|^2(x,s),
\cr}$$
for the weak solution $u$ of (1.2). It is easy
to see that $M_{R}^i(x,t)$ is non-decreasing with
respect to $R$ so that $M^i(x,t)=\lim_{R\downarrow 0}
M_R(x,t)$ exists and is upper semi-continuous
exists for any $(x,t)\in\Omega\times {\Bbb R}^+$, for $i=1, 2$.
Let $\epsilon_0>0$ be as same as Corollary 2.5. For $t>0$, define
$\Sigma_t\equiv\Sigma_t^1\cup\Sigma_t^2
(\subset\Omega)$, where
$$\displaylines{
\Sigma_t^1 =\{x \in \Omega: M^1(x,t)\ge\epsilon_0^2\}\cr
\Sigma_t^2 =\{x \in \Omega: M^2(x,t)=\infty\},\cr}
$$  
let  $\Sigma=\cup_{t>0}\Sigma_t$. Then it follows that
$\Sigma$ is a closed subset of $\Omega\times {\Bbb R}^+$. \medskip

\noindent{\bf Claim.} $u\in C^{2,\alpha}(\Omega\times {\Bbb R}^+
\setminus\Sigma, {\Bbb R}^3)$ for some $\alpha\in (0,1)$. 
To prove this claim, Let $(x_0,t_0)\in\Omega\times {\Bbb R}^+
\setminus\Sigma$. By definition, there exists $r_0>0$ such that 
$$M_{r_0}^1(x_0, t_0)<\epsilon_0^2, \  \Lambda _0\equiv M_{r_0}^2(x_0,t_0)<\infty.$$
For such $r_0$, there exists $0<\delta_0\le r_0$ such that
$$\sup_{[t_0-\delta_0^2,t_0]}\int_{B_{r_0}(x_0)}|Du|^2(x,t)\,dx\le\epsilon_0^2,
$$
and 
$$\sup_{[t_0-\delta_0^2,t_0]}\int_{B_{r_0}(x_0)}|\partial_t u|^2(x,t)\,dx\le 
2\Lambda _0.$$
If we define the rescaled mappings $u_{\delta_0}: 
B_1\times (-1, 0]\to {\Bbb R}^3$
by $u_{\delta_0}(x,t)=u(x_0+\delta_0x,t_0+\delta_0^2 t)$, then
$u_{\delta_0}$ is a weak solution to (1.2) on $B_1\times (-1,0]$ and
satisfies 
$$\sup_{(-1,0]}\int_{B_1}|Du_{\delta_0}|^2(x,t)\,dx\le
\epsilon_0^2,$$
and
$$\sup_{(-1,0]}\int_{B_1}|\partial_tu_{\delta_0}|^2(x,t)\le 2\Lambda _0.$$
Hence Corollary 2.5 implies 
$$u_{\delta_0}\in C^{\alpha}(B_{1/2}\times [-{1\over 2},0],{\Bbb R}^3),
$$ 
for some $\alpha\in (0,1)$. This means that
$u\in C^{\alpha}(B(x_0,\delta_0)\times (t_0-\delta_0^2,t_0+\delta_0^2),
{\Bbb R}^3)$. Since
$(x_0,t_0)$ is arbitrary in $\Omega\times {\Bbb R}^+
\setminus\Sigma$, this shows that $u\in C^{\alpha}(\Omega\times {\Bbb R}^+
\setminus\Sigma,{\Bbb R}^3)$. It is well known that
$C^{\alpha}$ solutions to (1.2) is in $C^{2,\alpha}$ as well.  
\smallskip

Now we estimate the size of $\Sigma_t$
for a.e. $t>0$. Since $u\in H^1_{\rm loc}(\Omega\times {\Bbb R}^+)$,
the set 
$$A=\{t_0\in {\Bbb R}^+: \liminf_{t\uparrow t_0}\int_\Omega
|Du|^2(x,t)+|\partial_t u|^2\,dx=+\infty\}$$
has Lebesgue measure equal to zero, $|A|=0$.
For any $t_1\in {\Bbb R}^+\setminus A$, it is
easy to see that $\Sigma_{t_1}^2=\emptyset$. 
We claim that $\Sigma_{t_1}^1$ is
finite. In fact, let $\{x_1,\cdots, x_N\}$ be a finite subset of
$\Sigma_{t_1}^1$. Then we can choose ${\Bbb R}^0>0$ such that
$\{B_{R _0}(x_i)\}_{i=1}^N$ are mutually disjoint and
$$\limsup_{t\uparrow t_1}\int_{B_{{R}^0}(x_i)}|Du|^2(x,t)\,dx\ge
\epsilon_0^2, \quad 1\le i\le N\,.$$ 
Therefore,
$$\eqalignno{\liminf_{t\uparrow t_1}\int_{\Omega
\setminus\cup_{i=1}^N B_{{R}^0}(x_i)}|Du|^2 &\le
\liminf_{t\uparrow t_1}\int_{\Omega}|Du|^2-
\sum_{i=1}^N\limsup_{t\uparrow t_1}\int_{B_{{R}^0}(x_i)}|Du|^2\cr
&\le \liminf_{t\uparrow t_1}\int_{\Omega}|Du|^2-N\epsilon_0^2\,.\cr}
$$
Hence $N\le \epsilon_0^{-2}\liminf_{t\uparrow t_1}\int_{\Omega}|Du|^2$,
which implies $\Sigma_{t_1}^1$ is finite. It then follows
from Fubini's theorem  that $\Sigma$ Lebesgue measure equal to zero.
 \quad$\diamondsuit$

\bigskip 
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\bigskip \noindent
Changyou Wang \hfil\break
Department of Mathematics,  Loyola University of Chicago, Chicago, IL 60626. 
USA
\hfill\break
E-mail address: wang@math.luc.edu

\end