\documentclass[reqno]{amsart}
\usepackage{amssymb}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 90, pp. 1--24.\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 2004 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE-2004/90\hfil Partial compactness]
{Partial compactness for  the 2-D Landau-Lifshitz flow}

\author[P. Harpes\hfil EJDE-2004/90\hfilneg]
{Paul Harpes}

\address{Paul Harpes \hfill\break
ETH Z\"urich \\
R\"amistrasse 101, 8092 Z\"urich, Switzerland}
\email{pharpes@math.ethz.ch}

\date{}
\thanks{Submitted September 11, 2003. Published July 5, 2004.}
\subjclass[2000]{35B65, 35B45, 35D05, 35D10, 35K45, 35K50, 35K55}
\keywords{Partial compactness; partial regularity;  
Landau-Lifshitz  flow; \hfill\break\indent 
a priori  estimates; harmonic map flow; non-linear parabolic;
 Struwe-solution; approximations}

\begin{abstract}
  Uniform local $C^\infty$-bounds for Ginzburg-Landau type
  approximations for the Landau-Lifshitz flow on planar domains
  are proven. They hold outside an energy-concentration set of
  locally finite parabolic Hausdorff-dimension 2, which has
  finite times-slices. The approximations subconverge to a global
  weak solution of the Landau-Lifshitz flow, which is smooth away
  from the energy concentration set.  The same results hold for
  sequences of global smooth solutions of the 2-d Landau-Lifshitz flow.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemm}[theorem]{Lemma}
\newtheorem{prop}[theorem]{Proposition}
\newtheorem{coro}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}

\def\abs#1{|#1|}
\def\norm#1{\|#1\|}

\section{Introduction}

The Ginzburg-Landau approximations
$u_\epsilon: \overline{\Omega} \times \mathbb{R}_+
\to \mathbb{R}^3$
to the Landau-Lifshitz flow are solutions of
\begin{gather} \label{Eq_eps_LL}
\gamma_1 {\partial}_t u_\epsilon   -   \gamma_2  u_\epsilon \times {\partial}_t u_\epsilon
- \Delta u_\epsilon  = - \frac{1}{\epsilon^2} f(u_\epsilon)
 \quad \mbox{ in } \Omega \times \mathbb{R}_+ \\
 \label{Eq_bdry_eps_LL}
u_\epsilon = u_0 \quad \mbox{ on } \bigl( \Omega \times \{ 0 \} \bigr)
\cup \bigl( \partial \Omega \times \mathbb{R}_+ \bigr)  \,.
\end{gather}
where $ \gamma_1 > 0 $ and
$\gamma_2 \in \mathbb{R}$. "$\times$" denotes the usual vector product in
$\mathbb{R}^3$. The domain
$ \Omega \subset \mathbb{R}^2$ is open, bounded and smooth.
The initial and boundary data
$u_0$ is always assumed to map a.e.  into the standard sphere $S^2 \subset \mathbb{R}^3$ or an
embedded manifold $N \subset \mathbb{R}^n$ (see below).
For the definition of $f_\epsilon$ we distinguish two cases:

\noindent{\bf Case (I):} If $\gamma_2 \neq 0$,
the target is $S^2 \hookrightarrow \mathbb{R}^3$
and the right hand side is given by
$$
f(u_\epsilon) := - (1-\abs{u_\epsilon}^2)   u_\epsilon
= \frac{1}{4}   \frac{d}{du}
\bigl( 1-|u_\epsilon|^2\bigr)^2 \,.
$$
For small $\epsilon > 0$ the maps $u_\epsilon$ then approximate
the Landau-Lifshitz flow
\begin{equation} \label{LL}
\gamma_1   \partial_t u - \gamma_2
  u \times \partial_tu - \Delta u = \abs{\nabla u}^2 u
\quad \mbox{ in } \Omega \times \mathbb{R}_+  \,.
\end{equation}
For sufficiently regular solutions
$u:\overline{\Omega}\times\mathbb{R}  \to S^2$
equation \eqref{LL}
is equivalent to
\begin{equation}
\label{ LL-original}
\partial_tu = - \alpha  u \times (u\times\Delta u) + \beta    u \times
\Delta u
   \mbox{ in } \Omega \times \mathbb{R}_+  \,,
\end{equation}
where $ \alpha := \frac{\gamma_1}{\gamma_1^2+\gamma_2^2} > 0$ and
$ \beta := \frac{\gamma_2}{\gamma_1^2+\gamma_2^2} \in \mathbb{R}$.
This is the usual form of the Landau-Lifshitz equations known in physics.
(Compare \cite{LL} and
\cite{Guo-Hong1}, \cite{Harpes0}, \cite{Harpes1}.)


\noindent{\bf Case (II):} If $\gamma_2 = 0$, the target is a smooth, closed,
isometrically embedded manifold
$N \hookrightarrow \mathbb{R}^n$.
For small $\epsilon >0$, the map
$u_\epsilon: \overline{\Omega} \times\mathbb{R}_+ \to \mathbb{R}^n$
will then be an
approximation of a harmonic map flow
(compare \cite{Struwe-Chen})
and is defined to be a solution of
\begin{gather}\label{Eq_eps_HMF}
{\partial}_t u_\epsilon - \Delta u_\epsilon
 = - \frac{1}{2\epsilon^2} \frac{d}{du}\chi \bigl(\operatorname{dist}^2(u_\epsilon,N)\bigr)
 \quad \mbox{ in } \Omega \times \mathbb{R}_+ \\
 \label{Eq_bdry_eps_HMF}
u_\epsilon = u_0 \quad \mbox{ on } \bigl( \Omega \times \{ 0 \} \bigr)
\cup \bigl( \partial\Omega \times \mathbb{R}_+ \bigr) \,,
\end{gather}
That is, for the function $f(u_\epsilon)$ in \eqref{Eq_eps_LL}, we choose
$$
f(u_\epsilon):= \frac{1}{2} \frac{d}{du}\chi \bigl(\operatorname{dist}^2(u_\epsilon,N)\bigr)  \,,
$$
The cut-off function
$ \chi:\mathbb{R}_+ \to \mathbb{R}_+$
is smooth, non decreasing and satisfies
$ \chi (t) = t $ for $ 0 \leq t \leq \delta_N^2 $
and $ \chi (t) \equiv 2 \delta_N^2 $ for $ t \geq 4 \delta_N^2$.
The parameter $\delta_N >0$ is chosen in such a way that
the nearest neighbour projection $ U \ni x \mapsto  \pi_N(x) \in N$ is defined and smooth
on a tubular neighborhood $ U \subset \mathbb{R}^n $ of $N$ with uniform radius
radius $ 2 \delta_N >0 $.
(Such a $\delta_N > 0$ always exists if $N$ is closed. 
Compare \cite{Struwe-Chen} and \cite[Section 2.12.3 p.42]{Noebi}.)

For fixed $\epsilon >0$, smooth solutions
of \eqref{Eq_eps_LL}-\eqref{Eq_bdry_eps_LL}
or (\ref{Eq_eps_HMF})-(\ref{Eq_bdry_eps_HMF})
on $ \Omega \times \mathbb{R}_+ $ exist and
if $u_0 \in H^{1,2}(\Omega;N) \cap H^{3/2,2}(\partial\Omega;N)$,
they are unique in
$$
H^{1,2}_{\rm loc}\cap
L^\infty(H^{1,2}) :=
H^{1,2}_{\rm loc}( \overline{\Omega} \times \mathbb{R}_+;\mathbb{R}^n)\cap
L^\infty(\mathbb{R}_+;H^{1,2}(\Omega;\mathbb{R}^n))\,.
$$
(Compare \cite{Chen1},\cite{Struwe-Chen}.)
Existence is obtained by
Galerkin's method, regularity ($C^\infty$) follows from a standard bootstrap
argument and uniqueness may be proven as for the two dimensional
harmonic map flow
(see \cite{Struwe1} or \cite{Struwe2} (5${}^\circ$) p.234
in the proof of Theorem 6.6).

The total energy of the flow at time $t\geq 0$ is defined by
\begin{equation}
\label{def-en}
G_\epsilon \bigl(u_\epsilon (t)\bigr)
:= \int_\Omega  g_\epsilon (u_\epsilon)(x,t)   dx
\end{equation}
where
\begin{gather*}
\label{g_epsilon}
 g_\epsilon (u_\epsilon)
:= \frac{1}{2} \abs{\nabla u_\epsilon}^2 +\frac{1}{4 \epsilon^2}
(1-\abs{u_\epsilon}^2)^2  \quad \mbox{if } \gamma_2 \neq 0 \,,\\
g_\epsilon (u_\epsilon)
:= \frac{1}{2} \abs{\nabla u_\epsilon}^2 +\frac{1}{2\epsilon^2}
  \chi\bigl( \operatorname{dist}^2(u_\epsilon,N) \bigr)
\quad \mbox{if } \gamma_2 = 0  \,.
\end{gather*}
While the total energy of the "$\epsilon$-approximations" always decreases
(see  Lemma \ref{en-est-LL} below),
the local energy given by
\begin{equation}
\label{def-loc-en }
G_\epsilon \bigl( u_\epsilon(t)\,, B^\Omega_R(x_0) \bigr)  :=
\int_{B_R(x_0)\cap \Omega}
  g_\epsilon (u_\epsilon)(x,t) \,  dx  \,.
\end{equation}
may concentrate at space-time points $(x_0,t_0)$  as $\epsilon \searrow 0 $
either for fixed \mbox{$t=t_0$} or for variable $t \nearrow t_0$ or
$t \searrow t_0$. It characterizes the local
"asymptotic regularity behaviour" of the flow. Here asymptotic refers to
the limit $\epsilon \searrow 0$.

We will show that all the derivatives of
the family  of maps $\{ u_\epsilon \}_{\epsilon>0}$
are locally uniformly bounded on a regular set
$\mathop{\rm Reg}\bigl(\{u_\epsilon\}_{\epsilon>0} \bigr)$
consisting of all points
$$
z_0 = (x_0,t_0) \in\overline{\Omega} \times ]0,\infty[
$$
for which there is
$R_0 = R_0(z_0) >0 $, such that
\begin{equation}\label{reg-cond}
\limsup_{\epsilon \searrow 0}
\sup_{t_0 - R_0^2 < t < t_0 }
G_\epsilon \bigl( u_\epsilon(t)\,, B^\Omega_{R_0}(x_0) \bigr)
< \epsilon_0 \,,
\end{equation}
for a constant $\epsilon_0 >0$ that will be determined later.
The complement
$$
\mathbb{S}\bigl(\{u_\epsilon\}_{\epsilon>0} \bigr)
:= \bigl(\overline{\Omega} \times \mathbb{R}_+\bigr)
\smallsetminus \mathop{\rm Reg} \bigr( \{u_\epsilon\}_{\epsilon>0} \bigl)
$$
is referred to as the energy-concentration set. % We will show that
It is closed, has locally finite parabolic Hausdorff dimension two and
finite slices at fixed time.
The limits of converging subsequences  $ \{u_{\epsilon_j}\}_j$
are distributional solutions of the Landau-Lifshitz flow
(or harmonic map flow if $\gamma_2 =0$) on all
$ \Omega \times ]0,\infty[$
in
$H^{1,2}_{\rm loc}\cap L^\infty(H^{1,2})$.

Bubbling phenomena of the $\epsilon$-approximations
as $\epsilon \searrow 0 $ either for
fixed \mbox{$t=t_0$} or  for variable $t \nearrow t_0$ or $t \searrow t_0$
as described in
\cite{Harpes0} will be presented in \cite{Harpes3}.
Strong subconvergence of the harmonic map flow penalty-approximations in
$$
W^{1,0}_{2,\rm loc}\Bigl(\mathop{\rm Reg}(\{u_{\epsilon}\}_{\epsilon>0});
\mathbb{R}^n
\Bigr)
$$
to a global distributional
$ H^{1,2}_{\rm loc}\cap L^\infty(H^{1,2})$-solution
of the harmonic map flow
was already proved by M. Struwe and Y. Chen
in \cite{Struwe-Chen} for the case of
a closed domain manifold $\Omega =M$ with
dim$M = m \geq 2 $ or $ M = \mathbb{R}^m $.
($W^{1,0}_{2,\rm loc}$ refers to functions $f$, whose restriction to any
closed ball (in space-time) lies in $L^2$ as well as
the restriction of the space-gradient $\nabla f$.)

Struwe and Chen provided uniform local $L^\infty$-bounds for
$g_\epsilon (u_\epsilon)$ on $\mathop{\rm Reg}(\{u_{\epsilon}\}_{\epsilon>0})$.
Their result was extended to compact domains with boundary
by  Chen and  Lin in \cite {Chen-Lin1}.
The energy-concentration set
$ \mathbb{S}(\{u_\epsilon\}_{\epsilon >0})$ is known to have
locally finite $m (=\dim M)$-dimensional
Hausdorff measure in the case of the harmonic map flow
(see \cite{Struwe-Chen}).

X. Cheng investigates in \cite{Cheng} weak(*)
$H^{1,2}_{\rm loc}\cap L^\infty (H^{1,2})$-limits $u_*$ of
sequences of smooth solutions of the harmonic map flow
on the domain $ M = \mathbb{R}^m $
and shows that the time slice
$\mathbb{S}(\{u_k\}_k) \cap \bigl(\mathbb{R}^m \times \{ t \}\bigr)$
has finite $(m-2)$-dimensional Hausdorff measure.

Weak(*)-subconvergence in $H^{1,2}_{\rm loc}\cap L^\infty(H^{1,2})$
of the Landau-Lifshitz $\epsilon$-approx\-imations from closed surfaces
to a distributional
$ H^{1,2}_{\rm loc}\cap L^\infty(H^{1,2})$-solution
of the Landau-Lifshitz flow
was proven by B. Guo and M.C. Hong in \cite{Guo-Hong1}.

Guo and Ding also studied partial convergence of the
two dimensional Landau-Lifshitz penalty-approximations
in \cite{Ding-Guo1},\cite{Ding-Guo2} and \cite{Ding-Guo3}.
Their arguments however contain several gaps and inconsistencies.

\section{Energy-estimates}

In the case $\gamma_2 = 0$,
equation \eqref{Eq_eps_HMF} is the $L^2$-gradient flow of
the functional $ u \mapsto G_\epsilon(u)$.
\eqref{Eq_eps_LL} is not known to be a gradient flow, but
the total energy  still
decreases along the (smooth) flow
\eqref{Eq_eps_LL}-\eqref{Eq_bdry_eps_LL}.

\begin{lemm} \label{en-est-LL}
Let $ u_\epsilon $ be a solution of
\eqref{Eq_eps_LL}-\eqref{Eq_bdry_eps_LL}.
Then
\begin{gather} \label{en-est}
G_\epsilon \bigl(u_\epsilon(T)\bigr)
  +  \gamma_1\int_0^T \int_\Omega   \abs{\partial_t u_\epsilon }^2   dx  dt
  =  G_\epsilon \bigl(u_\epsilon(0)\bigr)
= E(u_0)=: E_0  \,, \\
\label{loc-en}
G_\epsilon \bigl(u_\epsilon(T_2), B_R^\Omega(x_0)\bigr)
  \leq  G_\epsilon \bigl(u_\epsilon(T_1), B_{2R}^\Omega(x_0)\bigr)
  +   \frac{C}{\gamma_1 R^2}
\int_{T_1}^{T_2}
G_\epsilon \bigl(u_\epsilon(t), B_{2R}^\Omega(x_0)\bigr)
  dt,
\end{gather}
for $0\leq T_1 <T_2 $.
Also for all $ \eta > 0 $,
there exist $ T_0 > 0 $ and $ R_0 > 0 $, such that for all
$ x_0 \in \Omega $ and all $ \epsilon > 0 $ we have
\begin{equation} \label{gs}
\sup_{0 \leq t \leq T_0} G_\epsilon \bigl(u_\epsilon(t), B_{R_0}^\Omega(x_0)\bigr)
  \leq   \eta  \,.
\end{equation}
\end{lemm}

\begin{proof}
Inequality \eqref{en-est} is obtained by multiplying
\eqref{Eq_eps_LL} with $ \partial_t u_\epsilon $.
Inequality \eqref{loc-en} follows by multiplying \eqref{Eq_eps_LL} with
$ \partial_t u_\epsilon   \phi^2 $
for an adequate cut-off function $\phi$
and then integrating by parts and absorbing.
Note that $ \partial_t u_\epsilon \equiv 0 $ on
$ \partial\Omega \times \mathbb{R}_+ $.
Inequality (\ref{gs}) follows from \eqref{loc-en},
if we set $T_1 =0$ and
$T_2 = T_0 = \frac{\gamma_1 R_0^2}{2 C E_0}$
for sufficiently small $R_0>0$, such that
$ G_\epsilon \bigl(u_\epsilon(0), B_{R_0}^\Omega(x_0)\bigr)
= E\bigl( u_0,  B_{R_0}^\Omega(x_0)\bigr) < \eta / 2$.
\end{proof}

The energy estimates imply  the penalty-approximations
subconverge weak(*) in
$H^{1,2}_{\rm loc} (\overline{\Omega}\times \mathbb{R}_+;\mathbb{R}^n)$
$\cap L^\infty \bigl(\mathbb{R}_+; H^{1,2}(\Omega; \mathbb{R}^n)\bigr)$.
This was already pointed out by B.Guo and M.C.Hong
in section 4 of \cite{Guo-Hong1}.


\section{Partial compactness}
\label{sec-high}

In this section we show that, under the uniform smallness
condition (\ref{reg-cond}) on the local energy, all higher
derivatives of $u_\epsilon$ are locally and uniformly  bounded.
Here ``uniform'' of course always means uniform in $\epsilon >0$.
In Section \ref{subsec-standard}, estimates for linear parabolic systems
that can be applied to \eqref{Eq_eps_LL}
as soon as $\nabla u_\epsilon$ is locally bounded are recalled.
In Section \ref{subsec-sup} we show that
$\nabla u_\epsilon$
is necessarily locally uniformly bounded,
whenever (\ref{reg-cond}) holds.
In Section \ref{subsec-high} we derive estimates that will provide bounds
for the right hand side of \eqref{Eq_eps_LL} and allow to combine
the previous estimates into a bootstrap argument.

\subsection{Some ``standard'' parabolic estimates}
\label{subsec-standard}

Equation \eqref{Eq_eps_LL} may be written as
\begin{equation} \label{para-LL}
L_\epsilon (u_\epsilon) :=  \partial_t u_\epsilon
  - M(u_\epsilon)   \Delta u_\epsilon
 =  - \frac{1}{\epsilon^2}M(u_\epsilon) f (u_\epsilon)
 =  f_\epsilon (u_\epsilon)  \,.
\end{equation}
The coefficient-matrix $ M(u) $ is smooth with respect to $ u $
and also strictly elliptic:
$$
\frac{\gamma_1}{\gamma_1^2 + \gamma_1^2}
   \abs{\xi}^2   <
 \xi^T M(u) \xi  =
\frac{1}{\gamma_1 (\gamma_1^2 + \gamma_2^2 \abs{u}^2)}
\Bigl( \gamma_1^2 \abs{\xi}^2 + \gamma_2^2 (u\cdot\xi)^2\Bigr)
  < \frac{1}{\gamma_1}  \abs{\xi}^2 ,
$$
for all $\xi \in \mathbb{R}^3 $
(See \cite[p.12]{Ding-Guo2}, \cite[p.316]{Guo-Hong1}, \cite[p.37]{Ding-Guo1}).
Note that for $\gamma_2=0$, we obtain $M(u) = \frac{1}{\gamma_1}  Id$.
The results of this section are indeed merely
interesting in the case $\gamma_2 \neq 0$,
where the left hand side of
\eqref{Eq_eps_LL}
is non-linear.
We will therefore restrict ourselves to the case $\gamma_2 \neq 0$.


For fixed $\epsilon >0$, the solution $ u_\epsilon $ of
\eqref{Eq_eps_LL}-\eqref{Eq_bdry_eps_LL}
is smooth and in particular continuous.
$u_\epsilon$ has the same regularity  up to the boundary
as the boundary data $u_0$.
The family of solutions $\{u_\epsilon\}_{\epsilon >0}$  is also
uniformly bounded in
$\epsilon > 0$,
since \mbox{$\abs{u_\epsilon (x,t)} \leq 1$}  $  \forall x,t $.
This follows from the Maximum Principle applied to the equation obtained by multiplying
\eqref{Eq_eps_LL} with $(1-\abs{u_\epsilon})$.
$L_\epsilon$ defines a strongly parabolic system in the
sense of Petrovskii
(Definition 2, p.599 in \cite{LadySolUral}) but satisfies as well all the other
(not necessarily equivalent) definitions of strong parabolicity for
general linear parabolic systems (Definitions 3-6)  in \cite{LadySolUral}.
The boundary-data operators also fullfill the required conditions.

First we have estimates in the $W^{2,1}_p$-Sobolev spaces with $p > 1$
(see \cite{LadySolUral} Chapter IV, Theorem 9.10 p.342 and (10.12) p.355
but also Chapter VII, Theorem 10.4 p.621,
for the generalization to parabolic systems).
Let $f_\epsilon \in L^p(\Omega\times [0,T];\mathbb{R}^n)$ and $u_0 \in H^{2,p}(\Omega;\mathbb{R}^n)$.
Then for any $ \delta \in ]0,1[ $,  $p> 3/2$ and
for $t_0 - R^2>0$
a solution of
$$
L_\epsilon(v) = f_\epsilon
\mbox{ in }  \Omega \times ]0,T[
\quad \mbox{and}\quad
v = u_0 \mbox{ on } \bigl( \Omega \times \{ 0 \} \bigr)
\cup \bigl( \partial \Omega \times ]0,T[ \bigr)
$$
satisfies
\begin{gather}\label{star1}
\norm{v}_{W^{2,1}_p(\Omega \times [0,T])}
  \leq
C_p(\Omega, T, \omega_{u_\epsilon})
\bigr(\norm{f_\epsilon}_{L^p(\Omega \times [0,T])} +
\norm{u_0}_{H^{2,p}(\Omega)}\bigl)  ,\\
\label{star2}
\begin{aligned}
\norm{v}_{ W^{2,1}_p (P_{\delta R}^\Omega(z_0)) }
& \leq \tilde{C}_p(R, \delta, \Omega, \omega_{u_\epsilon})
   \Bigr(\norm{f_\epsilon}_{L^p(P_R^{\Omega}(z_0))}
 +\norm{v}_{L^q(P_R^{\Omega}(z_0))} \\
&\quad +   \delta_{B_R\cap\partial\Omega}
\norm{u_0}_{H^{2-(1/p),p}(B_R^\Omega\cap\partial\Omega(z_0))}
\Bigl)  ,
\end{aligned}
\end{gather}
with $1\leq q \leq p$. Here
$$
P_R^{\Omega}(z_0) := (B_R(x_0) \times ]t_0 -R^2, t_0[)
\cap (\Omega \times ]0,\infty[)
$$
and
$\delta_{B_R\cap\partial\Omega} = 1 $ if $ B_R\cap\partial\Omega \neq \emptyset $
and $0$ otherwise.
The trace theorems of course imply
$$\norm{u_0}_{H^{2-(1/p),p}(\partial\Omega)} \leq
\norm{u_0}_{H^{2,p}(\Omega)}\,.$$
The constants
$ C_p $ and $ \tilde{C_p} $ depend on the indicated quantities
and additionally on the uniform lower and upper bounds
for the eigenvalues of $M(u_\epsilon)$,
which may be chosen independent of $ \epsilon > 0 $.
Note that the constants
$ C_p,   \tilde{C_p} $
also depend on the moduli of continuity of the coefficients
of the leading term,
i.e.  the modulus of continuity $\omega_{u_\epsilon}$
of $ u_\epsilon $.
The equation can also be written in divergence form,
$$
L_{\epsilon} (v) :=
 \partial_t v - div \bigl( M(u_\epsilon) \nabla v \bigr)
+ \bigl(DM(u_\epsilon) \partial_k u_\epsilon\bigr) \partial_k v
= f_\epsilon \,.
$$
If we assume in addition
\begin{equation} \label{ass-sup}
\limsup_{\epsilon\searrow 0}
\sup_{P_R^\Omega}    \abs{\nabla u_\epsilon} < \infty \,,
\end{equation}
then estimates for equations in divergence form imply
$ v \in C^{\gamma,(\gamma/2)}(P_{\delta R}^\Omega;\mathbb{R}^n)$
for some $ \gamma \in ]0,1[ $ and any $\delta \in ]0,1[$.
(See \cite{LadySolUral} Chapter VII Theorem 3.1 p.582
or Chapter V, Theorem 1.1, p.419.)
Indeed if the right hand side $f_\epsilon \in L^p(P_R^\Omega;\mathbb{R}^n)$ with $p>2$,
the following estimate for
the mixed H\"older-norm of $v$
on $P_{\delta R}^\Omega$ holds
\begin{equation} \label{cont-est}
\norm{v}_{C^{\gamma,\gamma/2}(P_{\delta R}^\Omega)}   \leq
C(f_\epsilon) \,.
\end{equation}
(See \cite{LadySolUral} p.7 for the definition
of the mixed H\"older-spaces denoted there by $H^{\gamma,\gamma/2}$)
The bound $C(f_\epsilon)$ depends on
the parabolicity constants,
on $0<\delta<1$, $\sup_{P_R^\Omega} \abs{ u_\epsilon}$,
$\norm{f_\epsilon}_{L^p(P_R^\Omega)}$,
bounds for the coefficients of the equations depending on
$ \sup_{P_R^\Omega} \abs{\nabla u_\epsilon} $
and also on
$ \norm{u_0}_{C^{\gamma}(B_R\cap\partial\Omega)}$
if $ B_R\cap\partial\Omega \neq \emptyset $.

Therefore, if (\ref{ass-sup}) holds and
$\norm{f_\epsilon}_{L^p(P_R^\Omega)}$
or
$ \sup_{P_R^\Omega} \abs{f_\epsilon} $
are uniformly bounded with respect to $\epsilon > 0 $,
then
estimate (\ref{cont-est}) holds for $u_\epsilon$
and is uniform in $\epsilon >0$.
Now the modulus of continuity of $ u_\epsilon $
on $P_{\delta R}^\Omega$ is bounded from above
(by an increasing function $ h $ with $ \lim_{t\searrow 0} h(t) = 0 $)
independently of
$ \epsilon > 0 $. We gain uniform bounds for the modulus of continuity
of $ u_\epsilon $ with respect to $t\geq 0$
and estimate $(\ref{star2})$ is now uniform in $ \epsilon > 0 $.

Further by Lemma 3.3 p.80 in Chapter II of \cite{LadySolUral}
for $p> m+2 (= 4)$ ($m$ being the dimension of the spatial domain, in our case
$m=2$), we have
$$
\norm{\nabla v}_{C^\lambda(P_R^\Omega)}
 \leq C(m,p, \lambda, \Omega )
\norm{v}_{W^{2,1}_p(P_R^\Omega)}
\quad \mbox{for } \lambda = 1 - (m+2)/p \,.
$$
Also if (\ref{ass-sup}) holds and
$\norm{f_\epsilon}_{L^p(P_R^\Omega)}$ is uniformly bounded,
then
$(\ref{star2})$ yields $\epsilon$-uniform estimates
for $\norm{\nabla u_\epsilon}_{C^\lambda(P_{\delta R}^\Omega)}$.

\subsection{The main sup-estimates for the energy-density}
\label{subsec-sup}

\subsubsection{An interior sup-estimate for
the Landau-Lifshitz-flow approximations}
\label{subsubsec-sup}

In this section we derive an interior sup-estimate for the energy density
in the
case $\gamma_2 \neq 0$, but the proof also works if $\gamma_2 =0$
and the target is $N$.
The proof of the interior estimate is much simpler than in the boundary case
and we therefore consider each case separately.
The estimate will result from a scaling argument
combined to the following higher estimates, that
will be proven in the next section.
Let
$$
P_R(z_0) := B_R(x_0) \times ]t_0 -R^2, t_0[
\quad \mbox{for } z_0=(x_0,t_0)\,.
$$

\begin{lemm} \label{Cinfty1}
Let $ u_\epsilon $ be a solution of
\eqref{Eq_eps_LL} for each $ \epsilon > 0 $.
Assume
$$
\limsup_{\epsilon \searrow 0}
 \sup_{P_R(z_0)} g_{\epsilon} (u_\epsilon)   \leq   C_0
$$
and
$ B_R(x_0) \subset \Omega $, $ 0<R^2<t_0$.
Then for any $ 0 < \delta < 1 $,
$$
\limsup_{\epsilon \searrow 0}
  \norm{ u_{\epsilon}}_{C^k (P_{\delta R} (z_0)) }
\leq   C_k \quad \mbox{ and } \quad
\limsup_{\epsilon \searrow 0}
\norm{\frac{1}{\epsilon^2}   (1 - \abs{u_\epsilon}^2)
}_{C^k (P_{\delta R} (z_0)) }
\leq   \tilde{C}_k
$$
for all $ k \geq 0 $.
The constants $ C_k,\tilde{C}_k $ depend on
$ C_0, k, R, \delta  > 0 $.  If $ \gamma_2 =0$ and the target is $N$,
they also depend on the geometry of $N$
(i.e.  the metric on $N$ and its derivatives).
\end{lemm}

We will now prove the following ``$\epsilon_1$-regularity'' result.

\begin{theorem}  \label{eps-reg-LL}
There are constants
$ C_1 = C_1(N),  \epsilon_1 = \epsilon_1(N) > 0 $,
such that if, for some $0<R_0<\min \{1,\sqrt{t_0} \} $ and
$x_0\in \Omega$
with $ B_{R_0}(x_0) \subset \Omega$,
a solution
$ u_\epsilon $ of
\eqref{Eq_eps_LL}
satisfies
$$
\sup_{t_0-R_0^2 < t < t_0} \int_{B_{R_0}(x_0)}
g_\epsilon (u_\epsilon )(x,t)   dx   < \epsilon_1  \,,
$$
then
$$
\sup_{P_{\delta R_0}(z_0)} g_\epsilon (u_\epsilon )
  \leq \frac{C_1}{(1-\delta)^2 R_0^2}
$$
for any $ \delta \in ]0,1[$.
\end{theorem}

In the proof we would like to consider points
$ z_\epsilon =(x_\epsilon,t_\epsilon) \in \overline{P_R} (z_0) $  such that
$ g_\epsilon(z_\epsilon) = \sup_{P_{R}(z_0)}
g_{\epsilon} $. Difficulties however arise if
$z_\epsilon \in \partial P_R(z_0)$,
since we then do not have uniform estimates on a neighborhood of
$  z_\epsilon $.
This is elegantly avoided by considering
$$
\max_{0 \leq \sigma \leq R_0 }
\bigl( (R_0 - \sigma)^2 \sup_{P_{\sigma}}
g_\epsilon  \bigr)  \,.
$$
This trick is initially due to R.~Schoen. (See  \cite{Schoen}, proof of Theorem 2.2.
Schoen's method was extended to the parabolic context in
\cite{Struwe3}, \cite{Struwe-Chen}.)

\begin{proof}[Proof of Theorem \ref{eps-reg-LL}]
Without loss of generality, let $ (x_0,t_0) = 0 $.
We set $P_R := P_R(0)$.
Since $ u_\epsilon $ is regular, there is some $\sigma_\epsilon \in [0,R_0[ $
such that
$$
(R_0 - \sigma_\epsilon)^2 \sup_{P_{\sigma_\epsilon}}g_\epsilon
  = \max_{0 \leq \sigma \leq R_0}
\Bigl( (R_0 - \sigma)^2 \sup_{P_{\sigma}}g_\epsilon  \Bigr)\,.
$$
Moreover, there is some $ z_\epsilon = (x_\epsilon, t_\epsilon)
\in \overline{P_{\sigma_\epsilon }} $,
such that
$e_\epsilon = g_\epsilon(u_\epsilon (z_\epsilon)) =
\sup_{P_{\sigma_\epsilon}}  g_\epsilon  $.
Set $ \rho_\epsilon := \frac{1}{2} (R_0 - \sigma_\epsilon).$
Since
$ P_{\rho_\epsilon }(z_\epsilon) \subset P_{\sigma_\epsilon +\rho_\epsilon }
 \subset P_{R_0}$, we have
\begin{align*}
\sup_{P_{\rho_\epsilon}(z_\epsilon)}  g_\epsilon
&\leq \frac{1}{(R_0 - (\sigma_\epsilon+\rho_\epsilon))^2}
(R_0 - (\sigma_\epsilon+\rho_\epsilon))^2
\sup_{P_{\rho_\epsilon+\sigma_\epsilon}}  g_\epsilon \\
&\leq \frac{4}{(R_0 - \sigma_\epsilon)^2}
(R_0 - \sigma_\epsilon)^2 e_\epsilon \leq 4 e_\epsilon \,.
\end{align*}
Set
$r_\epsilon := \sqrt{e_\epsilon} \rho_\epsilon $ and consider
the rescaled map
$$
v_\epsilon(y,s) := u(x_\epsilon + e_\epsilon^{-1/2}  y,
t_\epsilon + e_\epsilon{}^{-1}s)
\quad \mbox{ for } (y,s) \in P_{r_\epsilon} \,.
$$
By definition $ v_\epsilon $ satisfies \eqref{Eq_eps_LL}
on $P_{r_\epsilon}$ with
$\tilde{\epsilon} := \sqrt{e_\epsilon} \epsilon $
instead of $\epsilon $ and
$$
g_{\sqrt{e_\epsilon} \epsilon}(v_\epsilon)(0,0) = 1 \,,
\quad \sup_{P_{r_\epsilon}} g_{\sqrt{e_\epsilon} \epsilon}(v_\epsilon) \leq 4 \,.
$$
Now we claim $ r_\epsilon \leq 2 $.
This will prove the theorem, since by definition
of $r_\epsilon$, we then have
$ (R_0-\sigma_\epsilon)^2   e_\epsilon \leq 16$.

Assume $ r_\epsilon > 2 $.
Since $ B_{R_0}(x_0) \subset \Omega $,
all the higher derivatives of $ v_\epsilon$ are then bounded
on $P_{1} $ independently of $ \epsilon > 0 $.
Indeed
if
$ \liminf_{\epsilon\searrow 0} \sqrt{e_\epsilon} \epsilon   > 0 $,
the uniform estimates are immediate and if
$ \liminf_{\epsilon\searrow 0} \sqrt{e_\epsilon} \epsilon   = 0 $,
they follow  from Lemma \ref{Cinfty1}.
In particular
$$
\sqrt{\abs{\partial_t g_{\tilde{\epsilon}}(v_\epsilon)}},
\abs{\nabla g_{\tilde{\epsilon}}(v_\epsilon)} \leq
C < \infty
\mbox{ on } P_1
   \mbox{ (uniformly in $ \epsilon > 0 $)}
$$
and therefore,
$$
\inf_{ P_{r_0} }g_{\tilde{\epsilon}}(v_\epsilon) \geq \frac{1}{2}
\quad \mbox{ for } r_0 := \min \{ \frac{1}{4C},1 \} \,.
$$
Note that $ C $ is an absolute constant in the sense that it merely
depends on the radius $2$,
the factor $ \delta = \frac{1}{2}$, the $L^\infty$-bound $4$ and the
parabolicity constants and the geometry of $N$.
This lower bound implies
\begin{align*}
1 &= g_{\sqrt{e_\epsilon} \epsilon}(v_\epsilon)(0,0)   \leq
\frac{2}{\pi r_0^2} \sup_{-r_0^2 < s < 0 }   \int_{B_{r_0}}
g_{\sqrt{e_\epsilon} \epsilon}(v_\epsilon)(y,s)  \, dy \\
&\leq  C_*
\sup_{t_\epsilon - r_0^2 e_\epsilon^{-1} < t < t_\epsilon }
\int_{B_{e_\epsilon^{-1/2}r_0}(x_\epsilon)}
g_{ \epsilon}(u_\epsilon)(x,t)  \, dx \\
&\leq  C_*\sup_{-(\frac{r_0^2}{e_\epsilon} + \sigma_\epsilon^2)
 < t < 0 }
\int_{B_{\frac{r_0}{\sqrt{e_\epsilon}} + \sigma_\epsilon }(x_0)}
g_{ \epsilon}(u_\epsilon)(x,t)  \, dx\,.
\end{align*}
Set $ \epsilon_1 := \min \{\frac{1}{2}, \frac{1}{2C_*} \}$.
Since $ r_\epsilon = \sqrt{e_\epsilon} \rho_\epsilon > 2 > r_0 $,
we have
$ \frac{r_0}{\sqrt{e_\epsilon}} + \sigma_\epsilon
  \leq    \rho_\epsilon + \sigma_\epsilon
  \leq   R_0 $ and
$ (\frac{r_0}{\sqrt{e_\epsilon}})^2 + \sigma_\epsilon^2
  \leq    (\rho_\epsilon + \sigma_\epsilon)^2
  \leq   R_0^2 $.
Then the last estimate %, which gives an upper bound for 1,
yields a contradiction,
since the right hand side
is smaller than $\epsilon_1 \leq \frac{1}{2}$.
Therefore
$r_\epsilon =  \sqrt{e_\epsilon} \rho_\epsilon \leq 2 $
and
$$
(1-\delta)^2  R_0^2    \sup_{P_{\delta R_0}}
  g_\epsilon   \leq   16  \,.
$$
\end{proof}


\subsubsection{ A local boundary sup-estimate for the energy density}
\label{subsec-sup-bd}
Local $L^p$-estimates for $\nabla^3 u_\epsilon$
up to the boundary which are uniform in $\epsilon > 0$
cannot be expected, even if
$u_0 \in C^\infty(\Omega;S^2)$.
Indeed for fixed $ \epsilon > 0 $, $ u_\epsilon $
is smooth up to the boundary and we may thus evaluate
\eqref{Eq_eps_LL} at $ x \in \partial\Omega $ for any $t\geq0$.
This gives
$ \Delta u_\epsilon = 0 $ on $ \partial\Omega \times \mathbb{R}_+ $.
As we will see later, uniform estimates imply
the existence of a subsequence $ u_{\epsilon_i} $ converging to a
map $ u_* $, which is a smooth
solution of the Landau-Lifshitz
or harmonic map flow in $ \mathop{\rm Reg}(\{u_{\epsilon_i}\}) $
and satisfies
$$
- \Delta u_* = |\nabla u_*|^2 u_* \quad \mbox{ on }
(\partial\Omega \times \mathbb{R}_+) \cap \mathop{\rm Reg}(\{u_{\epsilon_i}\})  ,
$$
since $\partial_t u_* = 0$ on $\partial\Omega \times \mathbb{R}_+$.
However $L^p_{\rm loc}$-estimates for $ \nabla^3 u_\epsilon $ would imply
$$
0 = \Delta u_{\epsilon_i} \to \Delta u_*
\mbox{ in } L^p_{\rm loc}((\partial\Omega \times \mathbb{R}_+)
\cap \mathop{\rm Reg}(\{u_{\epsilon_i}\}) ; \mathbb{R}^3)
$$
by compactness of
the ``projection'' $ H^{1,p}(\Omega) \hookrightarrow L^p(\partial\Omega)$.
This is not possible unless $ u_* \equiv  const.$
on $ (\partial\Omega\times \mathbb{R}_+)
\cap \mathop{\rm Reg}(\{u_{\epsilon_i}\})$.
(Compare \cite{BBH1}, Remark 1 p.125 for a similar argument in the time
independent case.)

The following lemma will be proven in Section \ref{subsec-high}.

\begin{lemm} \label{Cinfty2}
Let $ u_\epsilon $ be a solution of
\eqref{Eq_eps_LL}-\eqref{Eq_bdry_eps_LL},
with \\
$ u_0 \in H^{1,2}(\Omega;S^2) \cap H^{2,p}(\partial\Omega;S^2)$
and $ p \geq2 $ for each $ \epsilon > 0 $.
Assume
$$
\sup_{P^\Omega_R(z_0)} g_{\epsilon}   \leq   C_0
$$
and  $B_R(x_0) \cap \partial\Omega \neq \emptyset$,
$ 0<R^2<t_0$.
Then for any $\delta \in ]0,1[$, we have
\begin{align*}
&\norm{ u_{\epsilon}}_{W^{2,1}_p (P^\Omega_{\delta R} (z_0))}\\
&\leq
C_1 \Bigl(\norm{\frac{1}{\epsilon^2} (1 - \abs{u_\epsilon}^2 )
}_{L^p (P^\Omega_{R} (z_0))}+   \norm{ u_{\epsilon}}_{L^{2}
(P^\Omega_{R} (z_0))}
+\norm{u_0}_{H^{2-(1/p),p}(B^\Omega_{R}(z_0)\cap\partial\Omega)}
\Bigr)  ,
\end{align*}
where the constant $C_1$ depends on
$C_0, p, R, \delta$ and $\Omega$.
Further we have for any $\delta \in ]0,1[$,
\begin{gather*}
\norm{ \frac{1}{\epsilon^2}
(1 - \abs{u_\epsilon} ^2)}_{L^p (P^\Omega_{\delta R} (z_0))}
  \leq C(p) \norm{g_\epsilon}_{L^p(P^\Omega_R )}
+  \epsilon^{2/p} C\bigl( \norm{g_\epsilon}_{L^p(P_R^\Omega)},
p, \delta, R \bigr) , \\
 \norm{ \frac{1}{\epsilon^2}
(1 - \abs{u_\epsilon} ^2)}_{L^\infty (P^\Omega_{\delta R} (z_0))}
  \leq 8  C_0 + o_\delta(\epsilon)  ,
\end{gather*}
where $\epsilon \mapsto o_\delta(\epsilon)$
is a function that depends on $\delta \in ]0,1[$
and $  \lim_{\epsilon \searrow 0} \epsilon^{-k} o_\delta(\epsilon) = 0$
for all $k \in \mathbb{N}$.
All the constants also depend on the parabolicity constants.
If $\gamma_2 = 0$ and the target is $N$, they
also depend on the geometry of $N$.
\end{lemm}

We now prove the following result.

\begin{theorem}  \label{eps-reg-LL-bd}
Consider $ u_0 \in H^{1,2}(\Omega;S^2) \cap
C^{2}(\partial\Omega;S^2)$.
Let $ u_\epsilon $ be
a solution of
\eqref{Eq_eps_LL}-\eqref{Eq_bdry_eps_LL}
for each $\epsilon >0$.
There are constants
$ C_0 =C_0(\norm{u_0}_{C^2(\partial\Omega)}, E_0, \Omega)$
and
$\epsilon_0$
$= \epsilon_0(\norm{u_0}_{C^2(\partial\Omega)}, E_0, \Omega)> 0 $,
such that if for some
$z_0 =(x_0,t_0)$ and $R_0 \in ]0,\min\{1,\sqrt{t_0}\}[$
$$
\limsup_{\epsilon\searrow 0}
\underset{t_0-R_0^2 < t < t_0}{\sup }
 \int_{B_{R_0}(x_0)\cap \Omega}
g_\epsilon (u_\epsilon )   dx    < \epsilon_0 \,,
$$
then
$$
\limsup_{\epsilon\searrow 0}
\sup_{P^\Omega_{\delta R_0}(z_0)}
g_\epsilon (u_\epsilon )
  \leq
\frac{C_0}{(1-\delta)^2 R_0^2}    ,
$$
for any $ \delta \in ]0,1[$.
\end{theorem}

If the target is $N$, the above constants $C_0$ and $\epsilon_0$ also
depend on the geometry of $N$.

\begin{proof}
Without loss of generality let $ (x_0,t_0) = 0 $.
Since $u_0 \in C^2(\partial\Omega)$ admits an extension
$w_0\in C^2(\overline{\Omega})$ and since $t_0 -R^2 >0$, we may assume
$u_0 \in C^2(\overline{\Omega})$.
We have
$ u_\epsilon \in C^{1,0}_{\alpha}(\overline{\Omega}\times \mathbb{R}_+;S^2)$
for any $ 0<\alpha <1$ and so
there are $\sigma_\epsilon \in [0,R_0[$
and
$ z_\epsilon = (x_\epsilon, t_\epsilon)
\in \overline{P^\Omega_{\sigma_\epsilon }}$,
such that
\begin{gather*}
(R_0 - \sigma_\epsilon)^2
\sup_{P^\Omega_{\sigma_\epsilon}}
g_\epsilon
  = \max_{0 \leq \sigma \leq R_0}
\Bigl( (R_0 - \sigma)^2 \underset{P^\Omega_{\sigma}}{ \sup }
g_\epsilon  \Bigr),\\
e_\epsilon = g_\epsilon(u_\epsilon (z_\epsilon)) =
\underset{P^\Omega_{\sigma_\epsilon}}{ \sup }  g_\epsilon \,.
\end{gather*}
Again for $ \rho_\epsilon := \frac{1}{2} (R_0 - \sigma_\epsilon)$, we have
$ \sup_{P^\Omega_{\rho_\epsilon}(z_\epsilon)}  g_\epsilon
\leq 4 e_\epsilon$.
Consider
the rescaled map
$$
v_\epsilon(y,s) := u(x_\epsilon + e_\epsilon^{-1/2}  y,
t_\epsilon + e_\epsilon^{-1}s) \,.
$$
By construction $ v_\epsilon $ satisfies
\begin{equation}
\gamma_1 \partial_t v_\epsilon - \gamma_2 v_\epsilon \times \partial_t v_\epsilon
- \Delta v_\epsilon =
\frac{1}{\tilde{\epsilon}^2} (1-|v_\epsilon|^2) v_\epsilon
\mbox{ on }
P_{r_\epsilon}^{\Omega_\epsilon}\,,\label{*}
\end{equation}
with
$\tilde{\epsilon} := \sqrt{e_\epsilon} \epsilon$,
$ r_\epsilon := \sqrt{e_\epsilon} \rho_\epsilon$,
$ \Omega_\epsilon := \sqrt{e_\epsilon} \bigl( \Omega - x_\epsilon \bigr) $
and
$$P_{r_\epsilon}^{\Omega_\epsilon} :=
\bigl( B_{r_\epsilon} \cap \Omega_\epsilon \bigr)\times
]-r_\epsilon^2, 0[\,.$$
\\
Further by construction,
\begin{equation}
g_{\tilde{\epsilon}}(v_\epsilon)(0,0) = 1
 \mbox{ and }
\sup_{ P_{r_\epsilon}^{\Omega_\epsilon} }
g_{\tilde{\epsilon}}(v_\epsilon) \leq 4 \,. \label{**}
\end{equation}
The boundary data are also rescaled.
Set
$v_{\epsilon,0}(y) :=
u_0(x_\epsilon + e_\epsilon^{-1/2}   y)$.
Then
$$
v_\epsilon (y,s) = v_{\epsilon,0}(y) \mbox{ on }
(\partial\Omega_\epsilon \cap B_{r_\epsilon}) \times ]-r_\epsilon^2,0[
$$
and
$$
\sup_{P_{r_\epsilon}^{\Omega_\epsilon}} |\nabla v_{\epsilon,0}|
\leq
  e_\epsilon^{-1/2}
\sup_{P_{R_0}^{\Omega}} |\nabla u_0|  \,, \quad
\sup_{P_{r_\epsilon}^{\Omega_\epsilon}} |\nabla^2 v_{\epsilon,0}|
\leq
  e_\epsilon^{-1}
\sup_{P_{R_0}^{\Omega}} |\nabla^2 u_0|  \,.
$$
Now we claim that for sufficiently small $\epsilon> 0$,
we have
$$
r_\epsilon \leq C_0
:= \max\{2,\tilde{C}(\Omega,\norm{u_0}_{C^2(\overline{\Omega})})\} ,
$$
where $\tilde{C}(\cdot)>0$ will be specified later.
Again by definition of $r_\epsilon$,
this will prove the theorem.


Assume by contradiction
$ r_\epsilon > C_0
\geq 2 $ for small $\epsilon >0$.
Then
$$
e_\epsilon^{-1/2}
= \rho_\epsilon/r_\epsilon
< R_0/(2C_0)
 \leq 1/(2C_0) ,
$$
since $0<R_0<1$.
First we claim that
$$
\liminf_{\epsilon\searrow 0} \sqrt{e_\epsilon} \epsilon =
\liminf_{\epsilon\searrow 0} \tilde{\epsilon}(\epsilon) = 0 \,.
$$
Indeed if $\liminf_{\epsilon\searrow 0} \sqrt{e_\epsilon} \epsilon > 0$,
the right hand side of \eqref{*} is uniformly bounded in
$\tilde{\epsilon} = \sqrt{e_\epsilon} \epsilon >0$
and together with \eqref{**} we obtain uniform bounds in
$C^\infty(P_{2}^{\Omega_\epsilon})$.
This however leads to a contradiction
as in the proof of Theorem \ref{eps-reg-LL}, if
$\epsilon_0$ is smaller than $\epsilon_1$.
Further if
$$
\limsup_{\epsilon\searrow 0}
\bigl(\sqrt{e_\epsilon} \mathop{\rm dist}(x_\epsilon,\partial\Omega)\bigr)
=
\limsup_{\epsilon\searrow 0}
\mathop{\rm dist}(0,\partial\Omega_\epsilon)
\geq \frac{1}{2}\,,
$$
we can also use uniform interior
estimates in $C^\infty(P_{1/4}^{\Omega_\epsilon})$ and proceed
as in the proof of Theorem \ref{eps-reg-LL} to get a contradiction,
if we choose $\epsilon_0$ sufficiently small.
So far the required upper bound on $\epsilon_0$
is universal in the sense that it only depends on
the geometry of $N$ and the parabolicity constants.
We therefore have
$$
\limsup_{\epsilon\searrow 0}
\mathop{\rm dist}(0,\partial\Omega_\epsilon)
< 1/2 \,,
$$
and in the sequel we consider sufficiently small $\epsilon >0$, such that
$\mathop{\rm dist}(0,\partial\Omega_\epsilon)< 1/2$.

Lemma \ref{Cinfty2}
combined to the embedding
$W^{2,1}_p(P_1) \hookrightarrow C^1(P_1)$
for $p>4$, implies
\begin{align*}
&\sup_{P_1^{\Omega_\epsilon}}|\nabla v_\epsilon|^2 \\
&\leq C \norm{v_\epsilon}_{W^{2,1}_p(P_1^{\Omega_\epsilon} )}^2\\
&\leq C(p,\Omega_\epsilon \cap B_2)  \Bigl(
\norm{\frac{1}{\tilde{\epsilon}^2}
(1-|v_\epsilon|^2)}_{L^p(P_2^{\Omega_\epsilon} )}^2
+\norm{v_\epsilon}_{L^2(P_2^{\Omega_\epsilon})}^2
+\norm{v_{\epsilon,0}}_{H^{2,p}(P_2^{\Omega_\epsilon})}^2
\Bigr)
\end{align*}
Note that $\Omega_\epsilon \cap B_2$ has uniformly bounded curvature
and so
$$ 0< C(p,\Omega_\epsilon \cap B_2) < C(p,\Omega)\,.$$
Since
$(1-|v_\epsilon|^2) \leq 1$
and %by construction
$\sup_{ P_2^{\Omega_\epsilon} }
g_{\tilde{\epsilon}}(v_\epsilon) \leq 4 $,
Lemma \ref{Cinfty2} implies
$$
\norm{\frac{1}{\tilde{\epsilon}^2}
(1-|v_\epsilon|^2)}_{L^p(P_2^{\Omega_\epsilon} )}^2
\leq C_{p}   \bigl( o(\epsilon_0) + o(\tilde{\epsilon}) \bigr)  \,,
$$
where $C_p = C(p,E_0)$ and $o(\tau)$ denotes a generic function
that satisfies
$$ \lim_{\tau \searrow 0} o(\tau)=0 \,.$$
A Poincar\'e inequality on $P_2^{\Omega_\epsilon}$ leads to
\begin{align*}
\norm{v_\epsilon}_{L^2(P_2^{\Omega_\epsilon})}^2
&\leq 2 \bigl(\norm{v_{\epsilon,0}}_{L^2(P_2^{\Omega_\epsilon})}^2
+ \norm{v_\epsilon - v_{\epsilon,0}}_{L^2(P_2^{\Omega_\epsilon})}^2\bigr) \\
&\leq 2   \norm{v_{\epsilon,0}}_{L^2(P_2^{\Omega_\epsilon})}^2
+C(\Omega)\bigl( \norm{\nabla v_{\epsilon,0}}_{L^2(P_2^{\Omega_\epsilon})}^2
+\norm{\nabla v_{\epsilon}}_{L^2( P_2^{\Omega_\epsilon})}^2   \bigr) \,.
\end{align*}
Again $\norm{\nabla v_{\epsilon}}_{L^2( P_2^{\Omega_\epsilon})}^2
\leq o(\epsilon_0)$.
Of course
\begin{align*}
\norm{v_{\epsilon,0}}_{H^{1,2}(B_2^{\Omega_\epsilon})}
&\leq C(p) \norm{v_{\epsilon,0}}_{H^{1,p}(B_2^{\Omega_\epsilon})}\\
&\leq C(p) \norm{v_{\epsilon,0}}_{H^{2,p}(B_2^{\Omega_\epsilon})}^2\\
&\leq C(p,\Omega)
\norm{v_{\epsilon,0}}_{C^2(B_2^{\Omega_\epsilon})}^2
\end{align*}
and we still need to estimate
$\norm{v_{\epsilon,0}}_{C^2(B_2^{\Omega_\epsilon})}^2$.

For each $\epsilon>0$ we may chose coordinates
for the target such that $ v_{\epsilon,0} (0) =0$.
Then
\begin{gather*}
\sup_{B_2^{\Omega_\epsilon} }|v_{\epsilon,0}|
  \leq  4  \sup_{B_2^{\Omega_\epsilon}} |\nabla v_{\epsilon,0}|\,, \\
\norm{v_{\epsilon,0}}_{C^2(B_2^{\Omega_\epsilon})}
\leq    C  e_\epsilon^{-1/2} \sup_{B_{R_0}^\Omega} |\nabla u_0|
+ e_\epsilon^{-1} \sup_{B_{R_0}^\Omega} |\nabla^2 u_0|\,.
\end{gather*}
The above estimates combined to the one for
$\norm{ \frac{1}{\tilde{\epsilon}^2}
(1 - \abs{v_\epsilon} ^2)}_{L^\infty (P^{\Omega_\epsilon}_1)} $
in Lemma \ref{Cinfty2} yield
\begin{align*}
1 &\leq \sup_{P_1^{\Omega_\epsilon}} g_{\tilde{\epsilon}}(v_\epsilon)\\
& \leq\sup_{P_1^{\Omega_\epsilon}}\frac{1}{2}|\nabla v_\epsilon|^2
+\tilde{\epsilon}^2 \sup_{P_1^{\Omega_\epsilon}}\bigl(
\frac{1}{\tilde{\epsilon}^2} (1 - \abs{v_\epsilon} ^2)\bigr)^2\\
&\leq C_1   \left(o(\epsilon_0) +o(\tilde{\epsilon}) +
  e_\epsilon^{-1}   \norm{\nabla u_0}_{C^1(B_{R_0}^\Omega)}^2\right) \,,
\end{align*}
where $C_1 = C_1(\Omega , E_0)$.
Now if both
$ o(\epsilon_0)< (1/4) C_1^{-1}$
and $o(\tilde{\epsilon}) < (1/4) C_1^{-1}$,
this leads to
$$
e_\epsilon  <  2  C_1
\norm{\nabla u_0}^2_{C^1(B_{R_0}^\Omega)} ,
$$
which is in contradiction with
$r_\epsilon > C_0 := \max\{2,2 C_1  \norm{u_0}_{C^2(\overline{\Omega})}\}$
and
$\sqrt{e_\epsilon} > 2C_0$.
Thus
$ r_\epsilon \leq C_0$
and by definition of $r_\epsilon$ also
$$
\frac{1}{4}  (R_0 - \delta R_0)^2
\sup_{P^\Omega_{ \delta R_0}}
g_\epsilon \leq C_0^2
 = C(\Omega, E_0, \norm{u_0}_{C^2(\overline{\Omega})})  \,.
$$
Since $t_0 -R^2 >0$, we could replace $u_0$ in the above
by any $w_0 \in C^2(\overline{\Omega})$ with
$w_0 = u_0$ on $\partial\Omega \cap B_{R_0}$.
Therefore the above constants merely depend on
$\norm{u_0}_{C^2(\partial\Omega)}$.
\end{proof}

\subsection{Higher estimates}
\label{subsec-high}

In this section,  we prove Lemmata \ref{Cinfty1} and \ref{Cinfty2},
for which the following uniform estimates will be needed.

\subsubsection{Uniform estimates in $ \epsilon > 0 $}
\label{subsec-comp}
The ``distance-to-the-target-function''
$\rho_\epsilon := 1-|u_\epsilon|^2$ satisfies
\begin{equation} \label{rho}
\gamma_1   \partial_t \rho_\epsilon - \Delta  \rho_\epsilon  +
\frac{2}{\epsilon^2}  \rho_\epsilon
  =
2 |\nabla u_\epsilon|^2 +  \frac{2}{\epsilon^2}  \rho_\epsilon^2 \,.
\end{equation}
Since $\gamma_1 >0$, we may assume $\gamma_1 = 1$ without loss of
generality.
We will now derive uniform a priori estimates for this equation.
Lemma \ref{f} extends a comparision argument from
\cite{BBH1} (Lemma 2, p.130)
to the time dependent case and to non-positive solutions.

The parabolic boundary
of $ P_R := B_R(0) \times ]-R^2,0[$
is denoted as
$$
\tilde{\partial} P_R := \bigl( B_R(0) \times \{ - R^2 \} \bigr)
\cup \bigl( \partial B_R(0) \times [-R^2,0 ] \bigr) \,.
$$

\begin{lemm} \label{f}
Let $ a > 0$, $R \in ]0, \frac{1}{4}[ $,
$\epsilon \in ]0, 1[ $ and $ g \in C^0(\overline{P_R})$
with
$\epsilon^2  \sup_{P_R} |g| \leq a$.
Let $ f \in C^0(\overline{P_R}) \cap C^2(P_R) $
be a solution of
\begin{gather*}
 \bigl( \partial_t f - \Delta f \bigr) + \frac{1}{\epsilon^2} f
  =    g \quad \mbox{in } P_R\,,\\
|f|   \leq    a \quad \mbox{on } \tilde{\partial} P_R \,.
\end{gather*}
Then for any $ \delta \in ]0,1[ $, we have
$$
\frac{1}{\epsilon^2 }   |f|   \leq
\underset{P_R }{ \sup } |g|
  +   \frac{2a}{\epsilon^2 }  e^{-\frac{1}{\epsilon } (1-\delta^2)^2 R^4 }
\mbox{ on }  P_{\delta R} \,.
$$
\end{lemm}

\begin{proof}
Consider $\omega(x,t)   =  2 a   e^{-\frac{1}{\epsilon}(R^2 - |x|^2)(R^2 + t )}$.
Then
\begin{gather*}
\epsilon^2 \bigl( \partial_t \omega - \Delta \omega \bigr) + \omega
 > 0\quad  \mbox{in }P_R  \,,\\
\omega  = 2 a \quad  \mbox{on } \tilde{\partial} P_R  \,.
\end{gather*}
For $ f_1 := f - \epsilon^2 \underset{P_R }{ \sup } |g| $
and $ f_2 := f + \epsilon^2 \underset{ P_R }{ \sup } |g| $,
we have
$$
|f_1| \leq 2 a \quad \mbox{and}\quad |f_2| \leq 2 a \mbox{ on }
\tilde{\partial}P_R\,,
$$
and hence
$$
f_1 - \omega \leq 0 , \quad    f_2 + \omega \geq 0 \quad \mbox{on }
\tilde{\partial} P_R \,.
$$
Moreover
$$
\epsilon^2 \bigl( \partial_t f_1 - \Delta f_1 \bigr) + f_1
 \leq 0  \,, \quad
\epsilon^2 \bigl( \partial_t f_2 - \Delta f_2 \bigr) + f_2
\geq 0 \quad \mbox{in } P_R  \,.
$$
The Maximum Principle now implies $ f_1 - \omega  \leq 0 $ and
$ f_2 + \omega  \geq 0 $ on $ P_R $, that is
$$
- \omega   -   \epsilon^2    \underset{ P_R }{ \sup }   |g|
  \leq    f   \leq   \omega   +   \epsilon^2
  \sup_{P_R } |g| \,.
$$
\end{proof}

The above lemma will yield interior estimates. If
$ B_R \cap \Omega \neq \emptyset $ and
$ f \equiv 0 $ on $ B_R \cap \partial\Omega $,
we still obtain a local estimate up to the boundary,
i.e.  on
$ P^\Omega_{\delta R} = (B_{\delta R}\cap \Omega) \times ]-(\delta R)^2,0[$.

\begin{coro} \label{bd-f}
Consider a smooth domain $\Omega \subset \mathbb{R}^2$,
$ a > 0$,   $R \in ]0, \frac{1}{4}[ $,
$\epsilon \in ]0, 1[ $ and
$ g \in C^0(\overline{P_R^\Omega})$
with
$\epsilon^2  \sup_{P_R} |g| \leq a$.
Let $ f \in C^0(\overline{P_R^\Omega}) \cap C^2(P_R^\Omega) $
be a solution of
\begin{gather*}
 \bigl( \partial_t f - \Delta f \bigr) + \frac{1}{\epsilon^2} f
 =  g \quad  \mbox{in } P^\Omega_R \,, \\
|f|  \leq   a \quad \mbox{on } \tilde{\partial} P_R \cap \Omega \,,
\\
f  =  0 \quad \mbox{on } \partial\Omega \cap P_R  \,.
\end{gather*}
Then  for any $ \delta \in ]0,1[$, we have
$$
\frac{1}{\epsilon^2 }   |f|   \leq
\sup_{ P^\Omega_R } |g|   +   \frac{2a}{\epsilon^2 }  e^{-\frac{1}{\epsilon }
 (1-\delta^2)^2  R^4 } \quad
\mbox{on }  P^\Omega_{\delta R} \,.
$$
\end{coro}

The proof of Lemma \ref{f} also applies  in this case.
The next interior-estimate-version of Lemma \ref{f} deals with
the case
$ B_R(x_0) \cap \Omega \neq \emptyset $ and
$ f \neq 0 $ on $ \partial B_R(x_0) \cap \Omega $.
The estimate then also depends on
$\operatorname{dist}(x,\partial\Omega)$. We formulate the following lemma in such a way
that it readily extends to the case $\Omega = M$ is a manifold.

\begin{coro} \label{fbdry} % subsolution 2
Let $ U \subset \mathbb{R}^2 $
be an open smooth neighborhood of $ 0 $ with $\mathop{\rm diam} U \leq 1 $
and set $P_{R,U} := U \times ]-R^2,0[ $.
Consider
$ a > 0 $,   $R \in ]0, \frac{1}{4}[ $,
$\epsilon \in ]0,\frac{1}{4} [ $ and
$ g \in C^0(\overline{P_{R,U}})$ with
$\epsilon^2  \sup_{P_{R,U}} |g| \leq a$.
Let $ f \in C^0(\overline{P_{R,U}}) \cap C^2(P_{R,U}) $
be a solution of
\begin{equation} \label{eq_rho_2}
\begin{gathered}
 \bigl( \partial_t f - \Delta f \bigr) + \frac{1}{\epsilon^2} f
 =  g \quad \mbox{in } P_{R,U} \,,\\
|f| \leq a \quad  \mbox{on } \tilde{\partial} P_{R,U} \,. \\
\end{gathered}
\end{equation}
Then there is a constant $ C = C(U) > 0 $, such that
for any $ \delta \in ]0,1[$ we have
$$
  \frac{1}{\epsilon^2}
|f (x,t)|   \leq \underset{P_{R,U}}{\sup}  |g|
  +     \frac{2a}{\epsilon^2} e^{-\frac{ R^2}{C \epsilon }
(1-\delta^2) \operatorname{dist}^2 (x,\partial U) }
\quad \mbox{ on }  P_{\delta R,U}\,.
$$
\end{coro}
Of course
$$
\tilde{\partial } P_{R,U} := \bigl( U \times \{ - R^2 \} \bigr)
\cup \bigl( \partial U \times [-R^2,0] \bigr) \,.
$$

\begin{proof}
Set $ d(x) := \operatorname{dist} (x, \partial U ) $,
$ C = C(U) := \max \{ 1,\norm{ \Delta d^2}_{ L^\infty (U) },
\norm{\nabla d^2}_{ L^\infty (U) } \} $.
Note that $ d(x) \leq 1 $ on $ U $ since $\mathop{\rm diam} U \leq 1 $.
We claim that
$$
\omega (x,t) :=
2 a   e^{-\frac{1}{C \epsilon}  d^2(x)   (R^2 + t )}
$$
is a supersolution of equation (\ref{eq_rho_2}),
if  $ 0 < R < \frac{1}{4} $
and $ 0 < \epsilon < \frac{1}{4} $.
Indeed
\begin{align*}
\epsilon^2   (\partial_t - \Delta )   \omega + \omega
&= \omega   \Bigl[1 - \frac{\epsilon }{C} d^2
 + \frac{\epsilon }{C} (R^2 + t) \Delta d^2
- \frac{\epsilon }{C}\frac{1}{\epsilon C} (R^2 + t)^2
\abs{\nabla d^2}^2 \Bigr] \\
& \geq \omega   \bigl[ 1 - \epsilon - \epsilon R^2 - R^4\bigr] \\
&\geq    \frac{1}{4}   \omega    >   0
\quad \mbox{on } P_{R,U}\,.
\end{align*}
The claim  now follows just  as in the proof of Lemma \ref{f}.
\end{proof}

We will also need a priori $L^p$-estimates for the above equation.

\begin{lemm} \label{l1}
Consider a smooth domain $\Omega \subset \mathbb{R}^2$,
$ g \in L^1(\Omega\times ]0,T[)$ and $\epsilon >0$.
Let $ f \in C^1(\overline{\Omega}\times [0,T]) \cap C^2(\Omega\times ]0,T[) $
be a solution of
\begin{gather*}
 \bigl( \partial_t f - \Delta f \bigr) + \frac{1}{\epsilon^2} f
 =    g \quad  \mbox{in } \Omega \times ]0,T[\,,\\
f  =  0 \quad \mbox{on } \Omega \times \{0\} \cup \partial\Omega \times ]0,T[ \,.
\end{gather*}
For $f\geq 0$, we only need to assume
\begin{gather*}
 \bigl( \partial_t f - \Delta f \bigr) + \frac{1}{\epsilon^2} f
    \leq    g \quad \mbox{ in } \Omega \times ]0,T[ \,, \\
f  =  0 \quad \mbox{on } \Omega \times \{0\} \cup \partial\Omega \times ]0,T[ \,.
\end{gather*}
Then
\begin{equation} \label{global-l1}
\norm{ \frac{1}{\epsilon^2}  f }_{L^1(\Omega \times ]0,T[)}
  \leq
\norm{g}_{L^1(\Omega \times ]0,T[)} \,.
\end{equation}
and for any $R,\rho >0$ and $z_0 =(x_0,t_0) \in \Omega \times ]0,T]$ with
$R^2+\rho^2 < t_0$,
\begin{equation} \label{1/2}
\int_{P^\Omega_{R}(z_0)}
 \frac{1}{\epsilon^2} |f|   dz
  \leq \int_{P^\Omega_{R+\rho}(z_0)}\Bigl( |g|  +
 \frac{C}{\rho^2}  |f| %+ \frac{C}{\rho}   |\nabla f|
\Bigr)   dz  \,.
\end{equation}
\end{lemm}

\begin{proof}
(i) Multiplication of the equation for $f$ by
$\frac{f}{\sqrt{f^2+\delta^2}}$
leads to
$$
\frac{\partial_t |f|   |f|}{\sqrt{f^2+\delta^2}} +
\frac{|\nabla f|^2}{\sqrt{f^2+\delta^2}}
\big(1 -\frac{f^2}{f^2+\delta^2}\big) +
\frac{1}{\epsilon^2} \frac{f^2}{\sqrt{f^2+\delta^2}}
= \frac{g f}{\sqrt{f^2+\delta^2}}
+ \Delta \sqrt{f^2+\delta^2}  \,.
$$
Now integrate over $\Omega \times ]0,t[$
for any $t\in ]0,T]$ and let $\delta \to 0$
to obtain
$$
\sup_{0\leq t \leq T} \int_\Omega   |f(x,t)| \,  dx +
\int_0^T \int_\Omega \frac{1}{\epsilon^2} |f|
\leq \int_0^T \int_\Omega  |g|  \, dx \,dt  \,.
$$
(ii) We multiply the equation by $f$ with
$$
\Bigl( \frac{f}{\sqrt{f^2+\delta^2}}\Bigr) (x,t)
\phi(x)    \eta(t)  \,.
$$
The cut-off function $\phi$
satisfies
$ 0 \leq \phi \in C^\infty_c(\mathbb{R}^2)$ with
spt$\phi \subset B_{R+\rho}(x_0) $ and
$ \phi \equiv 1 $ on $ B_{R}(x_0) $, whereas
$ \eta \in C^\infty(\mathbb{R}_+) $
with $ 0 \leq \eta (t)\leq 1$,   $\eta (t_0 - R^2 - \rho^2) = 0 $
and $ \eta (t) \equiv 1$ if $ t \geq t_0 - R^2$.
We may assume
$$
|\nabla \phi| \leq \frac{C}{\rho}  \,, \quad
|\nabla^2 \phi| \leq \frac{C}{\rho^2}
\quad \mbox{and} \quad |d_t \eta| \leq \frac{C}{\rho^2} \,.
$$
This leads to
\begin{align*}
&\frac{\partial_t \bigl(|f|  \phi^2  \eta\bigr) |f|}{\sqrt{f^2+\delta^2}} +
\frac{|\nabla f|^2   \phi^2  \eta}{\sqrt{f^2+\delta^2}}
\big(1 -\frac{f^2 }{f^2+\delta^2}\big) +
\frac{1}{\epsilon^2} \frac{f^2  \phi^2  \eta}{\sqrt{f^2+\delta^2}}\\
&= \frac{g f  \phi^2  \eta}{\sqrt{f^2+\delta^2}}
+ \operatorname{div} \big( \nabla f \frac{f \phi^2  \eta}{\sqrt{f^2+\delta^2}}
 \big) + \frac{f^2   \phi^2   \partial_t\eta }{2 \sqrt{f^2+\delta^2}}
- 2 \eta  \phi   \nabla \phi    \nabla \sqrt{f^2+\delta^2} \,.
\end{align*}
Of course
$$
\int_\Omega   \phi  \nabla \phi  \eta   \nabla \sqrt{f^2+\delta^2}\, dx
= - \int_\Omega   \eta    \sqrt{f^2+\delta^2}
\bigl(  \phi  \nabla^2\phi + |\nabla \phi|^2 \bigr)\, dx \,.
$$
After integrating (\ref{troet}) and letting $\delta \to 0$, we obtain
$$
\sup_{t_0-(R^2+\rho^2)< t <t_0}   \int_{B^\Omega_R}   \frac{1}{2}  |f|\,dx
+ \int_{P^\Omega_R}\frac{1}{\epsilon^2}   |f|  \, dz
\leq \int_{P^\Omega_{R+\rho}}
\Bigl( |g| + \frac{C}{\rho^2} |f| %+ \frac{C}{\rho}   |\nabla f|
\Bigr) \, dz\,.
$$
\end{proof}

\begin{lemm} \label{flp}
Consider a smooth domain $\Omega \subset \mathbb{R}^2$,
$ g \in L^1 \cap L^p(\Omega \times ]0, T[)$ for $p\geq2$  and $\epsilon >0$.
Let $ f \in C^1(\overline{\Omega}\times [0, T]) \cap C^2(\Omega\times ]0, T[) $
be a solution of
\begin{gather*}
 \bigl( \partial_t f - \Delta f \bigr) + \frac{1}{\epsilon^2} f
  =  g\quad \mbox{in } \Omega\times ]0, T[\,,\\
f  =  0 \quad \mbox{on } \partial\Omega \times ]0,T[\,.
\end{gather*}
For $f\geq 0$, we only need to assume
\begin{gather*}
 \bigl( \partial_t f - \Delta f \bigr) + \frac{1}{\epsilon^2} f
  \leq    g \quad \mbox{ in } \Omega\times ]0,T[ \,,\\
f = 0\quad \mbox{ on } \partial\Omega\times ]0,T[ \,.
\end{gather*}
(i) For any $\delta \in ]0,1[$ and $z_0 =(x_0,t_0) \in \Omega \times ]0,T]$
with $0<R^2< t_0$, we have
$$
\norm{\frac{1}{\epsilon^2} f}_{L^p(P_{\delta R}^\Omega (z_0))}
  \leq   C_1  \norm{g}_{L^p(P^\Omega_{R}(z_0) )}
+  \epsilon^{2/p}    C_2 \,,
$$
where $ C_1 = C_1(p)$ and
$C_2 = C_2 \bigl(\norm{g}_{L^p(P^\Omega_{R}(z_0))},
\norm{f}_{L^{2p-1}(P^\Omega_{R}(z_0))}, p, \delta, R \bigr)$.

\noindent (ii) The same bound as in (i) holds for
$$
\norm{\bigl(\frac{1}{\epsilon^2}\bigr)^{(1-\frac{1}{p})}
f}_{L^{2p}(P^\Omega_{\delta R}(z_0))}  \,,
\norm{\bigl(\frac{1}{\epsilon^2}\bigr)^{(1-1/p)}f}_{ L^\infty
\bigl( [t_0-R^2,t_0]; L^p(B^\Omega_{\delta R}(x_0))\bigr)}
$$
and
$$
\norm{\frac{1}{\epsilon} \nabla f}_{L^2(P^\Omega_{\delta R}(z_0))}\,.
$$
\end{lemm}

\begin{proof}
We multiply the equation for $f$ by
$f |f|^{2s-2}(x,t) \phi^2(x) \eta(t)$, where $ s\geq 1$.
The cut-off functions $\phi$ and $\eta$ are the same as in the proof of
Lemma \ref{l1}.
Then
\begin{align*}
&\frac{1}{2s} \partial_t \bigl( |f|^{2s}(x,t) \phi^2(x) \eta(t) \bigr)
+ \frac{2s-1}{s^2} \abs{\nabla |f|^s}^2 \phi^2 \eta
+ \frac{1}{\epsilon^2} |f|^{2s} \phi^2 \eta \\
&= - \operatorname{div}\bigl( \nabla f  f |f|^{2s-2} \phi^2 \eta \bigr)
+g f |f|^{2s-2} \phi^2 \eta + \frac{1}{2s} |f|^{2s} \phi^2 d_t\eta -
\nabla |f|  |f|^{2s-1} 2 \nabla \phi \phi \eta  \,,
\end{align*}
for any $s \geq 1$.
By Young's inequality
$a b \leq \delta^{-p} \frac{a^p}{p} + \delta^q \frac{b^q}{q}$
 for $p,q>1$ with $\frac{1}{q} + \frac{1}{p} = 1$
 and $ a,b,\delta >0$,
we have
$$
\left| |g| |f|^{2s-1} \right| \leq \frac{1}{2\epsilon^2}
\frac{2s-1}{2s} |f|^{2s} + (2\epsilon^2)^{(2s-1)} \frac{1}{2s} |g|^{2s}
$$
and
\begin{align*}
\left| \nabla |f| |f|^{2s-1} 2 \nabla \phi \phi \eta \right|
&=2 |(\frac{1}{s} \nabla |f|^s \phi)   (|f|^s \nabla\phi)   \eta |\\
&\leq \frac{2s-1}{2s^2} \abs{\nabla |f|^s}^2 \phi^2 \eta + \frac{2}{2s-1}
|f|^{2s} \abs{\nabla \phi}^2 \eta  \,.
\end{align*}
This leads to
\begin{equation} \label{pain}
\begin{aligned}
&\frac{1}{2s} \partial_t \bigl( |f|^{2s}(x,t) \phi^2(x) \eta(t) \bigr)
+ \frac{2s-1}{2s^2} \abs{\nabla |f|^s}^2 \phi^2 \eta
+ \frac{1}{2\epsilon^2} |f|^{2s} \phi^2 \eta \\
&\leq - \operatorname{div}\bigl( \nabla f  f |f|^{2s-2} \phi^2 \eta \bigr)
+ (2\epsilon^2)^{(2s-1)} \frac{1}{2s} |g|^{2s} \phi^2 \eta\\
&\quad + \frac{2}{2s-1}|f|^{2s} \bigl( \abs{\nabla \phi}^2 \eta
 + \phi^2 |d_t\eta| \bigr) \,.
\end{aligned}
\end{equation}
For notational ease we relabel the domain
as  $\Omega \times ]-T,0[$ and assume
$z_0 = (0,0) \in  \Omega \times ]-T,0[$ and
$ 0< R^2 + \rho^2 <T$. As always $P_R := P_R(0)$.

\noindent (i) Set $p = 2s$.
After multiplying (\ref{pain}) with
$\bigl(\frac{1}{\epsilon^2}\bigr)^{p-1}$ and integrating,
we get, for $p \geq 2$,
\begin{equation} \label{main-s}
\begin{aligned}
&\sup_{t \geq -R^2-\rho^2}
 \int_{B^\Omega_{R+\rho}}\bigl(\frac{1}{\epsilon^2} \bigr)^{p-1}
 |f|^p(x,t) \phi^2(x) \eta(t)  \,  dx\\
&\quad + \int_{P^\Omega_{R+\rho}}
 \bigl(\frac{1}{\epsilon^2} \bigr)^{p-1}
\abs{\nabla f^{p/2}}^2 \phi^2 \eta \, dz\Bigr) \\
&\quad +
\int_{P^\Omega_{R+\rho}}
  \bigl(\frac{1}{\epsilon^2}\bigr)^p  |f|^p \phi^2 \eta  \, dz\\
&\leq C(p)\Bigr(
\int_{P^\Omega_{R+\rho}}    |g|^p \phi^2 \eta   dz  +
 \bigl(\frac{1}{\epsilon^2}\bigr)^{p-1} \int_{P^\Omega_{R+\rho}}
|f|^p \bigl( |\nabla \phi|^2 \eta + \phi^2 |d_t \eta| \bigr)
dz \Bigl) \,.
\end{aligned}
\end{equation}
In particular we have
\begin{equation} \label{joker}
\int_{P^\Omega_{R}}
 \bigl(\frac{1}{\epsilon^2}\bigr)^{p} |f|^{p}   dz
  \leq
C(p)\Big(\int_{P^\Omega_{R+\rho}}    |g|^{p}  \,dz +   \epsilon^2   \frac{C}{\rho^2}
\int_{P^\Omega_{R+\rho}} \bigl(\frac{1}{\epsilon^2}\bigr)^p
|f|^p   dz  \Bigl) \,.
\end{equation}
Let $p = \frac{k}{2} + 1$ for $k \in \mathbb{N}$.
H\"older's inequality for $ q_1 = (2p-1)/(2p-2) $ and $q_2 = 2p-1$
implies
$$
\int_{P^\Omega_{R}}
 \bigl( \frac{1}{\epsilon^2} |f| \bigr)^{p-1} |f|   dz
  \leq \norm{\frac{1}{\epsilon^2} f}^{p-1}_{ L^{p-(1/2)} }
   \norm{f}_{L^{2p-1}} \,.
$$
Now (\ref{main-s}) leads to
$$
\norm{\frac{1}{\epsilon^2} f}_{L^p(P^\Omega_{R})}^p
  \leq   C(p) \Bigr(\norm{g}_{L^p(P^\Omega_{R+\rho} )}^p
+\frac{C}{\rho^2}
\norm{\frac{1}{\epsilon^2} f}^{p-1}_{ L^{p-(1/2)}(P^\Omega_{R+\rho}) }
   \norm{f}_{L^{2p-1}(P^\Omega_{R+\rho})} \Bigl) \,.
$$
An iteration combined either to
(\ref{1/2}) from Lemma \ref{l1}
after the $k$-th step yields
an estimate of the form
\begin{equation} \label{est1-bis}
\norm{\frac{1}{\epsilon^2} f}_{L^{p}(P^\Omega_{R})}
  \leq C\bigl(\norm{g}_{L^p(P^\Omega_{R+(k+1)\rho})},
\norm{f}_{L^{2p-1}(P^\Omega_{R+(k+1)\rho})},
p, \rho, R \bigr)\,.
\end{equation}
If we set $\rho := \frac{\delta R}{k+1}$
and insert (\ref{est1-bis}) into (\ref{joker}), we obtain
for any $\delta >0$
\begin{equation} \label{est2}
\norm{\frac{1}{\epsilon^2} f}_{L^p(P^\Omega_{R})}^p
  \leq   C(p) \norm{g}_{L^p(P^\Omega_{(1+\delta)R} )}^p
+   \epsilon^2 C_2 \,,
\end{equation}
where
$$
C_2 = C_2 \bigl(\norm{g}_{L^p(P^\Omega_{(1+\delta)R})},
\norm{f}_{L^{2p-1}(P^\Omega_{(1+\delta)R})},
p, \delta, R \bigr)\,.
$$
Claim (i) follows by setting $R_{new} = (1+\delta)R$ and
$\delta_{new} = \frac{1}{1+\delta}$,
i.e. $\delta_{new}  R_{new} =R$.

\noindent (ii) By applying the estimate
$$
\Bigl(\int_{[a,b]} \int_{B_R}   \abs{u}^4    dx dt \Bigr)^{\frac{1}{2}}
 \leq   C   \Bigl(\max_{t \in [a,b]} \int_{B_R}   \abs{u}^2    dx dt
  + \int_{[a,b]} \int_{B_R}   \abs{\nabla u}^2    dx dt \Bigr)
$$
(see Theorem 6.9 p.110 in \cite{Lieberman})
to $ u := f^{p/2}  \phi \sqrt{\eta}$,
we find that the expression
\begin{equation} \label{s4}
\bigl(\frac{1}{\epsilon^2} \bigr)^{p-1}
\Bigl( \int_{P^\Omega_R}
|f|^{2p}\phi^4 \eta^2   dz \Bigr)^{\frac{1}{2}}
+ \int_{P^\Omega_R}
\bigl(\frac{1}{\epsilon^2} \bigr)^p
|f|^p \phi^2 \eta   dz
\end{equation}
admits the same bound as (\ref{main-s})
with a different constant $C(p) >0$.
By combining (\ref{s4}) with (\ref{main-s}),
we see that the same bounds as in (\ref{est2})
also holds for
$$
\bigl(\frac{1}{\epsilon^2} \bigr)^{p-1}
\Bigl(\sup_{-R^2<t<0}  \int_{B^\Omega_R} |f|^{p}(x,t)  \, dx
+\int_{P^\Omega_R}  \abs{\nabla f^{p/2}}^2\,  dz \Bigr)
$$
and
$$
\bigl(\frac{1}{\epsilon^2} \bigr)^{p-1}
\Bigl(\int_{P^\Omega_R}   |f|^{2p}\,   dz \Bigr)^{\frac{1}{2}} \,.
$$
\end{proof}

\subsubsection{Higher estimates}
\label{subsec-bootstrap}
By considering
the flow equation \eqref{Eq_eps_LL} and
equation (\ref{rho}) for $ \rho_\epsilon := 1-|u_\epsilon|^2$ as a coupled system,
the uniform estimates from the previous paragraph
and parabolic estimates for
\eqref{Eq_eps_LL}
 can be combined in a standard bootstrap argument
to prove Lemma \ref{Cinfty1} and \ref{Cinfty2}.

In the case of the harmonic map flow
$d_\epsilon := \operatorname{dist}(u_\epsilon,N)$ replaces
$\rho_\epsilon$.  The corresponding equation is then
\begin{equation} \label{d_eps}
\partial_t d_\epsilon - \Delta  d_\epsilon + |\nabla \nu_\epsilon |^2
d_\epsilon  +
\frac{1}{\epsilon^2 } \chi '(d_\epsilon^2 )  d_\epsilon =
\Delta v_\epsilon \cdot \nu_\epsilon
\leq C | \nabla u_\epsilon |^2  \,,
\end{equation}
whenever $ u_\epsilon \in U $.
Here we decomposed $ u_\epsilon = v_\epsilon + d_\epsilon \nu_\epsilon $,
where $ v_\epsilon := \pi_N (u_\epsilon ) $ and
$\nu_\epsilon := \nu (u_\epsilon ) $ is the unit normal in
$ \bigl( T_{\pi_N (u_\epsilon)} N \bigr)^\perp $, whereas
$ d_\epsilon := dist (u_\epsilon ,N) $.
Remember that
$\pi_N: U \to N$ denotes the nearest neighbour projection
from a tubular neighbourhood $U \subset \mathbb{R}^n$ of $N$ onto $N$.

\section{ Towards characterizing the limits}
\label{sec-limits}

We start with alternative characterisations of the ``regular set''
$\mathop{\rm Reg}\bigl( \{u_\epsilon\}_\epsilon\bigr)$.
Remember that by Lemma \ref{en-est-LL}, we have for any $0\leq s < t$
\begin{equation} \label{reminder}
G_\epsilon \bigl(u_\epsilon(t), B_R^\Omega(x_0)\bigr)
  \leq
G_\epsilon \bigl(u_\epsilon(s), B_{2R}^\Omega(x_0)\bigr)
  +   \frac{C (t-s) E_0}{\gamma_1 R^2} \,.
\end{equation}
Set
$ \delta_0 := \frac{ \gamma_1 \epsilon_0}{2 C E_0}$,
where $ \epsilon_0$ is the constant from Theorem \ref{eps-reg-LL-bd}.
After increasing $C$ if necessary, we may assume $ 0<\delta_0<1$.

\begin{lemm} \label{equiv}
Let $ u_\epsilon $ be a solution of
\eqref{Eq_eps_LL}-\eqref{Eq_bdry_eps_LL}
with $ u_0 \in H^{1,2}(\Omega;S^2)\cap C^{2}(\partial\Omega;S^2)$
for each $ \epsilon > 0 $.
Then the following assertions are equivalent:

\noindent (i) \quad
$ z_0 =(x_0,t_0)\in \mathop{\rm Reg}\bigl( \{u_\epsilon\}_{\epsilon>0}\bigr)$.

\noindent
(ii) \quad $\exists \delta,R>0: \phantom{ppppp|} $
$ \limsup_{\epsilon\searrow 0}\sup_{t_0-\delta < t < t_0}
G_\epsilon \bigl(u_\epsilon (t) , B_R^\Omega (x_0)\bigr)  < \epsilon_0 $.

\noindent (iii) \quad $  \exists \delta>0:\lim_{R\searrow 0}
\limsup_{\epsilon\searrow 0}
\sup_{t_0-\delta < t < t_0}
G_\epsilon \bigl(u_\epsilon (t), B_R^\Omega (x_0)\bigr) = 0 $.

\noindent (iv) \quad
$\exists R>0: \limsup_{\epsilon\searrow 0}
\frac{1}{R^2}  \int_{t_0-R^2}^{t_0} \int_{B_R^\Omega (x_0)}
g_\epsilon (u_\epsilon)\,dx\,dt < \frac{1}{4}  \delta_0   \epsilon_0 $.

\noindent (v) \quad
$\exists \delta,R>0: \limsup_{\epsilon\searrow 0}
\sup_{t_0-\delta < t < t_0 + \delta}
G_\epsilon \bigl(u_\epsilon (t) , B_R^\Omega (x_0)\bigr)  < \epsilon_0  $.
\end{lemm}

\begin{proof}
``$(i) \Leftrightarrow (ii)$''    is obvious.\\
``$(ii) \Rightarrow (iii)$''    follows from Theorem \ref{eps-reg-LL-bd}
in Section \ref{subsec-sup-bd}.\\
``$(iii) \Rightarrow (iv)$''     is obvious.\\
``$(iv) \Rightarrow (ii)$'':
Assume $(iv)$ holds. By (\ref{reminder})
and the above choice of $\delta_0$,
we have for sufficiently small $\epsilon >0$,
\begin{align*}
&\sup_{t_0 - (1/2)\delta_0 R^2 < t < t_0}
G_\epsilon \bigl(u_\epsilon(t), B_{(1/2)R}^\Omega(x_0)\bigr)\\
&\leq \inf_{t_0 -\delta_0 R^2 < s < t_0 - (1/2)\delta_0 R^2}
 G_\epsilon \bigl(u_\epsilon(s), B_R^\Omega(x_0)\bigr)
+ \frac{C \delta_0 R^2 E_0}{\gamma_1 R^2} \\
&\leq \frac{2}{\delta_0 R^2}  \int_{t_0-\delta_0 R^2}^{t_0 - (1/2)\delta_0 R^2 }
  G_\epsilon \bigl(u_\epsilon(t), B_R^\Omega(x_0)\bigr)   \,dt
+  \frac{1}{2} \epsilon_0 \\
&< \frac{2}{\delta_0}   \frac{1}{4} \delta_0 \epsilon_0
+  \frac{1}{2} \epsilon_0  < \epsilon_0 \,.
\end{align*}
``$(v) \Rightarrow (ii)$'' is obvious.\\
``$(iii) \Rightarrow (v)$'':
Assume (iii) holds. Then there are $R,\delta>0$,
such that
$$
\limsup_{\epsilon\searrow 0} \sup_{t_0-\delta\leq t\leq t_0}
\int_{B_R(x_0)\cap \Omega}  g_\epsilon (u_\epsilon(x,t))   dx
< \epsilon_0/2 \,.
$$
On the other hand by (\ref{reminder}), we have for
$\delta_{new} := \frac{\epsilon_0 \gamma_1 R^2}{2C E_0} = \delta_0 R^2 $,
$$
\sup_{t_0\leq t\leq t_0 +\delta_{new}}
\int_{B_{\frac{1}{2}R}(x_0)\cap \Omega}  g_\epsilon (u_\epsilon(x,t))   dx
\leq
\int_{B_{R}(x_0)\cap \Omega}  g_\epsilon (u_\epsilon(x,t_0))   dx
+ \frac{\delta_{new} C E_0}{\gamma_1 R^2}\,.
$$
Now (v) holds for $\frac{1}{2}R$ and $\min \{ \delta, \delta_{new}\}$.
\end{proof}

\begin{coro} \label{prop-limits}
Let $ u_\epsilon $ be a solution of \eqref{Eq_eps_LL}-\eqref{Eq_bdry_eps_LL}
with $ u_0$ in $H^{1,2}(\Omega;S^2)\cap \break C^{2}(\partial\Omega;S^2)$
for each $ \epsilon > 0 $. Let $\{\epsilon_i\}_i$ be a sequence with
$\epsilon_i \searrow 0$ as $i\to\infty$. Then the following holds:

\noindent(i)
$\mathop{\rm Reg}(\{u_\epsilon\}_\epsilon)$
and $\mathop{\rm Reg}(\{u_{\epsilon_i}\}_i)$
are open in $ \overline{\Omega}\times \mathbb{R}_+$.

\noindent (ii) There is some $ T_0 > 0 $, such that
$ \overline{\Omega} \times [0,T_0[
\subset \mathop{\rm Reg}(\{u_\epsilon\}_\epsilon) $.
\end{coro}

\begin{proof}
(i) follows from Lemma \ref{equiv} (v).\\
(ii) The existence of $T_0$ immediately follows from
Lemma \ref{en-est-LL} (\ref{gs}).
\end{proof}

Set
$$
Q_R(z) := B_R(x) \times ]t-R^2,t+R^2[ \quad \mbox{ for } z = (x,t) \,.
$$
and let $\mathfrak{H}^2$ denote the 2-dimensional parabolic Hausdorff measure.

\begin{prop} \label{Hausdorff}
Let $ u_\epsilon $ be a solution of
\eqref{Eq_eps_LL}-\eqref{Eq_bdry_eps_LL}
with $ u_0\in H^{1,2}(\Omega;S^2)\cap C^{2}(\partial\Omega;S^2)$
for each $ \epsilon > 0 $.
Then the following holds:

\noindent
(i) $\mathbb{S}(\{u_\epsilon\}_\epsilon)$ has locally finite two dimensional
parabolic Hausdorff-measure. More precisely
there is a constant $K_1 =K_1(E_0,\epsilon_0)>0$, such that
for any compact intervall $I \subset \mathbb{R}_+$
$$
\mathfrak{H}^2 \bigl( \mathbb{S}(\{u_\epsilon\}_\epsilon)
\cap (\overline{\Omega} \times I) \bigr)
\leq    K_1  |I| \,.
$$

\noindent (ii)
There is a constant $ K_2 = K_2(E_0,\epsilon_0) > 0 $, such that
for any $t>0$ the set
$\mathbb{S}^t(\{u_\epsilon\}_\epsilon) := $
$\mathbb{S}(\{u_\epsilon\}_\epsilon) \cap (\overline{\Omega} \times \{t\}) $
consists of at most $K_2$ points.
\end{prop}

\begin{proof}
(i) By $(iv)$ of  Lemma \ref{equiv}, we have for any
$z_0 =(x_0,t_0) \in \mathbb{S}(\{u_\epsilon\}_\epsilon)$,
any $R>0$ and sufficiently small $0 <\epsilon \leq \epsilon (z_0)$
\begin{equation} \label{troet}
\frac{1}{R^2}  \int_{t_0-R^2}^{t_0} \int_{B_R^\Omega (x_0)}
g_\epsilon (u_\epsilon)   dx   dt > \frac{1}{4}  \delta_0   \epsilon_0 \,.
\end{equation}
Fix a compact intervall $I\subset \mathbb{R}_+$ and $\delta>0$.
By compactness and
Vitali's Covering Theorem
any covering of
$ \mathbb{S}(\{u_\epsilon\}_\epsilon) \cap
\bigl(\overline{\Omega} \times I\bigr) $
by parabolic cylinders $Q_{R}^\Omega (z)$
with $ 0<R^2<\delta $ and
$ z \in \mathbb{S}(\{u_\epsilon\}_\epsilon) \cap
\bigl(\overline{\Omega} \times I\bigr)$
contains a finite covering
$ \bigcup_j  Q_{5 R_j}^\Omega (z_j)
\supset \mathbb{S}(\{u_\epsilon\}_\epsilon) \cap
\bigl(\overline{\Omega} \times I\bigr) $,
such that the cylinders $Q_{R_j}^\Omega (z_j)$ are pairewise disjoint.
By (\ref{troet}) and the energy estimate, we obtain for
$0< \epsilon \leq \min_j\{\epsilon(z_j)\} $
$$
\sum_j   \omega_2  (5 R_j)^2
\leq 25  \frac{4 \omega_2 }{\delta_0 \epsilon_0}
\sum_j \ \int_{t_j-R_j^2}^{t_j} \int_{B_{R_j}^\Omega (x_j)}
g_\epsilon (u_\epsilon)   dx   dt
< \frac{100 \omega_2 }{\delta_0 \epsilon_0}  (|I|+ \delta)  E_0  \,.
$$
By letting $\delta \searrow 0$, we find
$$
\mathfrak{H}^2 \bigl( \mathbb{S}(\{u_\epsilon\}_\epsilon) \cap I  \bigr)
\leq \frac{100 \omega_2 E_0}{\delta_0 \epsilon_0} |I|  \,.
$$
(ii) Pick any $ (x_1,T) , \dots , (x_k,T) \in \mathbb{S}(\{u_\epsilon \}) $.
By assumption we have for
$$
\forall R , \delta ,  \gamma > 0 ,
\exists  \epsilon \in ]0,\gamma [ : \quad
\underset{T - \delta < t < T}{\sup}   \int_{B_R^\Omega (x_l)}
g_\epsilon (u_\epsilon (x,t))   dx \geq  \frac{\epsilon_0}{2}
\mbox{ for } 1 \leq l \leq k \,.
$$
We may choose $ R > 0 $, such that the
$ B_R^\Omega (x_l)    (1 \leq l \leq k ) $
are pairewise disjoint.
Choose $ \delta \in ] 0,\frac{\gamma_1 R^2 \epsilon_0}{4 C E_0}[ $,
where $C$ is the constant from Lemma \ref{en-est-LL} \eqref{en-est}
and
$ \epsilon\in ]0,\gamma [ $ as above.
Since
$ t \mapsto \int_{B_R^\Omega (x_l)}   g_\epsilon (u_\epsilon (x,t))   dx $
is continuous, we may find
$ t_\delta^l \in ]T-\delta , T[  $ such that
$$
\int_{B_R^\Omega (x_l)}   g_\epsilon (u_\epsilon (x,t_\delta^l ))   dx
 \geq \frac{\epsilon_0}{2}
\quad \mbox{ for } 1 \leq l \leq k  \,.
$$
The energy estimate and the local energy inequality,
Lemma \ref{en-est-LL},\eqref{en-est} and \eqref{loc-en}
now imply
\begin{align*}
E_0 &\geq \sum_{l=1}^k   \int_{B_R^\Omega (x_l)}
g_\epsilon (u_\epsilon (x,T-\delta ))  \, dx\\
&\geq \sum_{l=1}^k   \Bigl(   \int_{B_R^\Omega (x_l)}
g_\epsilon \Bigl(u_\epsilon \bigl(x,t_\delta^l \bigl) \Bigl)    dx
  -   \frac{C}{\gamma_1 R^2} \int_{T - \delta}^T
\int_{B_R^\Omega (x_l)}
 \abs{\nabla u_\epsilon (x,t)}^2     dx   dt     \Bigr).
\end{align*}
Thus
$E_0  \geq k \bigl( \frac{\epsilon_0}{2}  -  \frac{C E_0}{R^2} \delta \bigr)$.
Now since $ \delta < \frac{R^2 \epsilon_0}{4 C E_0} $,
this implies
$ k   \leq    \frac{8 E_0}{\epsilon_0} =: K_2  $.
(Compare \cite{Struwe3} and \cite{Struwe2} ($1^\circ$) of the proof of Theorem
6.6 p.229 for a similar argument in the case of the harmonic map flow.)
\end{proof}

\begin{theorem} \label{limits}
Let $ u_\epsilon $ be a solution of \eqref{Eq_eps_LL}-\eqref{Eq_bdry_eps_LL}
with $ u_0$ in $H^{1,2}(\Omega;S^2)\cap \break H^{3/2,2}(\partial\Omega;S^2)$
for each $ \epsilon > 0 $. Then the following holds: \\
There is at least one sequence
$ \{ \epsilon_i \}_i $, with
$ \epsilon_i \to 0 $ as $ i \to \infty $ and
$$
u_*\in H^{1,2}_{\rm loc}( \overline{\Omega} \times \mathbb{R}_+; S^2)
\cap L^\infty(\mathbb{R}_+;H^{1,2}(\Omega;S^2))  ,
$$
such that $ u_{\epsilon_i} \rightharpoonup u_* $ weakly in
$ H^{1,2}_{\rm loc}(\overline{\Omega} \times \mathbb{R}_+;\mathbb{R}^3) $
and weak* in $L^\infty(\mathbb{R}_+; H^{1,2}(\Omega;\mathbb{R}^3))$.

In addition:
(i) For any such sequence $ \{ u_{\epsilon_i} \}_i $,
we have
$$
\lim_{i\to\infty} u_{\epsilon_i } = u_*\quad \mbox{ in }
C^\infty (\mathop{\rm Reg}(\{u_{\epsilon_i}\}_i)
\cap (\Omega \times \mathbb{R}_+); \mathbb{R}^3)
$$
and
$\frac{1}{\epsilon^2} (1-|u_\epsilon|^2) \to |\nabla u_*|^2$
in $C^\infty (\mathop{\rm Reg}(\{u_{\epsilon_i}\}_i)
\cap (\Omega \times \mathbb{R}_+))$.
\\
(ii) $ u_*$ is a smooth solution of \eqref{LL} in
$ \mathop{\rm Reg}(\{u_{\epsilon_i}\}_i)\cap (\Omega \times \mathbb{R}_+) $
and a distributional solution in
$$
H^{1,2}_{\rm loc}(\overline{\Omega} \times \mathbb{R}_+)
\cap L^\infty \bigl(\mathbb{R}_+;H^{1,2}(\Omega;\mathbb{R}^n)\bigr)
$$
on all $\Omega \times \mathbb{R}_+$.
Further $ \lim_{t\searrow 0} u_*(.,t) = u_0 $ in $H^{1,2}(\Omega;\mathbb{R}^3)$
and
$$
 {u_*(.,t)}_{|\partial\Omega} = {u_0}_{|\partial\Omega} \mbox{ as a }
H^{2,2}(\Omega;\mathbb{R}^3)\mbox{-trace for a.e. } t>0\,.
$$
(iii) If $ u_* $ is regular at $ z_0 = (x_0,t_0) \in \overline{\Omega}\times \mathbb{R}_+ $
in the sense that
$$
\lim_{R\searrow 0}   \sup_{t_0-R^2 \leq t \leq t_0}  \int_{B_R(x_0)}
  |\nabla u_*|^2   dx = 0
$$
and if $z_0$ is parabolically isolated for $\{ u_{\epsilon_i} \}_i$,
i.e.
$$
B_{R_0}(x_0) \times ] t_0 - R_0^2,t_0[
\subset \mathop{\rm Reg}(\{u_{\epsilon_i}\})\mbox{ for some }R_0 >0\,,
$$
 then
$z_0 \in \mathop{\rm Reg}(\{u_{\epsilon_i}\})$.
(In particular
$ u_* $ cannot (backwards) concentrate energy and (backwards) bubble at $z_0$
as $ t\searrow t_0 $.  Compare \cite{Harpes1}, \cite{Harpes3})
\end{theorem}

\begin{proof}
(i) The convergence statements follow from the energy estimate
(Lemma \ref{en-est-LL}), \mbox{Theorem \ref{eps-reg-LL}} and Lemma \ref{Cinfty1}.

\noindent(ii) For the case $ \gamma_2 = 0 $ and
$ f(u_\epsilon) =
\frac{1}{2} \frac{d}{du}\chi \bigl(\operatorname{dist}^2(u_\epsilon,N)\bigr) $,
this is proven in \cite{Struwe-Chen} III p.95.
We will prove it in the case $\gamma_2 \neq 0$.
If we apply ``$ u_{\epsilon_i} \times\,. $'' from the left to
\eqref{Eq_eps_LL}
and pass to the limit $\epsilon_i \to 0$
on $ \mathop{\rm Reg}(\{u_{\epsilon_i}\})\cap (\Omega \times \mathbb{R}_+) $,
we obtain
\begin{equation}\label{bof}
\gamma_1 u_* \times {\partial}_t u_*
  -  \gamma_2 u_* \times( u_* \times {\partial}_t u_*)
- u_* \times \Delta u_* = 0 \,.
\end{equation}
Since $(1-|u_{\epsilon_i}|^2) \to 0$ smoothly, we also have
$$
|u_*(x,t)| = 1 \quad \mbox{ in }
\mathop{\rm Reg}(\{u_{\epsilon_i}\})\cap (\Omega \times \mathbb{R}_+).
$$
Now we use $ a \times (b\times c) = (ac) b - (ab) c $ and
$ |u_*| \equiv 1 $ while applying  ``$ u_* \times $.'' from the left to (\ref{bof}),
to obtain
\begin{equation}\label{goal}
\gamma_1   \partial_t u_* - \gamma_2
  u_* \times \partial_t u_* - \Delta u_* = \abs{\nabla u_*}^2 u_*
\quad \mbox{ in } \Omega \times \mathbb{R}_+ \,.
\end{equation}
In particular, since the left side of \eqref{Eq_eps_LL}
converges to the left side of (\ref{goal}), we have
$$
\frac{1}{\epsilon_i^2} (1-|u_{\epsilon_i}|^2) \to |\nabla u_*|^2
\mbox{ in } C^\infty
(\mathop{\rm Reg}(\{u_{\epsilon_i}\})\cap
(\Omega \times \mathbb{R}_+))\,.
$$
We now prove that $u_*$ is a distributional
$H^{1,2}_{\rm loc}\cap L^\infty(H^{1,2})$-solution of \eqref{LL}
on all $\Omega\times\mathbb{R}_+ $.
Note that the sequence
$\{ u_{\epsilon_i} \}_i$ converges weakly in
$H^{1,2}(\Omega\times \mathbb{R}_+; S^2)$
and smoothly on
$\mathop{\rm Reg}(\{u_{\epsilon_i}\})\cap (\Omega \times \mathbb{R}_+)$.
Further since
$\mathbb{S}^t (\{u_{\epsilon_i}\}) := $
$\mathbb{S}(\{u_{\epsilon_i}\}) \cap (\overline{\Omega} \times \{t\}) $
is finite for all $t\geq 0$, we have both
$u_{\epsilon_i} \to u_*$ pointwise a.e.  in
$\Omega \times \mathbb{R}_+$
and $u_{\epsilon_i}(\cdot,t) \to u_*(\cdot,t)$
pointwise a.e.  in $\Omega$ for all $t \in \mathbb{R}_+$.
Since
$ \int_0^\infty\int_\Omega   |\partial_t u_{\epsilon_i}|^2   dx dt
\leq E_0 $, by Fatou's Lemma the complement of
$$
 A :=\{ t \geq 0   |  \liminf_{\epsilon_i\searrow 0}
\int_\Omega   |\partial_t u_{\epsilon_i}|^2(x,t)   dx < \infty \}
$$
has measure 0. Pick $t_0 \in A$. Then there is a subsequence still denoted by
$u_{\epsilon_i}$, such that
$ \partial_t u_{\epsilon_i}(\cdot,t_0) \rightharpoonup \partial_t u_*(\cdot,t_0)$
weakly in $L^2(\Omega ; \mathbb{R}^3)$.
By the local energy estimate, we may assume that, for the same subsequence,
we also have
$u_{\epsilon_i}(\cdot,t_0) \rightharpoonup u_*(\cdot,t_0)$
weakly in $H^{1,2}(\Omega ; S^2)$.
By pointwise a.e.  uniqueness of the limit, the whole sequence
converges.
Also
$$
u_*(\cdot,t_0) \in H^{1,2}(\Omega;S^2)
\quad \mbox{and} \quad
\partial_t u_* (\cdot,t_0) \in L^2(\Omega;\mathbb{R}^3)
\quad \mbox{for all } t_0 \in A \,.
$$
Now
$$
-\Delta u_*(\cdot,t_0) = (|\nabla u_*|^2 u_*)(\cdot,t_0) + f \,,
$$
where
$$
f = - \gamma_1 \partial_t u_*(\cdot,t_0)
+ \gamma_2 u_* \times \partial_t u_*(\cdot,t_0) \in  L^2(\Omega;\mathbb{R}^3)
$$
and by a regularity result of T.Rivi\`ere
(see \cite{Tristan} Lemma p.3), we have
$$
u_*(\cdot,t_0) \in H^{2,2}(\Omega;S^2)
\quad \mbox{if} \quad
u_0 \in H^{3/2,2}(\partial\Omega;S^2) \cap H^{1,2}(\Omega;S^2)\,.
$$
This in particular implies
$ {u_*(.,t)}_{|\partial\Omega} = {u_0}_{|\partial\Omega} $ as a
$H^{2,2}(\Omega)$-trace for any $t\in A$.
Further since
$ \mathbb{S}^{t_0}(\{u_{\epsilon_i}\}_i) $
consists of finitely many points, it has vanishing $2$-capacity
in $\mathbb{R}^2$, i.e.
$$
Cap_2 \bigl(\mathbb{S}^{t_0}(\{u_{\epsilon_i}\})\bigr) = 0
$$
(see \cite{Evans-Gariepy}).
Therefore, there is a sequence
$\{\eta_k \}_k = \{ \eta_{k,q}\}_k \subset C^\infty_c(\mathbb{R}^2)$
with
$$
\eta_k(x) = 1      \forall   x \in \mathbb{S}^{t_0}(\{u_{\epsilon_i}\}_i)
\quad \mbox{ and }\quad
\norm{\eta_k}_{H^{1,2}(\mathbb{R}^2)} \overset{(k\to\infty)}{\to} 0
$$
(see \cite{Evans-Gariepy} 4.7.1).
For $ \phi \in C^\infty_c(\Omega)$, we may test equation
(\ref{bof}) with the cut-off function
$ (1-\eta_k)   \phi$,  which has support in
$\mathop{\rm Reg} (\{u_{\epsilon_i}\}_i)$.
After  passing to the limit $ k\to\infty $,
we find that for %all $ \phi \in C^\infty_c(\Omega)$ and
any $t\in A$,
\begin{align*}
&\int_\Omega  \gamma_1 \partial_t u_*(x,t)   \phi(x)
- \gamma_2   (u_* \times \partial_t u_*)(x,t) \phi(x)
+  \nabla u_* (x,t)   \nabla \phi(x) \,  dx \\
&=\int_\Omega   (|\nabla u_*|^2 u_*)(x,t)  \phi(x) \, dx   \,.
\end{align*}
This equation holds for a.e.  $t\geq0$.
On the other hand, we have
$u_*\in H^{1,2}(\Omega \times [0,T];S^2)$ for any $T>0$
and so both sides of the above equation are locally integrable on
$\mathbb{R}_+$.
Therefore we may multiply the equation
with $ \psi \in C^\infty_c([0,\infty[) $ and integrate over $\mathbb{R}_+$.
Moreover linear combinations
$\sum_k a_k \phi_k(x) \psi_k(t) $ with
$ \phi_k \in C^\infty_c(\Omega) $ and
$ \psi_k \in C^\infty_c([0,\infty[) $ are dense in
$ C^\infty_c(\Omega \times [0,\infty[) $ and so
\begin{align*}
&\int_0^\infty \int_\Omega \gamma_1 \partial_t
u_*(x,t)   \phi(x,t)
-\gamma_2   (u_* \times \partial_t u_*)(x,t) \phi(x,t)
+\nabla u_* (x,t)   \nabla \phi(x,t)   dx\, dt \\
&= \int_0^\infty \int_\Omega   (|\nabla u_*|^2 u_*)(x,t)  \phi(x,t)  dx \,dt \,,
\end{align*}
for any $\phi \in C^\infty_c(\Omega \times [0,\infty[) $.
Finally $ \lim_{t\searrow 0} u_*(.,t) = u_0 $ in $H^{1,2}(\Omega;S^2)$
immediately follows from
$ E(u_*(t_0)) \leq E(u_0) $, since we have weak convergence
as $ t\searrow 0$.
\\
(iii): By assumption there is $R>0$, such that
$$
\sup_{t_0-R^2\leq t \leq t_0}   \int_{B_R(x_0)\cap\Omega}   \frac{1}{2}
|\nabla u_* (x,t)|^2   dx < \frac{\epsilon_0}{4} \,.
$$
Set $\delta := \min\{ R^2,\frac{\gamma_1 R^2\epsilon_0}{2 C E_0}\} $
and $ s_0 := t_0 -\frac{1}{2} \delta $.
We may assume we have for the same $R>0$
$ P_{2R}^\Omega(z_0)\setminus \{z_0\} \subset
\mathop{\rm Reg}(\{u_{\epsilon_i}\}) $.
Then Theorem \ref{eps-reg-LL} and \ref{eps-reg-LL-bd} and
Lemma \ref{Cinfty1} and \ref{Cinfty2} imply
$$
\lim_{i\to\infty} \int_{B_R(x_0)\cap\Omega}
g_{\epsilon_i}(u_{\epsilon_i}(x,s_0))   dx =
\int_{B_R(x_0)\cap\Omega}   \frac{1}{2}
|\nabla u_* (x,s_0)|^2   dx < \frac{\epsilon_0}{4}\,.
$$
Now by Lemma
\ref{en-est-LL} \eqref{loc-en}, we have
\begin{align*}
\sup_{s_0\leq t\leq s_0 +\delta}
\int_{B_{\frac{1}{2}R}(x_0)\cap \Omega}
g_{\epsilon_i} (u_{\epsilon_i}(x,t)) \, dx
&\leq \int_{B_{R}(x_0)\cap \Omega}  g_{\epsilon_i} (u_{\epsilon_i}(x,s_0)) \,dx
+ \frac{\delta C E_0}{\gamma_1 R^2} \\
&< \frac{\epsilon_0}{2} +  \frac{\epsilon_0}{2}\,,
\end{align*}
for $i$ sufficiently large. Since by construction
$t_0 \in ]s_0,s_0+\delta[$, the
claim follows.
\end{proof}

By Theorem \ref{limits}, Corollary \ref{prop-limits} (ii) and
uniqueness of smooth solutions, we obtain the following.

\begin{remark} \label{unique-Struwe} \rm
There is $T_0 >0$, such that
$$
\lim_{\epsilon \searrow 0} u_\epsilon = u_*
\quad \mbox{in } C^\infty( \Omega \times ]0,T_0[; S^2)  ,
$$
where $u_*$ is the unique smooth solution of
\eqref{LL} with initial and boundary data $u_0$.
(Compare \cite{Harpes1}.)
\end{remark}

If the energy of a (sub-)limit $u_*$ was everywhere decreasing,
A.Freire's uniqueness result  \cite{Freire1} would imply that
$u_*$ is (globally) the Struwe-solution. However all we can say about the
energy of sublimits $u_*$ is the following Lemma \ref{en-jump}. In particular
extension $u_*$ after the maximal smooth existence time $T_0$
with backward bubbling cannot be excluded. (See \cite{Harpes3}.)

\begin{lemm} \label{en-jump}
Let $  u_\epsilon $ be a solution
of \eqref{Eq_eps_LL}-\eqref{Eq_bdry_eps_LL} for fixed $ \epsilon > 0 $
and assume
$ u_* =
\mbox{weak-}H^{1,2}\mbox{-}\lim_ {i \to \infty} u_{\epsilon_i}$
for a sequence $0<\epsilon_i \searrow 0$.
If $ s < t $ and
$ \mathbb{S}^s(\{u_{\epsilon_i}\}) :=
(\overline{\Omega}\times \{s\}) \cap \mathbb{S}(\{ u_{\epsilon_i}\}_i )
= \emptyset $ and
$ \mathbb{S}^t(\{u_{\epsilon_i}\}) \neq \emptyset $,
then
$$
\int_\Omega  \frac{1}{2}  |\nabla u_*|^2 (x,s)  \, dx   \geq
\int_\Omega  \frac{1}{2}  |\nabla u_*|^2 (x,\tau) \, dx
\quad \forall \tau > s  \,,
$$
and
$$
\int_\Omega  \frac{1}{2}  |\nabla u_*|^2 (x,s)  \, dx   \geq
\int_\Omega  \frac{1}{2}  |\nabla u_*|^2 (x,t)  \, dx
+  \epsilon_0  \,,
$$
where $ \epsilon_0>0 $ is the constant from
Theorem \ref{eps-reg-LL-bd}.
\end{lemm}

\begin{proof}
Set $ \overline{x} := (x_1, \ldots , x_K) $ if
$ \mathbb{S}^t(\{u_{\epsilon_i}\}) = \{ x_1, \ldots , x_K \} $  and\\
$ B_R(\overline{x}) := \bigcup_{j=1}^K   B_R(x_j) $.
Then
\begin{align*}
E(u_*(s),\Omega ) &:=
\int_\Omega   \frac{1}{2} |\nabla u_* |^2 (x,s) \,  dx\\
&= \lim_{i \to \infty}   \int_{\Omega}
 g_{\epsilon_i}(u_{\epsilon_i})(x,s)\,  dx\\
&\geq \limsup_{i \to \infty}
\int_{\Omega} g_{\epsilon_i}(u_{\epsilon_i})(x,\tau)  \, dx
\quad \forall \tau > s
\quad \mbox{ (by Lemma \ref{en-est-LL})} \\
&\geq \int_\Omega   \frac{1}{2} |\nabla u_* |^2 (x,\tau) \,  dx
\quad \forall \tau > s
\quad \mbox{(by weak lower semi-continuity)}
\end{align*}
Also
$$
E(u_*(s),\Omega )
 \geq \limsup_{i \to \infty}
\Bigl(\int_{\Omega \smallsetminus B_R(\overline{x})}
g_{\epsilon_i}(u_{\epsilon_i})(x,\tau) \,  dx
 +\int_{B_R(\overline{x})}
g_{\epsilon_i}(u_{\epsilon_i})(x,\tau) \,  dx   \Bigr) ,
$$
for all $\tau \in ]s,t]$.
Now for any $\delta \in ]0,1[$ and $R>0$,
there are sequences $s<t_i \nearrow t$ and
$0< \delta_i \searrow 0$, such that
$$
\int_{B_R(\overline{x})}
g_{\epsilon_i}(u_{\epsilon_i})(x,t_i)   dx
= \sup_{\delta_i<\tau<t}
\int_{B_R(\overline{x})}
g_{\epsilon_i}(u_{\epsilon_i})(x,\tau)   dx
\geq \delta \epsilon_0
$$
and so
\begin{align*}
E(u_*(s))
&\geq \limsup_{i \to \infty}\int_{\Omega\smallsetminus B_R(\overline{x})}
g_{\epsilon_i}(u_{\epsilon_i})(x,t_i)   dx
 + \delta \epsilon_0\\
& \geq \int_{\Omega\smallsetminus B_R(\overline{x})}
  \frac{1}{2} |\nabla u_* |^2 (x,t)   dx   +  \delta \epsilon_0
\quad \forall R > 0\,,\;   \delta \in ]0,1[ \,.
\end{align*}
Since the last inequality holds for any $R>0$ and $ \delta \in ]0,1[$,
the claim follows.
\end{proof}

Theorem \ref{limits} provides an alternative version of
the construction of the ``Struwe-solution'' (see \cite{Harpes1}).

\begin{coro} \label{existenceLL}
Let $ u_0 \in H^{1,2}(\Omega;S^2)$.
Then there is a global distributional solution
$
u \in H^{1,2}_{\rm loc}(\overline{\Omega}\times ]0,\infty[;S^2) \cap
L^\infty (]0,\infty[; H^{1,2}(\Omega;S^2))
\mbox{ with }
\partial_t u \in L^2 (\Omega\times ] 0,\infty[;\mathbb{R}^3)
$
of \eqref{LL} with initial and boundary data $u_0$,
which is smooth on $\Omega\times ]0,\infty[$ except at finitely many points
and has decreasing and right continuous
energy.
If in addition $u_0 \in H^{3/2,2}(\partial\Omega;S^2)$, then
$u$ is unique among the
solutions of \eqref{LL} with initial and boundary data $u_0$
which are smooth except for isolated singular points and with
$\lim_{t\searrow s} E(u(t)) < E(u(s)) +\epsilon_0$
for all $ s \geq 0$.
(It is also unique among the $ H^{1,2}_{\rm loc}$-solutions with decreasing
energy by Freire's result.)
\end{coro}

\begin{proof}
By Theorem \ref{limits} the $\epsilon$-approximation scheme provides
a smooth short time solution
$$
u \in C^\infty \bigl( \Omega \times ]0,T_0[; S^2 \bigr)
$$
to \eqref{LL} with boundary data $u_0$ and
$
\lim_{t\searrow 0} u(\cdot,t) = u_0$ in $H^{1,2}(\Omega;\mathbb{R}^3)$.
Also there are $ \{ x_1, \ldots, x_K\} \subset \Omega $, such that
$$
\lim_{t \nearrow T_0} u(\cdot,t) =  u(\cdot,T_0) \quad  \mbox{in }
C^\infty \bigl(\Omega \smallsetminus \{ x_1, \ldots, x_K\}, \mathbb{R}^3\bigr)
$$
and
$$
\norm{\nabla u(\cdot,T_0)}^2_{L^2(\Omega)} \leq
\liminf_{t \nearrow T_0} \norm{\nabla u(\cdot,t)}^2_{L^2(\Omega)}\leq 2 E_0 \,.
$$
In particular $u(\cdot,T_0) \in H^{1,2}(\Omega)$.
If we now set $ \tilde{u}_0 := u(\cdot,T_0) $ and repeat
the same procedure with $ \tilde{u}_0$ instead of $u_0$, we obtain
step by step a global solution with point singularities.
To see that $ \partial_t u \in L^2 (\Omega \times \mathbb{R}_+;\mathbb{R}^3)$,
we sum up the energy inequalities of each time intervall
$]t_k, t_{k+1}[$ on which $u$ is regular
and use that the energy is right continuous, whereas
$$
\limsup_{t\nearrow t_{k+1}} E(u(t)) \geq
E(u(t_{k+1})) + \epsilon_0
$$
by Lemma \ref{en-jump}. This yields
$$
\int_0^\infty \int_\Omega   |\partial_t u|^2   dx dt
\leq E_0 - \sum_k \epsilon_0
$$
and also shows that there can only be finitely many
``singular times'' $t_k$.

Now assume we have two solutions $u_1$ and $u_2$
of \eqref{LL} with initial and boundary data $u_0$ and both with finitely many
point singularities and
$\lim_{t\searrow s} E(u(t)) < E(u(s)) +\epsilon_0$
for all $ s \geq 0$.
By Remark \ref{unique-Struwe},
we have $u_1 = u_2$ on $\Omega \times [0,T_1[$, where
$T_1$ is the maximal commun smooth existence time,
i.e.  either $u_1$ or $u_2$ has point singularities at $T_1$.
However by Corollary 4 on the existence of smooth extensions in \cite{Harpes1},
if $u_1$ admits a smooth extension up to $T_1$, then so does $u_2$
and conversely. Moreover, since the criterion for
the existence of a smooth extension is local, both solutions
have the same singularities $x_1,\ldots, x_K$ at time $T_1$
and $u_1(\cdot,T_1) = u_2(\cdot,T_1)$
on $\Omega \setminus \{x_1,\ldots, x_K\}$.
By Theorem 6 in \cite{Harpes1},
and the assumption on the energy, the extension of $u_1$ and $u_2$
after $T_1$ is again unique ``for a short time''
and an iteration of the previous argument leads to the claimed uniqueness.
\end{proof}


\subsection*{Acknowledgement}
The author gratefully acknowledges encouragement and support from Michael Struwe.

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\end{document}