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\markboth{A mixed semilinear parabolic problem}
{ C. Lederman, J. L. Vazquez, \&  N. Wolanski }
\begin{document}
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\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
USA-Chile Workshop on Nonlinear Analysis, \newline
Electron. J. Diff. Eqns., Conf. 06, 2001, pp. 203--214.\newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)}
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 A mixed semilinear parabolic problem from combustion theory
% 
\thanks{ {\em Mathematics Subject Classifications:}  35K20,  35K60, 80A25.
 \hfil\break\indent 
{\em Key words:} mixed parabolic problem, semilinear parabolic problem,  non-cylindrical
\hfil\break\indent
space-time domain, combustion.
\hfil\break\indent
\copyright 2001 Southwest Texas State University. 
\hfil\break\indent Published January 8, 2001. 
\hfil\break\indent
C.L. and N.W. were supported by UBA  grants TX47,
by  CONICET grant PIP0660/98,
\hfil\break\indent
 and by grant BID802/OC-AR PICT03-00000-00137.
\hfil\break\indent
 J.L.V. was supported by DGICYT Project PB94-0153, 
\hfil\break\indent
and  by HCM  contract CHRX-CT94-0618. } } 

\date{}
\author{ Claudia Lederman, Juan Luis Vazquez, \&  Noemi Wolanski }
\maketitle
\begin{abstract}
We prove existence, uniqueness, and regularity 
of the solution to a mixed initial boundary-value problem. The
equation is semilinear uniformly parabolic with principal part in 
divergence form, in a non-cylindrical space-time domain. 
Here we extend our results in \cite{LVWmix} to a more general domain. 
As in \cite{LVWmix}, we assume only mild regularity on the coefficients,
on the non-cylindrical part of the lateral boundary (where the Dirichlet 
data are given), and on the Dirichlet data.

This problem is of interest in combustion theory, where  
the non-cylindrical part of the lateral boundary may be considered
as an  approximation of a flame front. 
In particular, the results in this paper are used in \cite{LVWdf} to 
prove the uniqueness of a ``limit'' solution to the combustion problem 
in a two-phase situation. 
\end{abstract}

\newtheorem{thm}{Theorem}[section]
\newtheorem{prop}[thm]{Proposition}

\section{Introduction} 

In this paper we prove existence, uniqueness, and regularity 
of the solution to the mixed initial boundary-value problem 
$$\displaylines{
\sum_{\textstyle{i,j}}\frac\partial{\partial x_i}\Bigl(a_{ij}\frac{\partial u}
{\partial x_j}\Bigr) + \sum_{\textstyle{i}} b_i\frac{\partial u}
{\partial x_i} + c\,u-u_t=\beta(x,t,u)\quad\mbox{in }{\cal{R}}\cr
\sum_{\textstyle {i,j}} a_{ij}\frac{\partial u}{\partial x_j}
\eta_i=0\quad\mbox{on }\partial_N{\cal{R}}\cr
u=\phi\quad\mbox{on }\partial_D{\cal{R}},
}$$
where ${\cal{R}}\subset{{\mathbb R}}^N\times(0,T)$ is a bounded non-cylindrical space-time domain,
 $\partial_N{\cal{R}}$ is an open subset of the parabolic
boundary, $\partial_p{\cal{R}}$, and $\partial_D{\cal{R}}=\partial_p{\cal{R}}\setminus
\partial_N{\cal{R}}$. 
This is a semilinear uniformly parabolic equation with 
principal part in divergence form, in a non-cylindrical space-time 
domain. 
We look for a weak solution $u\in C(\overline{\cal{R}})$, with 
$\nabla u\in C(\overline{\cal{R}})$. Here we extend our 
results in \cite{LVWmix} to a more general domain.

The non-cylindrical part of $\partial_p{\cal{R}}$ is $\partial_D{\cal{R}}\cap
\{t>0\}$. As in \cite{LVWmix}, we assume only mild regularity on the 
coefficients, and on $\partial_D{\cal{R}}\cap\{t>0\}$.
We also assume a minimum smoothness on the Dirichlet datum $\phi$.

This problem is of interest in combustion theory. In that situation, the
non-cylindrical part of the lateral boundary may be considered as an approximation
of a flame front. The second order part of the equation is the Laplace operator.
In particular, the results in this paper are used in \cite{LVWdf} to prove 
the uniqueness of a ``limit'' solution to the combustion problem in a two
phase situation. We point 
out that in \cite{LVWuf} --where we proved 
the uniqueness of a ``limit'' solution to the combustion problem in a one
phase situation-- both
the results in \cite{LVWmix} and in the present paper can be applied. 
However, in \cite{LVWdf}  --which is  a two phase situation-- the results in 
\cite{LVWmix} do not apply and we need to use the more general results we are 
presenting here.

In the combustion context of \cite{LVWuf} and \cite{LVWdf}  the initial datum $\phi(x,0)$ 
is only globally H\"older continuous with H\"older 
continuous spatial gradient near  the initial flame front 
$\overline{\partial_D{\cal{R}}\cap\{t>0\}}\cap\{t=0\}$.
The solution $u$ must satisfy that $\nabla u\in C(\overline
{\cal{R}}\cap\{t>0\})$ and $\nabla u$ must be continuous up to time $t=0$ near the flame
front $\overline{\partial_D{\cal{R}}\cap\{t>0\}}$. With that regularity of the datum, standard
Schauder or Sobolev type results cannot be applied, even if we had a cylindrical
space-time domain or $\partial_N{\cal{R}}=\emptyset$.  
In order to get our results, both in \cite{LVWmix} and here, we reduce the
problem posed in a non-cylindrical  space-time domain to a similar problem
in a domain which is a space-time cylinder. Once this is done, the
main point is the proof of the regularity of $\nabla u$ up to 
the boundary with mild regularity assumptions on the data.

We point out that we prove the existence of a weak solution 
$u\in C^{\gamma, \frac{\gamma}{2}}(\overline{\cal{R}})$ with 
$\nabla u\in C({\cal{R}}\cup \partial_N{\cal{R}})$ assuming that $\phi$ 
is only H\"older continuous. Further continuity of $\nabla u$ is obtained
in every neighborhood of a point in $\partial_D {\cal{R}}$ where $\phi$ is 
smooth enough.

We remark that there is a vast body of literature on 
mixed boundary-value problems for parabolic equations 
(see, for instance, \cite{BFO1,BFO2,BR,L,T}). However, 
the results we present here, and those in \cite{LVWmix}, 
cannot be derived from those papers.

The paper is organized as follows: In Section 2 we introduce the
notation and hypotheses to be used throughout the paper and, 
in particular, we define the non-cylindrical space-time domain we are 
going to work with. As a
previous step to the study of the mixed semilinear parabolic problem, we 
prove in Section 3 results on existence, uniqueness and regularity, as well as 
a priori estimates, for the corresponding linear problem. Section 4 is devoted
to the proof of the main result in this paper, i.e. Theorem \ref{theo1.1},
which is an existence, uniqueness and regularity result for the mixed semilinear
problem. Finally, we show in Section 5 how the results in this paper are
used to prove the uniqueness of a ``limit'' solution to the combustion problem.

\section{Notation and hypotheses}

Throughout this paper the spatial dimension is denoted by $N$,
and the following notation is used:

The symbol  $\nabla$ will denote the corresponding
operator in the space variables; the symbol ${\partial}_p$ applied
to a domain will denote parabolic boundary.

For an integer $m\ge 0$, $0<\alpha<1$, and a space-time cylinder 
$Q=\Omega\times (0,T)\subset{\mathbb R}^{N+1}$, $C^{m+\alpha,\frac{m+\alpha}2}(Q)$ 
will denote the parabolic H\"older space which is denoted by
$H^{m+\alpha,\frac{m+\alpha}2}(Q)$ in \cite{LSU}. 

For ${{\cal D}}\subset{\mathbb R}^{N+1}$ a general domain,
$C^{m+\alpha,\frac{m+\alpha}2}({{\cal D}})$ will denote  the space of functions in 
$C^{m+\alpha,\frac{m+\alpha}2}(Q)$ for every space-time cylinder $Q\subset{{\cal D}}$.

For ${{\cal D}}$ bounded, we will say that 
$u\in C^{m+\alpha,\frac{m+\alpha}2}(\overline{{\cal D}})$ if there exists a domain 
${{\cal D}} '$ with $\overline{{\cal D}}\subset{{\cal D}} '$ and a function 
$u'\in C^{m+\alpha,\frac{m+\alpha}2}({{\cal D}}')$   such that $u=u'$ in
$\overline{{\cal D}}$. And we will denote by 
$C^{{\rm dini}}(\overline{{{\cal D}}})$ 
the set of functions which are continuous
in $\overline{{{\cal D}}}$ and such that their modulus of continuity $\omega(r)$  
with respect to the parabolic norm $\|(x,t)\|=|x|+|t|^{\frac12}$  
satisfy the Dini condition
$$
\int_0^1\frac{\omega(r)}r\,dr<\infty.
$$

Throughout the paper we will let $\Omega={\mathbb R}\times\Sigma$ and 
$\Sigma\subset {\mathbb R}^{N-1}$ be a bounded Lipschitz domain with interior unit 
normal $\eta'$. We will denote 
by $\eta=(0,\eta')$ the interior unit normal to $\partial\Omega$.
 We will denote points
in $\overline\Omega$ by $x=(x_1,x')$ with $x_1\in{\mathbb R}$ and 
$x'\in\overline\Sigma$. 

On the other hand $p , q$ will be Lipschitz continuous functions in
$\overline\Sigma\times[0,T]$, and we will denote
$$
{{\cal D}}:=\{(x,t)\in\Omega\times(0,T)\,/\,p(x',t)<x_1<q(x',t)\}.
$$ 
We will assume,  in addition, that there exists a constant
$\mu_0>0$ such that $q(x',t)-p(x',t)\ge\mu_0$ in 
$\overline{\Sigma}\times[0,T]$.
 
We define, as usual, $\partial_p{{\cal D}}:=
\overline{\partial{{\cal D}}\setminus\{t=T\}}$ and let
\begin{eqnarray*}
&\partial_N{{\cal D}}:=\{(x,t)\in\partial_p{{\cal D}}\,/\, x'\in\partial\Sigma,
\,0<t\le T,\,p(x',t)<x_1<q(x',t)\},\\ 
&\partial_D{{\cal D}}:=\partial_p{{\cal D}}\setminus\partial_N{{\cal D}},\\
&\partial_S{{\cal D}}:=\{(x,t)\in\partial_p{{\cal D}}\, /\, 
x_1=p(x',t)\,\mbox{ or }\, x_1=q(x',t)\}. 
\end{eqnarray*}

For $R_0<\mu_0$, we define
$$
{{{\cal D}}_{R_0}}:=\{(x,t)\in\Omega\times(0,T)\,/\,p(x',t)<x_1<p(x',t)+R_0\}.
$$ 

\section{The linear problem}

We will prove in this paper an existence, uniqueness, and regularity result,  as well as a-priori estimates,
for a mixed initial-boundary value problem associated to a uniformly parabolic 
equation with principal part in divergence form, with
a nonlinear forcing term. This will be done on the non-cylindrical space-time domain ${\cal D}$ defined in the previous 
section. 
To do so, we devote this section to the study of the corresponding
linear problem.
In Proposition \ref{prop1.1} we prove existence and uniqueness for the linear problem,
with a minimum smoothness of the data. In Proposition \ref{prop1.2} we prove a 
regularity result for the linear problem. 
We first prove the following existence and uniqueness result for the linear 
problem:

\begin{prop} \label{prop1.1} Let ${{\cal D}},\, \partial_N{{\cal D}},\, \partial_D{{\cal D}}$ 
and $\partial_S{{\cal D}}$ be as above. For $i,j=1,\cdots,N$, let $a_{ij},\, b_i,\, 
c,\, g \in L^\infty({{\cal D}})$. Assume that 
$a_{ij}(x,t)\xi_i\xi_j$ $\ge\lambda|\xi|^2$ for some
$\lambda>0$ and every $\xi\in {\mathbb R}^N$, $(x,t)\in{{\cal D}}$. Let $\phi\in 
C^{\alpha,\frac\alpha2}(\overline{{\cal D}})$. 
Then there exists a unique function $u\in C(\overline{{\cal D}})$, with 
$\nabla u\in L^2_{{\rm loc}}(\overline{{\cal D}}\setminus\partial_S{{{\cal D}}})$, such that $u$ is a weak solution
to the following mixed initial boundary-value problem
\begin{eqnarray}  
&\sum_{{i,j}}\frac\partial{\partial x_i}\Bigl(a_{ij}\frac{\partial u}
{\partial x_j}\Bigr) + \sum_{{i}} b_i\frac{\partial u}
{\partial x_i} + c\,u-u_t=g\quad\mbox{in }{{\cal D}} &\label{1.1a} \\
&\sum_{ {i,j}} a_{ij}\frac{\partial u}{\partial x_j}
\eta_i=0\quad\mbox{on }\partial_N{{\cal D}} &\label{1.1b}\\
&u=\phi\quad\mbox{on }\partial_D{{\cal D}}. &\label{1.1c}
\end{eqnarray}
Moreover, there exist $0<\gamma\le\alpha$ and $C>0$, depending only on $\alpha$, $T$, $\lambda$, 
$\|a_{ij}\|_{L^\infty({{\cal D}})}$, $\|b_i\|_{L^\infty({{\cal D}})}$, 
$\|c\|_{L^\infty({{\cal D}})}$,
$\|\phi\|_{C^{\alpha,\frac\alpha2}(\partial_D{{\cal D}})}$,  
$\|g\|_{L^\infty({{\cal D}})}$, the domain
$\Sigma$ and the functions $p$ and $q$, such that 
$u\in C^{\gamma,\frac{\gamma}2}(\overline{{\cal D}})$ and
\begin{eqnarray}
\|u\|_{C^{\gamma,\frac{\gamma}2}(\overline{{\cal D}})}\le C.
\label{1.2}
\end{eqnarray}
Now let  $\psi_1(x',t)=\phi(p(x',t),x',t)$,\, $\psi_2(x',t)=\phi(q(x',t),x',t)$
 and assume, in addition, that ${\psi_i}_t\in L^2(\Sigma\times(0,T))$, $\nabla_{x'}\psi_i\in 
L^2(\Sigma\times(0,T))$ for $i=1,2$. Then $\nabla u\in L^2({{\cal D}})$.
\end{prop}

\noindent{\bf Proof:} Let $\psi_1,\, \psi_2$ be as in the statement. 
We will first prove the 
proposition with the extra assumption that 
${\psi_i}_t\in L^2(\Sigma\times(0,T))$, $\nabla_{x'}\psi_i\in 
L^2(\Sigma\times(0,T))$ for $i=1,2$. 


We straighten up both lateral boundaries by taking a new coordinate system. 
In fact, we let $y=H(x,t)$ be defined by
\begin{eqnarray}
&y_1={\displaystyle \frac{x_1-p(x',t)}{q(x',t)-p(x',t)}}\,,& \label{7.3a}\\
&y_i=x_i\quad\mbox{for }i>1. &\label{7.3b}
\end{eqnarray}
Then, for $(y,t)\in Q:=(0,1)\times\Sigma\times(0,T)$, we let
$\overline u(y,t)=u(x,t)$. Then, $u\in C(\overline{{\cal D}})$, with
$\nabla u\in L^2({{\cal D}})$, is a weak solution to (\ref{1.1a})--(\ref{1.1c}) 
if and only
if $\overline u\in C(\overline Q)$, with $\nabla \overline u\in L^2(Q)$, is a weak solution
to
\begin{eqnarray}
&\overline{\cal L}\overline u:=\sum_{{i,j}}\frac{\partial}{\partial y_i}
\Bigl(\bar a_{ij}
\frac{\partial\overline u}{\partial y_j}\Bigr)+\sum_{{i}}\bar b_i
\frac{\partial \overline u}{\partial y_i} +\bar c\,\overline u -\overline u_t=\bar g
\quad\mbox{in }Q, &\label{7.4a}\\
&\sum_{{i,j}}\bar a_{ij}\frac{\partial\overline u}{\partial y_j}\eta_i=0\quad\mbox{on }
\partial_N Q:=(0,1)\times\partial\Sigma\times(0,T],&\label{7.4b}\\
&\bar u=\bar \phi\quad\mbox{on }\partial_D Q:=\partial_p
 Q\setminus\partial _N Q, &\label{7.4c}
\end{eqnarray}
where $\overline g(y,t)=g(x,t)$, $\overline\phi(y,t)=\phi(x,t)$, $\bar c(y,t)=c(x,t)$,
\begin{eqnarray*}
{\bar a}_{ij}(y,t)&=&\sum_{\textstyle{k,l}} a_{kl}(x,t)\frac
{\partial H_i}
{\partial x_k}(x)\frac{\partial H_j}{\partial x_l}(x),\\ 
\bar b_i(y,t)&=&\sum_{\textstyle{j}} b_j(x,t)\frac{\partial H_i}{\partial x_j}(x,t)-
\frac{\partial H_1}{\partial t}(x,t)\delta_{1i}+\sum_{\textstyle{j}} \bar a_{ji}(y,t)\frac
{q_{x_j}-p_{x_j}}{q-p}(x',t).
\end{eqnarray*}

Note that the equation for $\overline u$ has bounded coefficients and right hand side.
On the other hand, it is uniformly parabolic (with parabolicity constant depending
only on $\lambda$  and the functions $p$ and $q$).

The existence and uniqueness of a function  
$\overline u\in C\left([0,T];L^2((0,1)\times\Sigma)\right)$ with 
$\nabla \overline u\in L^2(Q)$, which is a weak solution to (\ref{7.4a})--(\ref{7.4c}), can be obtained, for
instance, from Theorem X.9 in \cite{B}, by proceeding as in Proposition 1.1 in
\cite{LVWmix}.

Let us prove that there exist $0<\gamma'\le\alpha$
and $C>0$  depending only on the $L^\infty$ norm of the coefficients
of the equation in (\ref{7.4a})--(\ref{7.4c}),  the constants $\alpha$, 
$\lambda$, $T$, the domain $\Sigma$, $\|\overline\phi\|_{C^{\alpha,\frac\alpha2}
(\partial_D Q)}$, $\|\overline g\|_{L^\infty(Q)}$, $\|\overline u\|_{L^\infty(Q)}$
and the functions $p$ and $q$, such that
\begin{equation}
\|\overline u\|_{C^{\gamma',\frac{\gamma'}2}(\overline Q)}\le C.\label{holderQ}
\end{equation}

To prove (\ref{holderQ}) we first take 
the set $Q_{\delta}:=(0,1)\times\Sigma_\delta\times(0,T)$
where $\Sigma_\delta:=\{y'\in \Sigma\,/
\mathop{\rm dist}(y',\partial\Sigma)>\delta\}$
and $\delta>0$ is small to be fixed later. 
Then we can get estimate $(\ref{holderQ})$ in 
$\overline {Q_\delta}$
by applying Theorem 10.1, Chap. III in \cite{LSU}.

Next, let $(y_0,t_0)\in[0,1]\times\partial\Sigma\times[0,T]$.  
We will straighten up $\partial_N Q$.
For that purpose we denote $y_0=(y_{0_1}, y_0')$, with 
$y_{0_1} \in[0,1]$ and $y_0'\in\partial\Sigma$ and we take 
${\cal O}\subset{\mathbb R}^{N-1}$ a neighborhood 
 of $y_0'$  such that $[0,1]\times(\partial\Sigma\cap{\cal O})$ 
is parameterized in the variables $(z_1,\cdots,z_{N-1})$ by
\begin{eqnarray}
&y_1=z_1, \quad 0\le z_1\le 1,& \label{param1}\\
&y'=\sigma'(z_2,\cdots,z_{N-1}), \quad (z_2,\cdots,z_{N-1})
\in{\cal N}\subset{\mathbb R}^{N-2}. &\label{param2}
\end{eqnarray}
Here ${\cal N}$ is the ball  in ${\mathbb R}^{N-2}$ with center in the origin and radius $r=1$.
Since $\Sigma$ is a Lipschitz domain we may assume that, in the neighborhood
${\cal O}$, $\sigma'$ is the graph of a Lipschitz function $G$ in the direction
$y_N$ and that
every point $y\in[0,1]\times{\cal O}$  can be written in a unique way as
$y=h^{-1}(z)$, where
\begin{eqnarray}
&y_1=z_1 & \nonumber\\
&y_i=z_i ,\quad 2\le i \le N-1,& \label{h^-1(z)}\\
&y_N=z_N + G(z_2,\dots, z_{N-1}), &\nonumber 
\end{eqnarray}
with $z\in [0,1]\times{\cal N}\times\{|z_N|<2\delta\}$, for some 
$\delta>0$, and $h$ a Lipschitz invertible function with 
non-vanishing Jacobian in  $[0,1]\times\overline{\cal O}$
and $h([0,1]\times{\cal O})=[0,1]\times{\cal N}\times\{|z_N|<2\delta\}$ 
and  $h([0,1]\times({\cal O}\cap\Sigma))= [0,1]
\times{\cal N}\times\{0<z_N<2\delta\}$.

Here $\delta$ can be chosen independent of $(y_0,t_0)$,  and will remain
fixed from now on. 
Let $\overline{\overline u}(z,t):=\overline u(y,t)$ for $z_N\ge0$ and
$\tilde Q_+:=\{(z,t)\in\tilde Q\,/\,z_N>0\}$, where 
$$
\tilde Q:=(0,1)\times{\cal N}\times\{|z_N|<2\delta\}\times(0,T).
$$ 
Then 
$\overline{\overline u}\in C\left([0,T];L^2((0,1)\times{\cal N}\times\{0<z_N<2\delta\})
\right)$, with $\nabla\overline{\overline u}\in L^2(\tilde Q_+)$, is a weak solution  
in $\tilde Q_+$ to 
\begin{equation}
\overline{\overline{\cal L}}\,\overline{\overline u}:=\sum_{\textstyle{i,j}}
\frac\partial{\partial z_i}\left(\bar{\bar a}_{ij}
\frac{\partial \overline{\overline u}}{\partial z_j}\right) +\sum_{\textstyle{j}} 
\bar{\bar b}_j\frac{\partial \overline{\overline u}}{\partial z_j}+ \bar{\bar c}\overline{\overline u}
-\overline{\overline u}_t=\bar{\bar g}
\label{1.7}
\end{equation}
which is a uniformly parabolic equation with principal part in divergence form with 
bounded coefficients and free term. Here
$$
\bar{\bar a}_{ij}(z,t)=\sum_{\textstyle{k,l}}\bar a_{kl}(y,t)\frac
{\partial h_i}
{\partial y_k}(y)\frac{\partial h_j}{\partial y_l}(y). 
$$

We extend $\overline{\overline u}$ to $\{z_N<0\}$ by reflection. This is,
we define for $z_N<0$
$$
\overline{\overline u}(z,t)=\overline{\overline u}(z_1,z_2,\cdots,-z_N,t).
$$
In this way, 
$\overline{\overline u}\in C\left([0,T];L^2((0,1)\times{\cal N}\times\{|z_N|<2\delta\})\right)$, 
with $\nabla\overline{\overline u}\in L^2(\tilde Q)$, becomes a weak solution in the domain 
$\tilde Q$ of equation (\ref{1.7}), where we have, for $z_N<0$,
$$
\bar{\bar a}_{ij}(z,t)=
\left\{
\begin{array}{ll}
\bar{\bar a}_{ij}(z_1,\cdots,z_{N-1},-z_N,t) &\mbox{if }i<N,
j<N, \mbox{  or }\ i=j=N,\\
-\bar{\bar a}_{ij}(z_1,\cdots,z_{N-1},-z_N,t) &\mbox{if }i=N,
j<N,  \mbox{  or }\ i<N,j=N,
\end{array}
\right.
$$
$$
\bar{\bar b}_{j}(z,t)=
\left\{
\begin{array}{ll}
\bar{\bar b}_{j}(z_1,\cdots,z_{N-1},-z_N,t) &\mbox{if }j<N,\\
-\bar{\bar b}_{j}(z_1,\cdots,z_{N-1},-z_N,t) &\mbox{if }j=N,
\end{array}
\right.
$$
and
$$
\bar{\bar c}(z,t)=\bar{\bar c}(z_1,\cdots,z_{N-1},-z_N,t),\quad 
\bar{\bar g}(z,t)=\bar{\bar g}(z_1,\cdots,z_{N-1},-z_N,t).
$$
Thus, $\overline{\overline u}$ is a weak solution in  $\tilde Q$ of a uniformly parabolic 
equation with principal part in divergence form with bounded coefficients 
and free term. 

We apply again Thm. 10.1, Chap. III in \cite{LSU} to conclude that there 
exist $0<\gamma'\le\alpha$ and $C>0$ such that
$$
\|\overline{\overline u}\|_{C^{\gamma',\frac{\gamma'}2}(\overline{\tilde Q}_{\frac12})}\le C.
$$
Here $\tilde Q_{\frac12}=
(0,1)\times{\cal N}_{\frac12}\times\{|z_N|<\delta\}\times(0,T)$, where
${\cal N}_{\frac12}$ is the ball  in ${\mathbb R}^{N-2}$ with center in the origin and 
radius $r=1/2$.

The constants $\gamma'$ and $C$ 
depend only on $\alpha$, $\lambda$, $T$, $\Sigma$, 
the functions $p$ and $q$, 
the $L^\infty$ norm of the coefficients of the equation in 
(\ref{1.7}), and free term in $\tilde Q$, 
$\|\bar{\bar u}\|_{L^\infty(\tilde Q)}$,
$\|\overline{\overline\phi}\|_{C^{\alpha,\frac\alpha2}(\overline
{\tilde Q}_+\cap(\{t=0\}\cup \{z_1=0\}\cup \{z_1=1\}))}$. Here 
$\overline{\overline\phi}(z,t):=\overline\phi(y,t)$
for $(y,t)\in\partial_D Q$.
Therefore (\ref{holderQ}) holds.

Since $\|u\|_{L^\infty({{\cal D}})}$
is bounded by a constant depending only on $T$,
$\|\phi\|_{L^\infty(\partial_D{{\cal D}})}$,  $\|c\|_{L^\infty({{\cal D}})}$ and 
$\|g\|_{L^\infty({{\cal D}})}$, 
we  conclude that (\ref{1.2}) holds.

Finally, the proof of the results in the statement without the extra assumption  that
${\psi_i}_t\in L^2(\Sigma\times(0,T))$, 
$\nabla_{x'}\psi_i\in  L^2(\Sigma\times(0,T))$ for $i=1,2$, as well as the  
proof of the uniqueness of solution follow as in Proposition 1.1 in \cite{LVWmix}. 


We next prove a regularity result for the linear problem.

\begin{prop}\label{prop1.2} Let ${{\cal D}},\, \partial_N{{\cal D}},\, 
\partial_D{{\cal D}},\, a_{ij},\, b_i,\, c,\, g,\, \phi,\, \psi_i$ as in Proposition 
\ref{prop1.1}. 
Let $u\in C^{\gamma,\frac\gamma 2}(\overline{{\cal D}})$, with 
$\nabla u\in L^2({{\cal D}})$, be the unique weak solution to 
(\ref{1.1a})--(\ref{1.1c}).
Now assume that $\Sigma\in C^3$, $p, q\in C^1(\overline\Sigma\times[0,T])$,
$\nabla_{x'}p,\nabla_{x'}q\in C^{{\rm dini}}(\overline\Sigma\times[0,T])$,
and that $\nabla_{x'} p(x',t)\cdot\eta'=0$ and 
$\nabla_{x'} q(x',t)\cdot\eta'=0$  on $\partial\Sigma\times(0,T)$.
Assume also that $a_{ij}\in C^{{\rm dini}}(\overline{{{\cal D}}})$ and 
$a_{ij}=\delta_{ij}$ on $\partial_N{{\cal D}}$. 
Then, $\nabla u\in  C({{\cal D}}\cup\partial_N{{\cal D}})$. 

If, in addition, $\psi_1(x',t)\in C^1(\overline\Sigma\times
(0,T])$, with $\nabla_{x'} \psi_1\in C^{{\rm dini}}(\overline\Sigma\times(0,T])$
and $\frac{\partial\psi_1}{\partial\eta'}=0$ on $\partial\Sigma \times(0,T)$, 
there holds that $\nabla u$ is continuous in $\overline{{\cal D}}\cap\{
x_1<q(x',t)\}\cap\{t>0\}$.

If, moreover, $\psi_1(x',t)\in C^1(\overline\Sigma\times
[0,T])$, with $\nabla_{x'} \psi_1\in C^{{\rm dini}}(\overline\Sigma\times[0,T])$, 
and $\nabla\phi\in C^{{\rm dini}}(\overline{{\cal D}}_{R_0}\cap\{t=0\})$, with $\frac{\partial \phi}
{\partial\eta}=0$ on $\overline{\partial_N{{\cal D}}_{R_0}}\cap\{t=0\}$, there
holds that $\nabla u\in C(\overline{{\cal D}}_{R_0/2})$ and
there exist a constant $C>0$ and an increasing function $\omega(r)$,
with $\omega(0^+)=0$, 
such that
\begin{eqnarray}
&\|\nabla u\|_{L^\infty({{\cal D}}_{R_0/2})}\le C,&
\label{1.5a}\\ 
&|\nabla u(x,t)-\nabla u(y,s)|\le \omega(|x-y|+|t-s|^{1/2}),\
(x,t),\,(y,s)\in\overline{{\cal D}}_{R_0/2}.&\label{1.5b}
\end{eqnarray}
With the same regularity of $\psi_1$ and no regularity assumptions on 
$\phi(x,0)$, for every $\tau>0$, (\ref{1.5a})--(\ref{1.5b}) holds in 
${{\cal D}}_{R_0/2}\cap\{t\ge\tau\}$ with
$C$ and $\omega$ independent of $\phi(x,0)$ but depending on $\tau$.

Analogously, if $\psi_2(x',t)\in C^1(\overline\Sigma\times
(0,T])$, with $\nabla_{x'} \psi_2\in C^{{\rm dini}}(\overline\Sigma\times(0,T])$ 
and $\frac{\partial\psi_2}{\partial\eta'}=0$ on $\partial\Sigma
\times(0,T)$, and  with no regularity assumptions on $\psi_1$ and on $\phi(x,0)$,
there holds that $\nabla u$ is continuous in $\overline{{\cal D}}\cap\{
x_1>p(x',t)\}\cap\{t>0\}$.

Also, if $\psi_i\in C^1(\overline\Sigma\times
[0,T])$, with $\nabla_{x'} \psi_i\in C^{{\rm dini}}(\overline\Sigma\times[0,T])$,  
and $\frac{\partial\psi_i}{\partial\eta'}=0$ on $\partial\Sigma
\times(0,T)$ for $i=1,2$
and $\nabla\phi\in C^{{\rm dini}}(\overline{{{\cal D}}}\cap\{t=0\})$ with $\frac{\partial\phi}{\partial
\eta}=0$ on $\overline{\partial_N{{\cal D}}}\cap\{t=0\}$, there holds that $\nabla u\in
C(\overline{{\cal D}})$.

If $a_{ij}\in C^{1+\mu,\frac{1+\mu}2}({{\cal D}})$, $b_i,\, c,\,g\in
C^{\mu,\frac\mu2}({{\cal D}})$, $u$ is a classical 
solution in the
sense that $u\in C^{2+\mu,1+\frac{\mu}2}({{\cal D}})$.
\end{prop}

\noindent{\bf Proof:} 
In this proof we use the same notation as in the proof of 
Proposition \ref{prop1.1}.
Since we have assumed that $\Sigma$ is a $C^3$ domain, we may take as $\sigma'$ 
in (\ref{param1})-(\ref{param2}) a
$C^{3}$ regular parameterization.
Also $\eta'$, the interior unit normal to $\Sigma$, is a $C^{2}$ function 
of the point $y'\in\partial\Sigma$. Then, instead of taking $y=h^{-1}(z)$ as 
in (\ref{h^-1(z)}), we take  $y=h^{-1}(z)$ in the following way:
\begin{eqnarray}
&y_1=z_1 & \label{h^-1(z)b1} \\
&y'=\sigma'(z_2,\cdots,z_{N-1})+\eta'(\sigma'(z_2,\cdots,z_{N-1}))\,z_N, &
\label{h^-1(z)b2}
\end{eqnarray}
so now $h$ is, in addition, a $C^2$ function.

Let us now assume that $a_{ij}\in C^{{\rm dini}}(\overline{{{\cal D}}})$ and 
$a_{ij}=\delta_{ij}$ on $\partial_N{{\cal D}}$. 
In order to prove that 
$\nabla u\in C({{\cal D}}\cup\partial_N{{\cal D}})$ we consider a point $(y_0,t_0)\in
(0,1)\times\partial\Sigma\times[0,T]$ and the corresponding function $\overline{\overline u}
(z,t)$ which is defined and continuous in $\overline{\tilde Q}$ with $\nabla\overline{\overline u}\in
L^2(\tilde Q)$. Also, $\overline{\overline u}$ is a weak solution to (\ref{1.7}) in $\tilde Q$.

Let us see that the principal coefficients in (\ref{1.7}), $\bar{\bar a}_{ij}$, belong
to $C^{{\rm dini}}(\overline{\tilde Q})$. In fact, using that 
$\nabla_{x'}p,\nabla_{x'}q\in C^{{\rm dini}}(\overline\Sigma\times[0,T])$, we get that 
$\bar{\bar a}_{ij}$ are Dini continuous
in $\{z_N\ge 0\}\cup\{z_N\le 0\}$. Then, we only need to 
verify that $\bar{\bar a}_{iN}(z_1,\cdots,z_{N-1},0,t)=0$ for $i<N$. 
We observe that, for $y'\in\partial\Sigma$, 
$$\displaylines{
{\bar a}_{ij}(y,t)=\delta_{ij} \quad i,j>1,\cr
{\bar a}_{1j}(y,t)=\frac{p_{y_j}(y_1-1)-q_{y_j}y_1}{q-p} (y',t) \quad j>1.
}$$
Therefore,
\begin{eqnarray*}
\bar{\bar a}_{iN}(z_1,\cdots,z_{N-1},0,t)
&=&\nabla h_i\cdot\nabla h_N +
\frac{\partial h_N}{\partial y_1}\left(- \frac{\partial h_i}{\partial y_1} + 
\sum_{\textstyle{k\ge 1}} {\bar a}_{k1}\frac{\partial h_i}
{\partial y_k}\right)\\
&&+\frac{\partial h_i}{\partial y_1}\left(\sum_
{\textstyle{k>1}}\left(\frac{p_{y_k}(y_1-1)-q_{y_k}y_1}{q-p} \right)\frac{\partial h_N}{\partial y_k}\right).
\end{eqnarray*}
  From the fact that $h\bigl(z_1,\sigma'(z_2,\cdots,z_{N-1})+\eta'(\sigma'
(z_2,\cdots,z_{N-1}))z_N\bigr)=z$ we deduce that,
on $(0,1)\times\partial\Sigma$, $\nabla h_i$ is tangent to $(0,1)\times\partial\Sigma$
for $i<N$ and $\nabla h_N=\eta$. Therefore, $\nabla h_i\cdot\nabla h_N=0$
on $(0,1)\times\partial\Sigma$ for $i<N$.
Since $\eta_1=0$, there holds that $\frac{\partial h_N}{\partial y_1}=0$. 
Finally, we use the fact that $\nabla_{y'}p\cdot\eta'=0$ and 
$\nabla_{y'}q\cdot\eta'=0$ on $\partial\Sigma
\times(0,T)$ and conclude that we have $\bar{\bar a}_{iN}=0$ on $\{z_N=0\}$
for $i<N$.
We can now apply Theorem 1.3.1  in \cite{CK} in $\tilde Q$ to deduce that
$\nabla \overline{\overline u}$ is continuous in 
$(0,1)\times{\cal N}\times\{|z_N|<2\delta\}\times(0,T]$.


On the other hand, a direct application of Theorem 1.3.1 in \cite{CK} to $u$ gives
the continuity of $\nabla u$ in ${{\cal D}}$.

The rest of the proof follows as that of Proposition 1.2 in \cite{LVWmix}.
Namely, under further assumptions on the Dirichlet data  $\psi_i$ and/or $\phi$ 
we obtain the continuity of $\nabla u$ up to the corresponding subset of  
the Dirichlet boundary by suitably applying the results in \cite{CK}. 

 From classical Schauder estimates we deduce that, when 
$a_{ij}\in C^{1+\mu,\frac{1+\mu}2}({{\cal D}})$ and
$b_i, c, g\in  C^{\mu,\frac{\mu}2}({{\cal D}})$, there holds that
$u\in  C^{2+\mu,1+\frac{\mu}2}({{\cal D}})$.
This completes the present proof. 

\section{The semilinear problem}

In this section we prove the main result in the paper,   
Theorem \ref{theo1.1}, which is an existence, uniqueness and regularity 
result for the mixed semilinear problem.

\begin{thm} \label{theo1.1} Let ${{\cal D}},\, \partial_N{{\cal D}},\,\partial_D{{\cal D}},\,
 \partial_S{{\cal D}},\, a_{ij},\, b_i,\, c$ and $\phi$ as in Proposition 
 \ref{prop1.1}.
 Let $\beta(x,t,u)\in L^\infty({{\cal D}}\times{\mathbb R})$ be
such that $\beta(x,t,\cdot)$ is locally Lipschitz continuous in ${\mathbb R}$ uniformly
for $(x,t)\in {{\cal D}}$.
There exists 
a unique function $u\in C^{\gamma,\frac\gamma 2}(\overline{{\cal D}})$ for some 
$0<\gamma\le\alpha$, with 
$\nabla u\in L^2_{{\rm loc}}(\overline{{\cal D}}\setminus\partial_S{{\cal D}})$, 
such that $u$ is a weak solution to the  following problem
\begin{eqnarray}
&\sum_{{i,j}}\frac\partial{\partial x_i}\Bigl(a_{ij}
\frac{\partial u}{\partial x_j}\Bigr) + \sum_{{i}} b_i
\frac{\partial u}{\partial x_i} + c\,u - u_t
=\beta(x,t,u)\quad\mbox{in }{{\cal D}} &\label{1.8a}\\
&\sum_{ {i,j}} a_{ij}\frac{\partial u}{\partial x_j}
\eta_i=0\quad\mbox{on }\partial_N{{\cal D}} &\label{1.8b}\\
&u=\phi\quad\mbox{on }\partial_D{{\cal D}}. &\label{1.8c}
\end{eqnarray}
Now let  $\psi_1(x',t)=\phi(p(x',t),x',t)$, 
$\psi_2(x',t)=\phi(q(x',t),x',t)$
 and assume, in addition, that ${\psi_i}_t\in L^2(\Sigma\times(0,T))$, 
$\nabla_{x'}\psi_i\in 
L^2(\Sigma\times(0,T))$ for $i=1,2$. Then $\nabla u\in L^2({{\cal D}})$.

Moreover, Propositions \ref{prop1.1} and \ref{prop1.2} apply to $u$. 
In particular, further 
assumptions on $\Sigma$, $p$, $q$, 
the coefficients $a_{ij}$ and  on the Dirichlet data  $\psi_i$ and/or 
$\phi$ give regularity
results for $\nabla u$ up to the corresponding subset of the Dirichlet 
boundary, $\partial_D{{\cal D}}$. 

If, in addition, $\beta(\cdot,\cdot,u)\in C^{\mu,\frac\mu2}({{\cal D}})$ 
uniformly
for $u$ in compact subsets of ${\mathbb R}$, $a_{ij}\in C^{1+\mu,\frac{1+\mu}2}({{\cal D}})$, $b_i,\ c\in C^{\mu,\frac\mu2}({{\cal D}})$, there 
holds that $u\in C^{2+\mu,1+\frac{\mu}2}({{\cal D}})$.
\end{thm}

\noindent{\bf Proof:} The proof of existence and uniqueness of the 
solution 
is analogous to that of  Theorem 1.1 in \cite{LVWmix} and follows
by using  Schauder's fixed point Theorem and the result of 
Proposition \ref{prop1.1}.  We include the proof for the sake of 
completeness.  

Let $B=\|\beta\|_{L^\infty}$, and let $\gamma$ and $C$ be the constants given by 
Proposition \ref{prop1.1} when $\phi$ is
fixed and $\|g\|_{L^\infty({{\cal D}})}\le B$. 
Let $0<\nu<\gamma$ and let $K>0$
be such that $\|\cdot\|_{C^{\nu,\frac\nu2}(\overline{{\cal D}})}\le K
\|\cdot\|_{C^{\gamma,\frac\gamma2}(\overline{{\cal D}})}$. Let us consider the set
$$
{\cal B}=\{v\in C^{\nu,\frac\nu2}(\overline{{\cal D}})\,/\,\|v\|_{C^{\nu,\frac\nu2}(\overline{{\cal D}})}
\le K\,C\}.
$$
Let ${\cal T}$ be defined on ${\cal B}$ by ${\cal T}v:=u$ where $u$ is the 
unique solution 
given by Proposition \ref{prop1.1} when $g(x,t)=\beta(x,t,v(x,t))$. Then
$$
\|{\cal T}v\|_{C^{\nu,\frac \nu2}(\overline{{\cal D}})}\le K\|{\cal T}v\|_{C^{\gamma,\frac \gamma2}(\overline{{\cal D}})} 
\le K\,C.
$$

Therefore, ${\cal T}$ maps  ${\cal B}$  continuously into a compact subset of ${\cal B}$. 
So that ${\cal T}$ has
a fixed point $u$ which clearly is a solution to (\ref{1.8a})--(\ref{1.8c}).

To prove uniqueness, we let $u_1$ and $u_2$ be solutions to 
(\ref{1.8a})--(\ref{1.8c}).
Then $w=u_1-u_2$ is a solution to (\ref{1.1a})--(\ref{1.1c}) with a different coefficient $c$ (which
depends on $u_1$ and $u_2$), and with $g=\phi=0$. By Proposition \ref{prop1.1}, $w=0$.

If, in addition, $\beta(\cdot,\cdot,u)\in C^{\mu,\frac\mu2}({{\cal D}})$ uniformly
for $u$ in compact subsets of ${\mathbb R}$, there holds that $u$ is a solution
of (\ref{1.1a})--(\ref{1.1c}) with $g\in C^{\gamma',\frac{\gamma'}2}({{\cal D}})$, with $\gamma'=\min\{\mu,\gamma\}$. 
Then, if $a_{ij}\in C^{1+\mu,\frac{1+\mu}2}({{\cal D}})$, 
$b_i,\ c\in C^{\mu,\frac\mu2}({{\cal D}})$, there 
holds that $u\in C^{2+\gamma',1+\frac{\gamma'}2}({{\cal D}})$, so that  
$g\in C^{\mu,\frac{\mu}2}({{\cal D}})$ and we deduce that 
$u\in C^{2+\mu,1+\frac{\mu}2}({{\cal D}})$. 

\section{The combustion problem}

The purpose of this section is to show how the results in this paper apply
in \cite{LVWdf} to a problem in combustion theory.
In \cite{LVWdf} the following two phase free boundary problem is considered: 
find a function 
$u(x,t)$, defined in ${\cal D}\subset {{\mathbb R}}^N\times(0,T)$, satisfying that
\begin{eqnarray}
&\Delta u+\sum a_i(x,t)\,u_{x_i}-u_t=0 \quad\mbox{in }
\{u>0\}\cup\{u\le 0\}^\circ,&\\
&u=0\,,\quad\  |\nabla u^+|^2-|\nabla u^-|^2= 2M
\quad\mbox{on }\partial\{u>0\},&
\end{eqnarray}
where $u^+=\max(u,0)$, $u^-=\max(-u,0)$, $M$ is a positive constant and
$a_i$ are bounded. We will refer to this free boundary problem as Problem
${\cal P}$.

This free boundary problem arises in several contexts (cf. \cite{V}). 
The most important
motivation to date has come from  combustion theory, where it
appears   as a limit situation  in the description of the propagation
of premixed  equi-diffusional deflagration flames.  In this case, 
$u$ is the limit, as $\varepsilon\to 0$, of solutions $u^\varepsilon$ to
\begin{equation}
\Delta u^\varepsilon +\sum a_i(x,t)\,u^\varepsilon_{x_i}- u^\varepsilon_t={\beta}_{\varepsilon}(u^\varepsilon),
\end{equation}
with $\varepsilon>0$, $\beta_\varepsilon\ge 0$, $\beta_\varepsilon(s)=\frac1\varepsilon\beta(\frac s\varepsilon)$,
support\,$\beta$=[0,1] and $\int\beta(s)\,ds=M$. We call this equation 
${\cal P}_{\varepsilon}$.   

Problem ${\cal P}$ admits {\it classical} solutions only for good data and 
for small times. Different generalized concepts of solution have been 
proposed, among them the concepts of {\it limit} solution (that is, 
$u=\lim u^\varepsilon$) and {\it viscosity} solution, cf. \cite{CV}, \cite{CLW}, resp. 
The purpose of \cite{LVWdf} is to investigate
conditions under which the three concepts agree and produce a unique
solution.


The results in \cite{LVWdf} can be summarized as saying that --under appropriate
conditions-- {\it if  a classical  solution 
of problem ${\cal P}$ exists, then it is at the same time  the unique  
classical 
solution, the unique limit solution and also the unique  viscosity 
solution.}

The results of \cite{LVWdf} extend those in \cite{LVWuf}, where similar conclusions
are obtained for the one phase version of this problem (i.e., under the 
assumption that $u \ge 0$).

One of the main results in \cite{LVWdf} is Theorem 6.1, which
gives simultaneously the uniqueness of {\it classical} and {\it limit}
solution.
The main tool in the proof of this theorem is the following basic
result of \cite{LVWdf}:

\begin{thm} \label{teo5.1} (Theorem 5.1 in \cite{LVWdf})  
Let $\Sigma\subset {\mathbb R}^{N-1}$ a bounded $C^3$ domain, 
$\Omega=(0,d)\times\Sigma$,
$Q=\Omega\times (0,T)$, $\partial_NQ=(0,d)\times\partial\Sigma\times(0,T]$. Let $w$ be a classical subsolution to ${\cal P}$
in
$Q$, with $\frac{\partial w}{\partial \eta}=0$ on $\partial_NQ$.
Assume, in addition, that there exists $\delta_0>0$ such that
$$|\nabla w^+|^2-|\nabla w^-|^2=2M+\delta_0\quad\mbox{ on }
Q\cap\partial\{w>0\}.$$
Then, there
exists a family $v^\varepsilon\in C(\overline Q)$, with
$\nabla v^\varepsilon\in L^2_{{loc}}(\overline Q)$,
of weak subsolutions to ${\cal P}_\varepsilon$ in
$Q$,
with $\frac{\partial v^\varepsilon}{\partial \eta}=0$ on
$\partial_N Q$,
such that, as $\varepsilon\to 0$, $v^\varepsilon\to w$ uniformly in
$\overline Q$.
\end{thm}

For the precise hypotheses and definitions, and detailed 
proofs of these results, we refer the reader to \cite{LVWdf}.

The results of the present paper are needed in Theorem \ref{teo5.1} for 
the construction of the family $v^\varepsilon$ which is constructed as follows:

Let $A$ be the constant in Lemma 4.1 of \cite{LVWdf} and let $\varepsilon>0$ be 
small. 

Let $p_\varepsilon, q_\varepsilon\in C^1(\overline\Sigma\times[0,T])$ with
$\nabla_{x'}p_\varepsilon,\nabla_{x'}q_\varepsilon\in 
C^{\alpha,\frac\alpha2}(\overline\Sigma\times[0,T])$ be such that 
$\{w>A\varepsilon\}$ is given
by $ x_1<p_\varepsilon(x',t)$ and $\{w<-A\varepsilon\}$ is given by
$ x_1>q_\varepsilon(x',t)$. 

Let the domain be  
$$
{{\cal D}}^{\,\varepsilon}=\{(x,t)\in Q\,/\,p_\varepsilon(x',t)<x_1<q_\varepsilon(x',t)\}.
$$

Let $w^\varepsilon$ be the solution to ${\cal P}_\varepsilon$ in ${{\cal D}}^{\,\varepsilon}$ with 
boundary conditions
\begin{eqnarray*}
&w^\varepsilon(x,t)=
\left\{
\begin{array}{ll}  
 A\varepsilon & \mbox{ on } x_1=p_\varepsilon(x',t),\\
-A\varepsilon & \mbox{ on }x_1=q_\varepsilon(x',t),
\end{array} \right.&\\
&\frac{\partial w^\varepsilon}{\partial\eta}=0\quad \mbox{on } 
\partial_N {\cal D}^{\,\varepsilon}:=\partial{\cal D}^{\,\varepsilon}\cap\partial_N Q,&
\end{eqnarray*}
and initial datum $w_0^{\varepsilon} \in C^{\alpha}(\overline{\cal D}^{\,\varepsilon}\cap\{t=0\})$
(for the choice of the function $w_0^{\varepsilon}$ we refer to \cite{LVWdf}). 

Then, the family $v^\varepsilon$ is defined by
$$
v^\varepsilon= \left\{
\begin{array}{l@{\quad}l} 
w & \mbox{ in }\{|w|\geq A\varepsilon\},\\
w^\varepsilon & \mbox{ in }{\cal D}^{\,\varepsilon}.
\end{array}
\right.
$$

Now we point out where the results in the present paper apply.
  From Theorem \ref{theo1.1}, there exists a 
 unique solution $w^\varepsilon\in C^{\gamma,\frac\gamma2}(\overline{\cal D}^{\,\varepsilon})$ with
$\nabla w^\varepsilon\in C(\overline{\cal D}^{\,\varepsilon}\cap\{t>0\}) \cap L^2({\cal
D}^{\,\varepsilon})$. Moreover, since $w^\varepsilon_0\in C^{1+\alpha}$ in
a subset of  $\overline{\cal D}^{\varepsilon}\cap\{t=0\}$, further continuity of
$\nabla w^\varepsilon$ can be derived from Proposition \ref{prop1.2}.


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\noindent{\sc Claudia Lederman}\\
Departamento de Matem\'atica, 
Facultad de Ciencias Exactas \\
Universidad de Buenos Aires \\
(1428) Buenos Aires  - Argentina\\
e-mail: clederma@dm.uba.ar \smallskip


\noindent{\sc Juan Luis Vazquez}\\
 Departamento de Matem\'aticas,
Universidad Aut\'onoma de Madrid\\
28049 Madrid - Spain\\
e-mail: juanluis.vazquez@uam.es  \smallskip

\noindent{\sc Noemi Wolanski}\\
Departamento de Matem\'atica, Facultad de Ciencias Exactas\\
Universidad de Buenos Aires\\
(1428) Buenos Aires  - Argentina\\
e-mail: wolanski@dm.uba.ar
\end{document}