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\begin{document}
\title[Maximal regularity for parabolic equations]{Maximal regularity for
parabolic equations \linebreak[1] with inhomogeneous boundary conditions 
\linebreak[1] in Sobolev spaces
with mixed $L_p$-norm} 

\author{Peter Weidemaier}
\address{Fraunhofer-Institut Kurzzeitdynamik, Eckerstr. 4, D-79104 Freiburg,
Germany} 
\email{weide@emi.fhg.de}
                                                        
\subjclass[2000]{Primary 35K20, 46E35; Secondary 26D99}
\date{October 16, 2002}
\commby{Michael E. Taylor}
\keywords{Maximal regularity, inhomogeneous boundary conditions, trace theory,
mixed norm, Lizorkin-Triebel spaces} 

\begin{abstract}
We determine the exact regularity of the trace of a function $ u \in 
L_{q}\,(0,T;\, W_{p}^{2}(\Omega)) $ $ \cap \,  W^{1}_{q}\,(0,T;\,
{L_{p}\,(\Omega))} $ and of the trace of its spatial gradient on $\partial
\Omega \times (\,0,T\,) $ in the regime $ p \le q $. While for $ p=q $ both the
spatial and temporal regularity of the traces can be completely characterized
by fractional order Sobolev-Slobodetskii spaces, for $ p \neq q $ the
Lizorkin-Triebel spaces turn out to be necessary for characterizing the sharp
temporal regularity.
\end{abstract}

\maketitle

\section*{Introduction}

The space $ \wpq \omT := \lux q 0 T {\wu 2 p \om} \cap \wux 1 q 0 T {\lud p
\om} $, $ \omT :=\om \cro \oiv 0 T $, $ \om \subset \mathbb{R}^n $, is often
employed in the theory of evolution equations which are of first order in time
and second order in space; see von Wahl  \cite{vWa82} for parabolic equations,
Sohr \cite{Soh01}, Iwashita \cite{Iwa89} for the Navier-Stokes equation, and
Cl\'ement and Pr\"uss \cite{ClP92} for parabolic Volterra equations. For the
heat equation (as model problem) this space corresponds to maximal regularity
if the inhomogeneous part in the equation belongs to $ \lux q 0 T {\lud p \om}
$. Results of maximal regularity type have been established under various
conditions (\cite{CaV86}, \cite{CoD00}, \cite{HiP97}, \cite{Kry02}), but always
for {\it homogeneous} boundary conditions or the Cauchy problem. Combining
these results with Theorems \ref{trace_thm} and \ref{isp_om} stated below,
maximal regularity follows also for problems with {\it inhomogeneous} boundary
conditions.

\section{Trace theory in the classical case $ p=q \in \oiv 1 \infty $}
The trace of $ u \in \wpp \omT $ (space denoted $ \mywp \omT $ in \cite{LaSU68}) 
belongs to the space $ \trpd \GamT $, where $ W^{\alpha,\beta}_p(\GamT) := \lux
p 0 T {\wu \alpha p \Gam} \cap \wux \beta p 0 T {\lud p \Gam} $,  $ W^s_p $
denoting the Sobolev-Slobodetskii spaces and $ \GamT := \Gam \cro \oiv 0 T$,
$ \Gam:= \bom$. This result is sharp. The analogous result holds for the trace
of a spatial derivative $ \pa j u $, $ j \in \{ 1,\ldots,n\}$, which is an
element of $ \trpn \GamT $. For these results see
Ladyzhenskaya, Solonnikov, and Ural'tseva \cite[Chapter II, Lemma 3.4]{LaSU68}
and Grisvard \cite[Th\'eor\`eme 4.2]{Gri66}. While the Russian authors used the
``method of integral representation'', Grisvard applied interpolation theory.

\section{Trace theory in the case $ 1 < p \le q < \infty $}
The present author \cite{Wei94} has shown  continuity of the map
\begin{eqnarray*}
&& \wpq \omT \ni u \mapsto u \rest \GamT  \in \lux q 0 T {\wu {2-1/p} p \Gam}
\cap \wux {(2-1/p)/2} q 0 T {\lud p \Gam}, 
\end{eqnarray*}
but had to leave open whether this map is onto. In a retrospective view, this
target space was not the optimal one. It turned out that the sharp result is
obtained if the regularity of the trace in the time variable is described by
the Lizorkin-Triebel space $ \litrx {(2-1/p)/2} q p 0 T {\lud p \Gam} $, whose
definition follows.

\begin{definition}
 A function $ g \in \lux q 0 T {\lud p \Gam} $
belongs to the class 
\[\litrx \beta q p 0 T {\lud p \Gam}\;,\,\beta \in \oiv 0 1\]
iff
\begin{eqnarray}
\label{def_litr}
&& \dintpp 0 T 0 {T-t} {\hsp{-0.1cm} h^{-1-p\beta}
\myno{g(\dt,t+h)-g(\dt,t)}_{\lud p \Gam}^p} 
                h t {q/p} {1/q}  < \infty\, .
\end{eqnarray}
\end{definition}

\begin{remark} The Lizorkin-Triebel spaces  $F^\beta_{q, p}
(\mathbb{R}, \mathbb{R}) $
defined by Fourier analysis have a finite difference characterization of the
type occurring in the previous definition (see Triebel \cite[2.5.10
Theorem]{Tri83}). This fact motivated us to introduce the Lizorkin-Triebel spaces on
the bounded domain $ \oiv 0 T $ as above. Since $ \beta < 1 $, one can use
first order discrete differences in the definition instead of second order
ones, which the reader might have expected. \end{remark}

Let us introduce
\begin{eqnarray*} 
\label{def_wf}
&& \wf \alpha \beta p q {\GamT} :=
  \lux q 0 T {\wu \alpha p \Gam } \cap \litrx \beta q p 0 T {\lud p \Gam}
\end{eqnarray*}
and endow this space with the norm
\begin{eqnarray*}
&&  \myno{g}_{\wf \alpha \beta p q {\GamT}} = \myno{g}_{\lux q 0 T {\wu \alpha
p \Gam }} + 
\abs{g}_{\litrx \beta q p 0 T {\lud p \Gam}},
\end{eqnarray*}
where $ \abs{g}_{\litrx \beta q p 0 T {\lud p \Gam}} $ is given by the integral
in \fo{def_litr}.                    

We are going to formulate our two main theorems, in which we assume $ \om
\subset \mathbb{R}^n $ is an open subset (not necessarily bounded) with a compact
boundary of the class $ C^{1,1}$ (see \cite[6.2.2 Definition]{KuJF77} for
details ). Moreover, $ \nu $ denotes the vectorfield of outer unit normals on $
\Gam $.

\begin{theorem}  
\label{trace_thm}
i) For $ 3/2 
0$ 
and that\\
\vsp{-0.1cm}
$\!\!\!\!\renewcommand{\arraystretch}{1.2} \begin{array}{ll}
{\rm (A_{\mbox{D}})} & \!\! (a_{ij}(x,t))_{1\le i,j\le n} \mbox{ is\:symmetric\:and\:positive\:definite\:uniformly\:in\:} \cl \om \cro \civ 0 T,\\
                       & \!\! a_{ij}(x,t) \in \civx 0 0 T {\cd 0 {\cl \om}}, \
                          a_i(x,t) \in \civx 0 0 T {\lud r \om} \mbox{ with } r>n,                  \\
                       &  \!\!a_0(x,t) \in \civx 0 0 T {\lud s \om} \mbox{ with }  s>\rat n 2;       
   \\
{\rm(E_{\mbox{D}})}    & \!\!3/2 < p \le q < \infty;                                                     \\
{\rm(F)}               & \!\!f \in \lux q 0 T {\lud p \om};                                             \\
{\rm(\g_{\mbox{D}})}   & \!\!\g \in \trwfpqd \GamT;                                                      \\
{\rm(Iv)}              & \!\!u_0 \in \bpq \om;                                                     \\
{\rm(C_{\mbox{D}})}    & \!\! u_0(\dt) = \g(\dt,0) \mbox{ on } \Gam.                                               
\renewcommand{\arraystretch}{1.0} \end{array} $\\[0.3cm]
Then $\prob D {u_0} f \g$ has a unique solution $u \in \wpq \omT$ and there is
a constant $ c_D^*(p,q,T) $ with  
\[
\myno u_{\wpq \omT} \le c_D^* \dt (\myno {u_0}_{\bpq \om} +  \myno f_{\lux q 0
T {\lud p \om}} + \myno \g_{\trwfpqd \GamT}). 
\]
\end{theorem}

In the next theorem $ \bivx 0 T X $ denotes the bounded $X$-valued functions.
\begin{theorem}
\label{ex_con}
Consider \fo {cv_1}, \fo {cv_2}, \fo {cv_4}. Let $\om$ be as in the previous
theorem.                                                                                                       
Let {\rm (F), (Iv)} from the previous theorem hold unaltered and assume further
that\\                                 
\vsp{-0.1cm}
$\!\!\!\!\renewcommand{\arraystretch}{1.2} \begin{array}{ll}
{\rm(A_{\mbox{N}})} & \!\!(a_{ij}(x,t))_{1\le i,j\le n} \mbox{ is\:symmetric\:and\:positive\:definite\:uniformly\:in } \cl \om \cro \civ 0 T, \\
    &  \!\!a_{ij}(x,t) \in \civx 0 0 T {\cd \mu {\cl \om}} \cap \bivx 0 T {\cd {1+\eps} \Gam } \cap \civx \alpha 0 T {\cd 0 \Gam},      \\
    &  \!\!b_0 \in \bivx 0 T {\cd \mu \Gam} \cap \civx \alpha 0 T {\cd 0 \Gam}  \mbox{\:for\:some\:} \eps >0 ,\, \alpha > \rat 1 2 (1-1/p), \\
                    &  \!\!\mu > 1-1/p, \mbox{ and } a_i(x,t), a_0(x,t) 
\mbox{ are as specified in the previous theorem};             \\
{\rm(E_{\mbox{N}})} & \!\!3