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%_ ************************************************************************** %_ * The TeX source for AMS journal articles is the publishers TeX code * %_ * which may contain special commands defined for the AMS production * %_ * environment. Therefore, it may not be possible to process these files * %_ * through TeX without errors. To display a typeset version of a journal * %_ * article easily, we suggest that you retrieve the article in DVI, * %_ * PostScript, or PDF format. * %_ ************************************************************************** % Author Package file for use with AMS-LaTeX 1.2 \controldates{18-DEC-2002,18-DEC-2002,18-DEC-2002,18-DEC-2002} \RequirePackage[warning,log]{snapshot} \documentclass{era-l} \issueinfo{8}{06}{}{2002} \dateposted{December 19, 2002} \pagespan{47}{51} \PII{S 1079-6762(02)00104-X} \usepackage{upref} \copyrightinfo{2002}{American Mathematical Society} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{xca}[theorem]{Exercise} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \numberwithin{equation}{section} \newcommand{\abs}[1]{|#1|} \newcommand{\myno}[1]{\|#1\|} \newcommand{\oiv}[2]{(\,#1,#2\,)} 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} \newcommand{\pro}[2]{\prod_{\scriptstyle #1=1 \atop \scriptstyle #1 \not=#2}^{n+1}} \newcommand{\prob}[4]{{\mathcal P}_{\mbox{{\scriptsize #1}}}\,(\,#2,#3,#4)} \newcommand{\mintqnp}[1]{\mint\limits_{Q^n \oiv 0 {{#1}^{\us\sig}}}} \newcommand{\mintqn}{\mintqnp \yn} \mathchardef\scrL="024C \newcommand{\D}{{\mathcal D}} \newcommand{\J}{{\mathcal J}} \newcommand{\cu}[3]{Q^{\,#1}(#2,\,#3)} \newcommand{\cud}[3]{{\infl Q}^{\,#1}(#2,\,#3)} \newcommand{\ccu}[3]{\overline{Q^{\,#1}}(#2,#3)} \newcommand{\cun}[2]{Q^{\,n}(#1,#2)} \newcommand{\cunp}[2]{Q^{\,n}_+(#1,#2)} \newcommand{\ccunp}[2]{\overline{Q^{\,n}_+(#1,#2)}} \newcommand{\cunm}[1]{Q^{\,n-1}(#1)} \newcommand{\mywp}[1]{W^{2,1}_p(#1)} \newcommand{\wpp}[1]{W^{2,1}_{p,p}(#1)} \newcommand{\trpd}[1]{W^{2-1/p,(2-1/p)/2}_p(#1)} \newcommand{\trpn}[1]{W^{1-1/p,(1-1/p)/2}_p(#1)} \newcommand{\wpq}[1]{W^{2,1}_{p,q}(#1)} \newcommand{\bpq}[1]{B^{\,2(1-1/q)}_{p,q}(#1)} \newcommand{\trpqe}[3]{W^{#1,#2}_{p,q}(#3)} \newcommand{\trwfpqd}[1]{WF^{2-1/p,(2-1/p)/2}_{p,q}(#1)} \newcommand{\trwfpqn}[1]{WF^{1-1/p,(1-1/p)/2}_{p,q}(#1)} \newcommand{\wf}[5]{WF^{#1,#2}_{#3,#4}(#5)} \newcommand{\M}[6]{{\mathcal M}_{(#1,#2)\,,\,#3,\,#4\,}^{#5}(#6)} \newcommand{\N}[4]{{\mathcal N}_{#1,#2,#3}(#4)} \newcommand{\mya}{a} \newcommand{\myb}{b} \newcommand{\G}{G} \newcommand{\g}{g} \newcommand{\myr}{\us \sig \cdot (\us 1 +\us \lam + \us \mu\,)} \newcommand{\R}[2]{R_{#1}\left [#2\right ]} \newcommand{\mj}{{m_j}} \newcommand{\yn}{y_n} \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 $ 3/20$ 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