\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{mathrsfs} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 109, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2011 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2011/109\hfil Resolvent estimates for elliptic systems] {Resolvent estimates for elliptic systems in function spaces of higher regularity} \author[R. Denk, M. Dreher\hfil EJDE-2011/109\hfilneg] {Robert Denk, Michael Dreher} % in alphabetical order \address{Robert Denk \newline Department of Mathematics and Statistics, University of Konstanz, 78457 Konstanz, Germany} \email{robert.denk@uni-konstanz.de} \address{Michael Dreher \newline Department of Mathematics and Statistics, University of Konstanz, 78457 Konstanz, Germany} \email{michael.dreher@uni-konstanz.de} \thanks{Submitted January 11, 2011. Published August 25, 2011.} \subjclass[2000]{35G45, 47D06} \keywords{Parameter-ellipticity; Douglis-Nirenberg systems; \hfill\break\indent analytic semigroups} \begin{abstract} We consider parameter-elliptic boundary value problems and uniform \emph{a~priori} estimates in $L^p$-Sobolev spaces of Bessel potential and Besov type. The problems considered are systems of uniform order and mixed-order systems (Douglis-Nirenberg systems). It is shown that compatibility conditions on the data are necessary for such estimates to hold. In particular, we consider the realization of the boundary value problem as an unbounded operator with the ground space being a closed subspace of a Sobolev space and give necessary and sufficient conditions for the realization to generate an analytic semigroup. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \newcommand{\norm}[2]{\|#1\|_{#2}} \section{Introduction} The aim of this paper is to establish resolvent estimates for parameter-elliptic boundary value problems in $L^p$-Sobolev spaces of higher order. \emph{A~priori} estimates involving parameter-dependent norms for parameter-elliptic or parabolic systems are known since long; classical works are, e.g., Agmon \cite{Agmon62}, Agranovich-Vishik \cite{AgranovichVishik64} for scalar equations, and Geymonat-Grisvard \cite{GeymonatGrisvard}, Roitberg-Sheftel \cite{roitberg-sheftel67} for systems. Further results on the $L^p$-theory for mixed-order systems were obtained, e.g., by Faierman \cite{faierman06}. For pseudodifferential boundary value problems, we refer to the parameter-dependent calculus developed by Grubb \cite{grubb96}. Parameter-dependent \emph{a~priori} estimates are motivated by their connection to operator theory: In the ground space $L^p$, the estimate immediately implies a uniform resolvent estimate for the $L^p$-realization of the boundary value problem. In particular, if the sector of parameter-ellipticity is large enough, i.e., if the problem is parabolic in the sense of Petrovskii, then the operator generates an analytic semigroup in $L^p$. Moreover, spectral properties and completeness of eigenfunctions can be obtained, see Denk-Faierman-M\"oller \cite{denk-faierman-moeller02} and Faierman-M\"oller \cite{faierman-moeller07}. If the equation is given in the whole space, we obtain the generation of an analytic semigroup in the whole scale of Sobolev spaces. In fact, the operator even admits a bounded $H^\infty$-calculus which was shown for general mixed-order systems of pseudodifferential operators in Denk-Saal-Seiler \cite{DSS09}. Consider the boundary value problem \begin{equation}\label{1-1} \begin{gathered} (A-\lambda) u = f,\quad\text{in }\Omega,\\ B_j u = g_j,\quad\text{on }{\partial}\Omega, \; j=1,\dots,M, \end{gathered} \end{equation} in a bounded smooth domain $\Omega\subset{\mathbb{R}}^d$. Here $A$ is a system of differential operators, and the $B_j$ form a vector of differential operators, and the number $M$ of boundary conditions is determined by the order and the dimension of the system $A$ (see below for details). In the present paper we study the question under which additional (compatibility) assumptions on the right-hand side this boundary value problem has a unique solution satisfying uniform (in $\lambda$) \emph{a~priori} estimates. In particular, for $s\ge 0$ and $10$, however, the situation is more complicated. As an example, one may consider the Dirichlet-Laplacian $\Delta_D$ in $Y=W_p^1(\Omega)$ with domain $D(\Delta_D) = \{ u\in W_p^3(\Omega)\colon u|_{{\partial}\Omega}=0\}$. This operator does not generate an analytic semigroup in $Y$; in fact, its resolvent decays as $|\lambda|^{-1/2-1/2p}$ as $|\lambda|\to\infty$ (see Nesensohn \cite{NesensohnDipl}). Roughly speaking, additional compatibility conditions have to be incorporated into the basic space $Y$ in order to obtain a decay of $|\lambda|^{-1}$. Therefore, the question is to find equivalent conditions on $Y$ for which ${\mathcal {A}}$ generates an analytic semigroup on $Y$. This question is fully answered by Theorem~\ref{Theo4} below, originating from a general criterion for the validity of a broad range of resolvent estimates in Theorem~\ref{GammaTheo}. We also study compatibility conditions for which the problem with inhomogeneous boundary data \eqref{1-1} is uniquely solvable with suitable \emph{a~priori} estimate for the solution. As a ground space, we consider subspaces of integer or non-integer Sobolev spaces both of Besov type and of Bessel potential type. The question of generation of an analytic semigroup for parabolic equations was also studied by Guidetti \cite{guidetti96} where higher order scalar equations are considered. Writing such an equation as a first order system, in \cite{guidetti96} necessary and sufficient conditions for the unique solvability of the non-stationary problem are given. Roughly speaking, in \cite{guidetti96} the author observes that the order of the boundary operators has to be sufficiently large. This coincides with our conditions as in this case the trace conditions given in Theorem~\ref{Theo4} are empty. Whereas the equations in \cite{guidetti96} have more general coefficients, the mixed-order system is of special structure (arising from a higher order equation), and the basic space is fixed. Our paper considers general mixed-order systems and the whole scale of Sobolev spaces. \section{Notation and auxiliary results} Let $\Omega$ be a bounded domain in ${\mathbb{R}}^{d}$, $d\geq2$, with boundary $\Gamma={\partial}\Omega\in C^{\infty}$. The Besov spaces are denoted by $B^s_{p,q}(\Omega)$, for $s\in{\mathbb{R}}$ and $1\leq p,q\leq\infty$, and the Bessel potential spaces are called $H^s_p(\Omega)$, for $s\in{\mathbb{R}}$ and $11/p$, we define the space ${\mathcal {K}}^{s-1/p}_{p;\Gamma}$ of traces of functions from ${\mathcal {K}}^s_p(\Omega)$ at the boundary $\Gamma={\partial}\Omega$: \[ {\mathcal {K}}^{s-1/p}_{p,\Gamma} := \begin{cases} B^{s-1/p}_{p,q}({\partial}\Omega), & {\mathcal {K}}^s_p(\Omega)=B^s_{p,q}(\Omega), \\ B^{s-1/p}_{p,p}({\partial}\Omega), & K^s_p(\Omega)=H^s_p(\Omega). \end{cases} \] To simplify later formulae, we set ${\mathcal {K}}^0 _{p,\Gamma}:=L^p(\Omega)$, although this is not the space of traces of functions from $H^{1/p}_p(\Omega)$ or $B^{1/p}_{p,q}(\Omega)$, except when $q=1$. The trace operator on ${\partial}\Omega$, mapping functions from $C^{\infty}(\overline{\Omega})$ to their boundary values, is called $\gamma_0$. We will write $[\cdot,\cdot]_{\theta}$ for the complex interpolation method, and $(\cdot,\cdot)_{\theta,q}$ for the real interpolation method, where $0\leq\theta\leq1$ and $1\leq q\leq\infty$. Then ${\partial}_{x}^{\alpha}$ maps continuously from ${\mathcal {K}}^s_p(\Omega)$ into ${\mathcal {K}}^{s-|\alpha|}_p(\Omega)$, for all $s\in{\mathbb{R}}$ and all $p\in(1,\infty)$, and $\{{\mathcal {K}}^s_p(\Omega)\}_{s\in{\mathbb{R}}}$ forms an interpolation scale with respect to the complex interpolation method: \[ [ {\mathcal {K}}^{s_0}_p(\Omega),{\mathcal {K}}^{s_1}_p(\Omega) ]_{\theta} = {\mathcal {K}}^{s_{\theta}}_p(\Omega), \quad s_{\theta}=(1-\theta)s_0+\theta s_1, \quad 0\leq\theta\leq1. \] We will also make free use of the following: if a Banach space $X_{\theta}$ is an interpolation space of the pair $(X_0,X_1)$ of order $\theta$, then \[ {\varrho}^{1-\theta}\norm{f}{\theta} \leq C(\norm{f}{1}+{\varrho}\norm{f}{0}), \quad {\varrho}\in {\mathbb{R}}_{+}, \; f\in X_0\cap X_1. \] For detailed representations of the theory of function spaces, we refer the reader to Bergh-L{\"o}fstr{\"o}m~\cite{BerghLoefstroem} and Triebel~\cite{Triebel78}. \begin{lemma}\label{TraceInterpolLem} Suppose $0\leq\sigma_0<1/p<\sigma_1$. Then we have the estimates \begin{gather} {\varrho}^{1-\theta}\norm{\gamma_0u}{L^p({\partial}\Omega)} \leq C \left( \norm{u}{{\mathcal {K}}^{\sigma_1}_p(\Omega)} + {\varrho}\norm{u}{{\mathcal {K}}^{\sigma_0}_p(\Omega)} \right), \quad \theta=\frac{1/p-\sigma_0}{\sigma_1-\sigma_0}, \label{BdryInterpolation1} \\ {\varrho}^{1-\theta}\norm{\gamma_0u}{L^p({\partial}\Omega)} \leq C \left( \norm{u}{B^{\sigma_1}_{p,q}(\Omega)} + {\varrho}\norm{u}{L^p(\Omega)} \right), \quad \theta=\frac{1}{p\sigma_1}, \label{BdryInterpolation2} \end{gather} for all $u\in{\mathcal {K}}^{\sigma_1}_p(\Omega)$ and all ${\varrho}\in[1,\infty)$. \end{lemma} \begin{proof} In~\cite[Theorem~4.7.1]{Triebel78}, we find $\gamma_0\in{\mathscr{L}}(B_{p,1}^{1/p}(\Omega),L^p({\partial}\Omega))$, hence we conclude that \[ \norm{\gamma_0u}{L^p({\partial}\Omega)} \leq C\norm{u}{B^{1/p}_{p,1}(\Omega)}. \] Now we have, for the above $\sigma_0$, $\sigma_1$, \[ \left( B^{\sigma_0}_{p,q}(\Omega), B^{\sigma_1}_{p,q}(\Omega) \right)_{\theta,1} = \left( H^{\sigma_0}_p(\Omega), H^{\sigma_1}_p(\Omega) \right)_{\theta,1} = B^{1/p}_{p,1}(\Omega), \quad \theta=\frac{1/p-\sigma_0}{\sigma_1-\sigma_0}, \] which brings us \eqref{BdryInterpolation1}. And \eqref{BdryInterpolation2} follows from \[ B^0 _{p,\min(2,p)}(\Omega) \hookrightarrow L^p(\Omega) \hookrightarrow B^0 _{p,\max(2,p)}(\Omega) \] and the interpolation identity $(B^0 _{p,r}(\Omega),B^{\sigma_1}_{p,q}(\Omega))_{\theta,1} =B^{1/p}_{p,1}(\Omega)$ for $r\in\{2,p\}$. \end{proof} \section{Main results} \subsection{Systems of uniform order} First, let $A=(a_{jk}(x,D_{x}))_{j,k=1,\dots,N}$ be a matrix differential operator with $\operatorname{ord} a_{jk}\leq m$ for all $j,k$. The coefficients of $a_{jk}$ are smooth on a neighborhood of $\overline{\Omega}$. If $A$ is a parameter-elliptic matrix differential operator in $\Omega$, then $mN\in2{\mathbb{N}}$, see Agranovich and Vishik~\cite{AgranovichVishik64}. Next let us be given differential operators $B_{j}=B_{j}(x,D_{x})$ for $j=1,\dots,mN/2$, with $\operatorname{ord} B_{j}=r_{j}\leq m-1$. For $\lambda$ from a sector ${\mathcal{L}}\subset{\mathbb{C}}$ with vertex at the origin, we consider the boundary value problem \begin{equation}\label{BVPOmega} \begin{gathered} (A-\lambda)u =f,\quad \text{in }\Omega,\\ \gamma_0B_{j}u =g_{j}, \quad\text{on }{\partial}\Omega, \; j=1,\dots,mN/2, \end{gathered} \end{equation} and its variant with homogeneous boundary data: \begin{equation}\label{BVPOmegaHom} \begin{gathered} (A-\lambda)u =f,\quad \text{in }\Omega,\\ \gamma_0B_{j}u =0, \quad\text{on }{\partial}\Omega, \; j=1,\dots,mN/2. \end{gathered} \end{equation} We suppose that the operators $(A,B_1,\dots,B_{mN/2})$ constitute a parameter-elliptic boundary value problem on $\Omega$ in the open sector ${\mathcal {L}}$. \begin{proposition} \label{prop3.1} Let $u$ be any function from ${\mathcal {K}}^{s+m}_p(\Omega)$ with $s\in[0,\infty)$ but $s\not\in{\mathbb{N}}+1/p$, and take $\lambda\in{\mathbb{C}}$ arbitrarily. Define $f$ and $g_{j}$ by the right-hand sides of \eqref{BVPOmega}. Then we have the inequality \begin{align*} & \norm{f}{{\mathcal {K}}^s_p(\Omega)} + \sum_{j=1}^{mN/2} \Big( \norm{g_{j}}{{\mathcal {K}}_{p,\Gamma}^{s+m-r_{j}-1/p}} + |\lambda|^{1+\frac{1}{m}\min(s-r_{j}-1/p,0)} \norm{g_{j}}{{\mathcal {K}}_{p,\Gamma}^{\max(s-r_{j}-1/p,0)}} \Big) \\ & \leq C \big( \norm{u}{{\mathcal {K}}_p^{s+m}(\Omega)} + |\lambda|\norm{u}{{\mathcal {K}}_p^s(\Omega)} \big), \end{align*} with some constant $C$ independent of $u$ and $\lambda$. \end{proposition} \begin{proof} We clearly have the estimates \begin{gather*} \norm{f}{{\mathcal {K}}^s_p(\Omega)} \leq C \left( \norm{u}{{\mathcal {K}}^{s+m}_p(\Omega)} + |\lambda|\norm{u}{{\mathcal {K}}^s_p(\Omega)} \right), \\ \norm{g_{j}}{{\mathcal {K}}_{p,\Gamma}^{s+m-r_{j}-1/p}} \leq C\norm{u}{{\mathcal {K}}^{s+m}_p(\Omega)}, \end{gather*} and now it suffices to establish the inequalities \begin{gather} |\lambda|\norm{g_{j}}{{\mathcal {K}}_{p,\Gamma}^{s-r_{j}-1/p}} \leq C|\lambda|\norm{u}{{\mathcal {K}}^s_p(\Omega)}, \quad (s-r_{j}-1/p>0), \label{UpperEst3} \\ \begin{aligned} &|\lambda|^{1+\frac{1}{m}(s-r_{j}-1/p)} \norm{g_{j}}{L^p({\partial}\Omega)}\\ &\leq C \left( \norm{u}{{\mathcal {K}}_p^{s+m}(\Omega)} + |\lambda|\norm{u}{{\mathcal {K}}^s_p(\Omega)} \right), \quad (s-r_{j}-1/p<0). \end{aligned} \label{UpperEst4} \end{gather} For $0\frac{m+r_{j}+1/p-s}{m}. \end{equation} \end{theorem} \begin{remark} \label{rmk3.4} \rm In case of the Dirichlet Laplacian $\Delta_{D}$, considered in the space $W_p^{1}(\Omega)$, the condition \eqref{CondGamma} turns into \[ \gamma>\frac{p+1}{2p}, \] which matches the result by Nesensohn~\cite{NesensohnDipl}, where the resolvent estimate from below, \[ \norm{(\Delta_{D}-\lambda)^{-1}}{{\mathscr{L}}(W_p^{1}({\mathbb{R}}^{n}_{+}))} \geq \frac{C}{|\lambda|^{(p+1)/(2p)}}, \quad C>0, \] was proved. \end{remark} Now we come to the proof of Theorem~\ref{GammaTheo}. \begin{proof} Choose such a $j$. By $\gamma\leq1$ and the condition \eqref{CondGamma}, we obtain $s-r_{j}>1/p$, and therefore $\gamma_0B_{j}f=\gamma_0B_{j}Au\in{\mathcal {K}}^{s-r_{j}-1/p}_{p,\Gamma}$ exists. By parameter-ellipticity in ${\mathcal {L}}$, there is a number $\lambda_{\ast}\in{\mathcal {L}}$ with $|\lambda_{\ast}|\geq\lambda_0+1$ such that ${\mathcal{A}}-\lambda_{\ast} \colon D({\mathcal{A}}) \cap W_p^{\sigma+m}(\Omega) \to W_p^{\sigma}(\Omega)$ is a continuous isomorphism, for all $\sigma\in{\mathbb{N}}_0$. By interpolation, ${\mathcal {A}}-\lambda_{\ast} \colon D({\mathcal {A}})\cap{\mathcal {K}}^{s+m}_p(\Omega) \to {\mathcal {K}}^s_p(\Omega)$ then is a continuous isomorphism, too. Then we have \[ u=({\mathcal {A}}-\lambda_{\ast})^{-1}(f+(\lambda-\lambda_{\ast})u), \] hence $\norm{u}{{\mathcal {K}}^{s+m}_p(\Omega)} \leq C(\norm{f}{{\mathcal {K}}^s_p(\Omega)} + |\lambda|\norm{u}{{\mathcal {K}}^s_p(\Omega)}) \leq C|\lambda|^{1-\gamma}\norm{f}{{\mathcal {K}}^s_p(\Omega)}$, by $|\lambda|\geq1$. Now we obtain \begin{align*} |\lambda|^{\frac{s-r_{j}}{m}(1-\frac{1}{p(s-r_{j})})} \norm{\gamma_0B_{j}f}{L^p({\partial}\Omega)} &= |\lambda|^{\frac{s-r_{j}}{m}(1-\frac{1}{p(s-r_{j})})} \norm{\gamma_0B_{j}Au}{L^p({\partial}\Omega)} \\ & \leq C \Big( \norm{B_{j}Au}{{\mathcal {K}}^{s-r_{j}}_p(\Omega)} + |\lambda|^{\frac{s-r_{j}}{m}}\norm{B_{j}Au}{{\mathcal {K}}^0 _p(\Omega)} \Big) \\ & \leq C \Big( \norm{u}{{\mathcal {K}}^{s+m}_p(\Omega)} + |\lambda|\norm{u}{{\mathcal {K}}^s_p(\Omega)} \Big) \\ & \leq C|\lambda|^{1-\gamma}\norm{f}{{\mathcal {K}}^s_p(\Omega)}. \end{align*} Per \eqref{CondGamma}, the left-hand side has a higher power of $|\lambda|$ than the right-hand side. Send $\lambda\to\infty$ in ${\mathcal {L}}$. \end{proof} \begin{corollary}\label{FirstCor} Let \eqref{BVPOmegaHom} be parameter-elliptic in ${\mathcal {L}}$. Fix $p\in(1,\infty)$ and $s\in[0,m]$. Then the following two statements are equivalent for $f\in{\mathcal {K}}_p^s(\Omega)$. \begin{enumerate} \item there are positive constants $\lambda_0$ and $C_0$ such that all solutions $u$ to \eqref{BVPOmegaHom} with $\lambda\in{\mathcal {L}}$, $|\lambda|\geq\lambda_0$ enjoy the following estimate: \[ \norm{u}{{\mathcal {K}}_p^{s+m}(\Omega)} + |\lambda|\norm{u}{{\mathcal {K}}_p^s(\Omega)} \leq C_0\norm{f}{{\mathcal {K}}_p^s(\Omega)}, \] \item $\gamma_0B_{j}f\equiv0$ for all $j$ with $s-r_{j}>1/p$. \end{enumerate} \end{corollary} \begin{proof} The second statement follows directly from the first, by Theorem~\ref{GammaTheo}. Conversely, suppose statement no.2. Define $X=L^p(\Omega)$ and ${\mathcal {A}}\colon D({\mathcal {A}})\to X$ by \[ D({\mathcal {A}}) := \{u\in W_p^{m}(\Omega)\colon \gamma_0B_{j}u=0,\quad j=1,\dots,mN/2\}, \quad {\mathcal {A}} u:=Au, \] and set \begin{align*} Y_{s} &:=\begin{cases} [ L^p(\Omega),D({\mathcal {A}}) ]_{s/m}, & {\mathcal {K}}_p^{\bullet}(\Omega)=H_p^{\bullet}(\Omega), \\ \big( L^p(\Omega),D({\mathcal {A}}) \big)_{s/m,q} , & {\mathcal {K}}_p^{\bullet}(\Omega)=B_{p,q}^{\bullet}(\Omega) \end{cases} \\ & = \{ u\in{\mathcal {K}}^s_p(\Omega)\colon \gamma_0B_{j}u\equiv0, \; \forall j \text{ with } s-r_{j}>1/p \}. \end{align*} Then $D({\mathcal {A}})\hookrightarrow Y_{s}\hookrightarrow X$ with dense embeddings. From Geymonat-Grisvard~\cite{GeymonatGrisvard} we quote the estimate \[ \norm{u}{W_p^{m}(\Omega)} + |\lambda|\norm{u}{L^p(\Omega)} \leq C\norm{f}{L^p(\Omega)}, \quad f\in X, \] for $u=({\mathcal {A}}-\lambda)^{-1}f$ and $\lambda\in{\mathcal {L}}$, $|\lambda|\geq\lambda_0$. And for $f\in D({\mathcal {A}})$, we have $u=({\mathcal {A}}-\lambda)^{-1}f\in D({\mathcal{A}}^{2})$, hence \[ \norm{u}{W_p^{2m}(\Omega)} + |\lambda|\norm{u}{W_p^{m}(\Omega)} \leq C\norm{f}{W_p^{m}(\Omega)}, \quad f\in D({\mathcal {A}}). \] Interpolating between these two estimates then implies \[ \norm{u}{{\mathcal {K}}_p^{s+m}(\Omega)} + |\lambda|\norm{u}{{\mathcal {K}}_p^s(\Omega)} \leq C\norm{f}{{\mathcal {K}}_p^s(\Omega)}, \quad f\in Y_{s}, \] for $s\in[0,m]$. \end{proof} \begin{theorem}\label{Theo3} Let \eqref{BVPOmegaHom} be parameter-elliptic in the sector ${\mathcal {L}}$, and fix $p\in(1,\infty)$ and $s_{\rm max}\in[0,\infty)$. Then the following two statements are equivalent, for $f\in{\mathcal {K}}_p^{s_{\rm max}}(\Omega)$. \begin{enumerate} \item there are positive constants $\lambda_0$ and $C_0$ such that all solutions $u$ to \eqref{BVPOmegaHom} with $\lambda\in{\mathcal {L}}$, $|\lambda|\geq\lambda_0$ satisfy the collection of estimates \[ \norm{u}{{\mathcal {K}}_p^{s+m}(\Omega)} + |\lambda|\norm{u}{{\mathcal {K}}_p^s(\Omega)} \leq C_0\norm{f}{{\mathcal {K}}_p^s(\Omega)}, \] for all $s\in[0,s_{\rm max}]$. \item for each pair $(j,k)\in\{1,2,\dots,m\}\times{\mathbb{N}}_0$ with $s_{\rm max}-r_{j}>mk+1/p$, we have $\gamma_0B_{j}A^{k}f\equiv0$. \end{enumerate} \end{theorem} \begin{proof} A proof for the case $s_{\rm max}\in[0,m]$ was given in Corollary~\ref{FirstCor}, whose notations we adopt here. And the proof of the first statement from the second is very similar to the proof of Corollary~\ref{FirstCor}, so we skip it. Therefore we may assume $s_{\rm max}\geq m$. We suppose now the statement no.1, and proceed by induction on $s_{\rm max}$ of step size $m$. Choosing $s=m$, we find $\gamma_0B_{j}f\equiv0$ for all $j$, hence $f\in D({\mathcal {A}})$, and then also ${\mathcal {A}} u\in D({\mathcal {A}})$. Choose $\lambda_{\ast}$ as in the proof of Theorem~\ref{GammaTheo}, and put $\tilde{u}:=({\mathcal {A}}-\lambda_{\ast})u$, $\tilde{f}:=({\mathcal {A}}-\lambda_{\ast})f$, and note that \begin{gather*} (A-\lambda)\tilde{u}=\tilde{f},\quad\text{in } \Omega,\\ \gamma_0B_{j}\tilde{u}=0, \quad\text{on } {\partial}\Omega, \end{gather*} with $\tilde{f}\in{\mathcal {K}}_p^{s_{\rm max}-m}(\Omega)$. For $0\leq s\leq s_{\rm max}-m$ and $\lambda\in{\mathcal {L}}$, $|\lambda|\geq\lambda_0$, we then have \begin{align*} \norm{\tilde{u}}{{\mathcal {K}}_p^{s+m}(\Omega)} + |\lambda|\norm{\tilde{u}}{{\mathcal {K}}_p^s(\Omega)} & \leq C \left( \norm{u}{{\mathcal {K}}_p^{m+(s+m)}(\Omega)} + |\lambda|\norm{u}{{\mathcal {K}}_p^{s+m}(\Omega)} \right) \\ & \leq C\norm{f}{{\mathcal {K}}_p^{s+m}(\Omega)} = C\norm{({\mathcal {A}}-\lambda)^{-1}\tilde{f}}{{\mathcal {K}}_p^{s+m}(\Omega)} \\ & \leq \tilde{C}_0\norm{\tilde{f}}{{\mathcal {K}}_p^s(\Omega)}. \end{align*} By induction, we know that $\gamma_0B_{j}A^{k}\tilde{f}\equiv0$ for all pairs $(j,k)\in\{1,\dots,m\}\times{\mathbb{N}}_0$ with $(s_{\rm max}-m)-r_{j}>mk+1/p$. The definition of $\tilde{f}$ then brings us to $\gamma_0B_{j}A^{k}f\equiv0$ for all $(j,k)$ with $s_{\rm max}-r_{j}>mk+1/p$. \end{proof} \begin{theorem}\label{Theo4} Let \eqref{BVPOmegaHom} be parameter-elliptic in a sector ${\mathcal {L}}$ that is greater than the right half-plane. For $s\geq0$ and $1mk+1/p$. \end{enumerate} \end{theorem} \begin{proof} The domain of a generator of a $C_0$ semigroup is always dense in the ground space. Under the assumptions on ${\mathcal {L}}$, $Y$ and $D({\mathcal {A}})$, the analyticity of the semigroup is equivalent to the resolvent estimate \[ \norm{({\mathcal {A}}-\lambda)^{-1}}{{\mathscr{L}}(Y)} \leq \frac{C}{|\lambda|} \] for all $\lambda\in{\mathcal {L}}$ of large modulus. Now apply Theorem~\ref{Theo3}. \end{proof} \subsection{Systems of mixed order}\label{MixedOrderSubSec} In this section, $A$ shall be a matrix differential operator of mixed order: \[ A=(a_{jk}(x,D_{x}))_{j,k=1,\dots,N}, \quad \operatorname{ord} a_{jk}\leq s_{j}+m_{k}, \] for integers $s_{j}$ and $m_{k}$. The orders on the diagonal of $A$ shall be equal, \[ s_1+m_1=\dots=s_{N}+m_{N}=:m, \] and without loss of generality, we can set $\min_{j} m_{j}=0$. The principal part $a_{jk}^0 $ of $a_{jk}$ is that part with degree exactly equal to $s_{j}+m_{k}$ (if such a part exists, otherwise $a_{jk}^0 :=0$). Then we put $A^0 :=(a_{jk}^0 )_{j,k=1,\dots,N}$, and the operator $A$ is called parameter-elliptic in the sector ${\mathcal {L}}\subset{\mathbb{C}}$ if $\det(A^0 (x,\xi)-\lambda)\not=0$ for all $(x,\xi,\lambda) \in \overline{\Omega}\times{\mathbb{R}}^{d}\times{\mathcal {L}}$ with $|\xi|+|\lambda|>0$. Then (see~\cite{AgranovichVishik64}) $mN\in2{\mathbb{N}}$, and we can consider a matrix of boundary differential operators, \[ B=(b_{j,k}(x,D_{x}))_{j,k}, \quad 1\leq j\leq mN/2, \quad 1\leq k\leq N, \quad \operatorname{ord} b_{jk}\leq r_{j}+m_{k}, \] with integers $r_{j}\leq m-1$. We define the principal part $B^0 $ of $B$ in the same way as $A^0 $ was defined. We say that the Shapiro--Lopatinskii condition is satisfied if at each $x^{\ast}\in{\partial}\Omega$, after introducing a new frame of Cartesian coordinates with center at $x^{\ast}$ and the $x_d $--axis pointing along the inner normal vector at $x^{\ast}$, the system of ordinary differential equations \begin{gather*} (A^0 (x^{\ast},\xi',D_{x_d })-\lambda)v(x_d )=0, \quad 0\leq x_d <\infty,\\ B^0 (x^{\ast},\xi',D_{x_d })v(x_d )=0, \quad x_d =0, \\ \lim_{x_d \to\infty}v(x_d )=0 \end{gather*} possesses only the trivial solution, for all $(\xi',\lambda)\in{\mathbb{R}}^{n-1}\times{\mathcal {L}}$ with $|\xi'|+|\lambda|>0$. Then the system $(A,B)$ is called a parameter-elliptic boundary value problem in the sector ${\mathcal {L}}\subset{\mathbb{C}}$ if $A$ is parameter-elliptic in ${\mathcal {L}}$, and the Shapiro-Lopatinskii condition holds. Write $B=(B_1,\dots,B_{mN/2})^{\top}$ as a column of rows, and consider the boundary value problem \begin{equation}\label{BVPMixed} \begin{gathered} (A-\lambda)u=f,\quad \text{in } \Omega,\\ \gamma_0B_{j}u=0,\quad \text{on } {\partial}\Omega, \; j=1,\dots,mN/2. \end{gathered} \end{equation} In Faierman~\cite{faierman06}, it has been shown that a number $\lambda_0$ exists such that, for all $\lambda$ from ${\mathcal {L}}$ with $|\lambda|\geq\lambda_0$, and for all $f\in W_p^{m_1}(\Omega)\times\dots\times W_p^{m_{N}}(\Omega)$, a unique solution $u\in W_p^{m+m_1}(\Omega)\times\dots\times W_p^{m+m_{N}}(\Omega)$ to \eqref{BVPMixed} exists, and the estimate \begin{align*} & \sum_{k=1}^{N} \left( \norm{u_{k}}{W_p^{m+m_{k}}(\Omega)} + |\lambda|^{1+m_{k}/m}\norm{u_{k}}{L^p(\Omega)} \right)\\ & \leq C\sum_{k=1}^{N} \left( \norm{f_{k}}{W_p^{m_{k}}(\Omega)} + |\lambda|^{m_{k}/m}\norm{f_{k}}{L^p(\Omega)} \right) \end{align*} holds, with $C$ depending only on $(A,B)$. Having secured the existence of $u$ for large $|\lambda|$, we can now ask under which conditions resolvent estimates for $A$ might exist. \begin{theorem}\label{thm3.8} If $f$ is such that for all $\lambda$ of large modulus the inequality \[ \sum_{j=1}^{N} \big(\norm{u_{j}}{W_p^{m+m_{j}}(\Omega)} +|\lambda|\norm{u_{j}}{W_p^{m_{j}}(\Omega)}\big) \leq C\sum_{j=1}^{N}\norm{f_{j}}{W_p^{m_{j}}(\Omega)} \] holds for all solutions $u$ to \eqref{BVPMixed}, with a constant $C$ independent of $\lambda$, then $\gamma_0B_{j}f\equiv0$ for all $j$ with $r_{j}\leq -1$. \end{theorem} \begin{proof} From $f_{k}\in W_p^{m_{k}}(\Omega)$ and $\operatorname{ord} b_{jk}\leq r_{j}+m_{k}$, we deduce that $B_{j}f\in W_p^{-r_{j}}(\Omega)$, and this has a trace at the boundary for $r_{j}\leq -1$. Pick such an index $j$. Now we can estimate as follows: \begin{align*} & |\lambda|^{\frac{1}{m}(1-\frac{1}{p})} \norm{\gamma_0B_{j}f}{L^p({\partial}\Omega)} = |\lambda|^{\frac{1}{m}(1-\frac{1}{p})} \norm{\gamma_0B_{j}Au}{L^p({\partial}\Omega)} \\ & \leq C \left( \norm{B_{j}Au}{W_p^{1}(\Omega)} + |\lambda|^{\frac{1}{m}}\norm{B_{j}Au}{L^p(\Omega)} \right) \\ & \leq C\sum_{k=1}^{N} \left( \norm{u_{k}}{W_p^{m+m_{k}+r_{j}+1}(\Omega)} + |\lambda|^{\frac{1}{m}}\norm{u_{k}}{W_p^{m+m_{k}+r_{j}}(\Omega)} \right) \\ & \leq C\sum_{k=1}^{N} \left( \norm{u_{k}}{W_p^{m+m_{k}}(\Omega)} + |\lambda|^{\frac{1}{m}}\norm{u_{k}}{W_p^{m+m_{k}-1}(\Omega)} \right) \\ & \leq C\sum_{k=1}^{N} \left( \norm{u_{k}}{W_p^{m+m_{k}}(\Omega)} + |\lambda|\norm{u_{k}}{W_p^{m_{k}}(\Omega)} \right) \\ & \leq C\sum_{k=1}^{N}\norm{f_{k}}{W_p^{m_{k}}(\Omega)}. \end{align*} Sending $\lambda$ to infinity in $\Omega$ then implies $\gamma_0B_{j}f\equiv0$. \end{proof} \section{Applications} As a first application, we mention the linear thermoelastic plate equations in a bounded and sufficiently smooth domain $\Omega\subset\mathbb{R}^n$. The equations have the form \begin{gather*} \partial_t^2 v + \Delta^2 v + \Delta \theta = 0\quad \text{ in } (0,\infty)\times \Omega, \\ \partial_t \theta -\Delta \theta -\Delta \partial_t v = 0 \quad \text{ in } (0,\infty)\times \Omega \end{gather*} subject to the initial conditions $v|_{t=0}=u_0$, $\partial_t v|_{t=0}=u_1$, $\theta|_{t=0}=\theta_0$ and Dirichlet boundary conditions \[ \gamma_0 v = \gamma_0\partial_\nu v = \gamma_0\theta = 0. \] Here $\partial_\nu$ denotes the derivative in the direction of the outer normal $\nu$. In the above system, $v=v(t,x)$ stands for a mechanical variable denoting the vertical displacement of a plate, while $\theta=\theta(t,x)$ stands for a thermal variable describing the temperature relative to a constant reference temperature (see, e.g., \cite{Lag89}, \cite{LT98}, \cite{DRS09}, and references therein). Setting in a standard way $u:= (v, \partial_t v, \theta)^{\top}$, we obtain the following first-order system for $u$: \begin{gather*} \partial_t u - A(D) u = 0 \quad \text{ in } (0,\infty)\times \Omega,\\ B(D) u = 0 \quad\text{ on } (0,\infty)\times\partial\Omega,\\ u|_{t=0}= u_0 \quad \text{in } \Omega, \end{gather*} where \[ A(D)=\begin{pmatrix} 0 & 1 & 0 \\ -\Delta^2 & 0 & -\Delta\\ 0 & \Delta & \Delta \end{pmatrix}, \quad B(D)=\begin{pmatrix} 1 & 0 & 0 \\ \partial_\nu & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}. \] This is a mixed-order system with $\operatorname{ord} a_{jk}(D)\le s_j+m_k$ for $s=(0,2,2)^{\top}$ and $m=(2,0,0)^{\top}$ and $\operatorname{ord} b_{jk}(D) \le r_j+m_k$ for $r=(-2,-1,0)^{\top}$. A natural choice for the $L^p$-realization of $(A(D),B(D))$ seems to be the operator $\mathcal{A}$ defined in the ground space $Y:= W_p^2(\Omega)\times (L^p(\Omega))^2$ by \[ D(\mathcal{A}) :=\{u\in W_p^{4}(\Omega)\times (W_p^{2}(\Omega))^2\colon \gamma_0 u_1= \gamma_0\partial_\nu u_1 = \gamma_0 u_3 = 0 \}, \quad \mathcal{A}u := A(D) u. \] \begin{corollary}\label{ApplicationCorollary} The operator $\mathcal{A}$ does not generate an analytic semigroup on $Y$. \end{corollary} \begin{proof} Assume $\mathcal{A}$ to generate an analytic semigroup on $Y$. Then, by Theorem~\ref{thm3.8}, we have $\gamma_0 B_j f=0$ for all $f\in Y$ and all $j$ with $r_j\le -1$. As $r=(-2,-1,0)^{\top}$, this implies $\gamma_0 f_1 =\gamma_0\partial_\nu f_1=0$ for all $f=(f_1,f_2,f_3)^{\top}\in Y$ which is a contradiction to the definition of the space $Y$. \end{proof} As we have seen in the last proof, Theorem~\ref{thm3.8} suggests to consider the ground space $Y_0$ defined by \[ Y_0:=\{ f\in Y\colon \gamma_0 f_1 = \gamma_0\partial_\nu f_1 = 0 \}. \] Therefore, we define the operator $\mathcal{A}_0$ by \begin{gather*} D(\mathcal{A}_0) :=\{u\in D(\mathcal{A})\colon \mathcal{A} u\in Y_0\} =\{ u\in D(\mathcal{A})\colon \gamma_0 u_2 = \gamma_0\partial_\nu u_2 = 0\}, \\ \mathcal{A}_0 u:= A(D)u. \end{gather*} In fact, this space is the ``correct'' one as can be seen from the following result which is taken from \cite{DRS09}. \begin{theorem} \label{thm4.2} The operator $\mathcal{A}_0$ generates an analytic semigroup on $Y_0$. \end{theorem} Our second application comes from semiconductor physics. The \emph{viscous model of quantum hydrodynamics} is a system of differential equations of the form \begin{equation}\label{QHDProb} \begin{gathered} \partial_{t}n-\operatorname{div} J =\nu\Delta n,\\ \partial_{t}J-\operatorname{div} \big(\frac{J\otimes J}{n}\big) -T\nabla n+n\nabla V+\frac{\varepsilon^{2}}{2}n\nabla \big(\frac{\Delta\sqrt{n}}{\sqrt{n}}\big) =\nu\Delta J-\frac{J}{\tau}, \\ \lambda_{D}^{2}\Delta V =n-C(x), \end{gathered} \end{equation} for $(t,x)\in (0,T_0)\times\Omega$, with $\Omega\subset\mathbb{R}^{d}$, being a domain with smooth boundary, $d=1,2,3$. The initial values are prescribed as $n|_{t=0}=n_0$ and $J|_{t=0}=J_0$. The unknown functions are the scalar valued electron density $n=n(t,x)$, the vector valued density of electrical currents $J=J(t,x)$, and the scalar electric potential $V=V(t,x)$. The scaled physical constants are the electron temperature $T$, the Planck constant $\varepsilon$, the Debye length $\lambda_{D}$, and constants $\nu$, $\tau$ characterizing the interaction of the electrons with crystal phonons. The known function $C=C(x)$ is the so--called doping profile which describes the density of positively charged background ions. An overview of models of this type is given in \cite{ChenDreherBook}. If we omit the terms with $\varepsilon$, $\nu$ and $\tau$, we obtain the well-known Euler equations of fluid dynamics, augmented by a Poisson equation. One choice of boundary conditions on $n$, $J$, $V$ are Dirichlet conditions: \[ \gamma_0n=n_{\Gamma},\quad \gamma_0J=0,\quad \gamma_0V=V_{\gamma}. \] To come to our standard way of writing a system, we define a vector function $u=(n, J^{\top})^{\top}$. Now we observe that $n\nabla\tfrac{\Delta\sqrt{n}}{\sqrt{n}} =\tfrac{1}{2}\nabla \Delta n - \tfrac{1}{2}\operatorname{div}(\frac{(\nabla n)\otimes(\nabla n)}{n})$, hence the construction of the principal part $A^0 $ as presented at the beginning of Section~\ref{MixedOrderSubSec} brings us to the matrix differential operator of size $(1+d)\times(1+d)$ \[ A^0 (D)=\begin{pmatrix} \nu\Delta & \operatorname{div}\\ \frac{\varepsilon^{2}}{4}\nabla \Delta & \nu\Delta I_d \end{pmatrix}, \] with $I_d $ being the $d\times d$ unit matrix. And the principal part $B^0 $ of the boundary conditions for $u$ is \[ B^0 (D)=\begin{pmatrix} 1 & 0 \\ 0 & I_d \end{pmatrix}. \] We find the order parameters as $(s_1,s_{2},\dots,s_{d+1})=(1,2,\dots,2)$, $(m_1,m_{2},\dots,m_{d+1})\\ =(1,0,\dots,0)$ and $(r_1,\dots,r_{d+1})=(-1,0,\dots,0)$. Similarly to the first application, it may seem natural to define an $L^p$-realization $\mathcal{A}^0 $ of $(A^0 (D),B^0 (D))$ in the ground space $Y:=W_p^{1}(\Omega)\times(L^p(\Omega))^{d}$ by \begin{gather*} D(\mathcal{A}^0 ) := \{ u\in W_p^{1}(\Omega)\times(L^p(\Omega))^{d} \colon \gamma_0 u_1=\gamma_0u_{2}=\dots=\gamma_0u_{d+1}=0 \}, \\ \mathcal{A}^0 u :=A^0 (D)u. \end{gather*} However, this operator $\mathcal{A}^0 $ does not generate an analytic semigroup on $Y$, and the proof of this fact runs along the same lines as the proof of Corollary~\ref{ApplicationCorollary}. 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