\documentclass[reqno]{amsart} \pdfoutput=1\relax\pdfpagewidth=8.26in\pdfpageheight=11.69in\pdfcompresslevel=9 \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 127, pp. 1--10.\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-2004/127\hfil The Korn inequality] {The Korn inequality for Jones domains} \author[R. G. Dur\'an \& M. A. Muschietti\hfil EJDE-2004/127\hfilneg] {Ricardo G. Dur\'an, Maria Amelia Muschietti} % in alphabetical order \address{Ricardo G. Dur\'an \hfill\break Departamento de Matem\'atica\\ Facultad de Ciencias Exactas y Naturales\\ Universidad de Buenos Aires\\ 1428 Buenos Aires, Argentina} \email{rduran@dm.uba.ar} \address{Maria Amelia Muschietti \hfill\break Departamento de Matem\'atica\\ Facultad de Ciencias Exactas\\ Universidad Nacional de La Plata\\ Casilla de Correo 172\\ 1900 La Plata, Provincia de Buenos Aires, Argentina} \email{mariam@mate.unlp.edu.ar} \date{} \thanks{Submitted March 1, 2004. Published October 27, 2004.} \thanks{The first author was supported by grant PICT 03-05009 from ANPCyT, by grant \hfill\break\indent PIP 0660/98 from CONICET, and by Fundaci\'on Antorchas \hfill\break\indent The second author was supported by grant 11/X228 from Universidad Nacional de La Plata \hfill\break\indent and by grant PIP 4723/96 from CONICET} \subjclass[2000]{74B05} \keywords{Korn inequality; Jones domains} \begin{abstract} In this paper we prove the Korn inequality, and its generalization to $L^p$, $1
0$, we call $\overline{x}$ its barycenter and for $v\in W^{1,\infty}(\Omega)^n$ we associate with $S$ and $v$ the affine vector field \begin{equation} \label{PS1} P_S(v)(x)=a + B (x-\overline{x}), \end{equation} where $a\in\mathbb{R}^n$ and $B=(b_{ij})\in\mathbb{R}^{n\times n}$ are defined by \begin{equation} \label{PS2} a=\frac1{|S|}\int_S v \quad \mbox{and} \quad b_{ij}= \frac1{2|S|}\int_S \Big(\frac{\partial v_i}{\partial x_j} - \frac{\partial v_j}{\partial x_i}\Big)\,. \end{equation} Observe that, since $b_{ij}=-b_{ji}$, $P_S(v)$ is an ``infinitesimal rigid motion'', i.e., it satisfies \begin{equation} \label{epsilon0} \varepsilon(P_S(v)) = 0\,. \end{equation} Moreover, $a$ and $B$ have been chosen in such a way that \begin{gather} \label{introt} \int_S \Big(\frac{\partial(v_i-P_S(v)_i)}{\partial x_j} - \frac{\partial(v_j-P_S(v)_j)}{\partial x_i}\Big)= 0\,,\\ \label{intv} \int_S (v-P_S(v))) = 0\,. \end{gather} Assume now that the inequality (\ref{korn2}) holds in $S$. Then, in view of (\ref{epsilon0}) and (\ref{introt}) we have \begin{equation} \label{kornenS} \|\nabla (v-P_S(v)))\|_{L^p(S)^{n\times n}}\le C \|\varepsilon(v)\|_{L^p(S)^{n\times n}} \end{equation} where the constant $C$ depends on $p$ and on the shape (but not on the scale!) of $S$ (see Remark \ref{rem1}). Now, from (\ref{intv}) and the Friedrichs-Poincar\'e inequality for functions with vanishing mean value and denoting with $d(S)$ the diameter of $S$, we have \begin{equation} \label{poinca} \|v-P_S(v)\|_{L^p(S)^n} \le C d(S) \|\nabla (v-P_S(v))\|_{L^p(S)^{n\times n}} \end{equation} and therefore, \begin{equation} \label{7} \|v-P_S(v)\|_{L^p(S)^n} \le C d(S) \|\varepsilon(v)\|_{L^p(S)^{n\times n}} \end{equation} where, again, the constant $C$ depends only on $p$ and on the shape of $S$. On the other hand, it is easy to see that \begin{equation} \label{7ymedio} \|\nabla P_S(v)\|_{L^{\infty}(S)^{n\times n}} \le \|\nabla v\|_{L^{\infty}(S)^{n\times n}} \end{equation} and so, \begin{equation} \|\nabla (v-P_S(v))\|_{L^{\infty}(S)^{n\times n}} \le 2 \|\nabla v\|_{L^{\infty}(S)^{n\times n}} \label{8} \end{equation} and therefore, using (\ref{poinca}) with $p=\infty$, we obtain \begin{equation} \|v-P_S(v)\|_{L^{\infty}(S)^n} \le C d(S) \|\nabla v\|_{L^{\infty}(S)^{n\times n}} \label{9} \end{equation} where, also here, the constant $C$ depends only on the shape of $S$. Our extension of $v$ will be constructed following the ideas developed in \cite{J} but using now the polynomials $P_S(v)$. Recall that any open set $\Omega\subset \mathbb{R}^n$ admits a Whitney decomposition into closed dyadic cubes $S_k$ (see \cite{W,S}) , i.e., $\Omega=\cup_k S_k$, such that, if $\ell(S)$ denotes the edgelength of a cube $S$, \begin{gather} \label{cubos1} 1\le\frac{\mathop{\rm dist}(S_k,\partial\Omega)}{\ell(S_k)}\le 4\sqrt{n} \quad\forall k \,,\\ \label{cubos2} S_j^0\cap S_k^0=\emptyset\quad \mbox{if } j\neq k\,, \\ \label{cubos3} \frac14\le\frac{\ell(S_j)}{\ell(S_k)}\le 4 \quad \mbox{if } S_j \cap S_k\neq\emptyset\,. \end{gather} Let $W_1=\{S_k\}$ be a Whitney decomposition of $\Omega$ and $W_2=\{Q_j\}$ one of $(\Omega^c)^0$. We define $$ W_3=\big\{Q_j\in W_2 \, : \, \ell(Q_j) \le \frac{\epsilon\delta}{16n} \big\} $$ It was shown by Jones (see Lemmas 2.4 and 2.8 of \cite{J}) that, for each $Q_j\in W_3$, it is possible to choose a ``reflected'' cube $Q^*_j=S_k \in W_1$ such that $$ 1\le \frac{\ell(S_k)}{\ell(Q_j)} \le 4 \quad \mbox{and} \quad d(Q_j,S_k)\le C\ell(Q_j) $$ and moreover, if $Q_j , Q_k \in W_3$ and $Q_j\cap Q_k \neq \emptyset$, there is a chain $F_{j,k} = \{Q^*_j=S_1,S_2,\cdots,S_m=Q^*_k\}$ (i.e, $S_j\cap S_{j+1} \neq \emptyset$) of cubes in $W_1$, connecting $Q^*_j$ to $Q^*_k$ and with $m\le C(\epsilon,\delta)$. It is known that, associated with a Whitney decomposition, there exists a partition of unity $\{\phi_j\}$ such that $\phi_j\in C^\infty(\mathbb{R}^n)$, $supp\,\,\phi_j\subset \frac{17}{16} Q_j$, $0\le\phi_j\le 1$, $$ \sum_{Q_j\in W_3} \phi_j \equiv 1 \quad \mbox{on} \ \ \bigcup_{Q_j\in W_3} Q_j $$ and $$ |\nabla\phi_j|\le C \ell(Q_j)^{-1} \quad \forall j $$ (see \cite{S,W}). Now, given $v\in W^{1,\infty}(\Omega)^n$, let $P_j=P_{Q^*_j}(v)$ defined as in (\ref{PS1}) and (\ref{PS2}) with $S=Q^*_j$. Then, we define $Ev$, the extension to $\mathbb{R}^n$ of $v$, in the following way, \begin{gather*} Ev=\sum_{Q_j\in W_3} P_j\phi_j \quad \mbox{in } (\Omega^c)^0 ,\\ Ev=v \quad \mbox{in } \Omega . \end{gather*} Since $|\partial\Omega|=0$ (see Lemma 2.3 in \cite{J}), it follows that $Ev$ is defined $p.p.$ in $\mathbb{R}^n$. The arguments of the following lemmas are similar to those in \cite{J}. In particular we will make repeated use of Lemma 2.1 of \cite{J} which says, \begin{lemma} \label{equivalencia} Let $Q$ be a cube and $F , G\subset Q$ be two measurable subsets such that $|F|, |G| \ge \gamma |Q|$ for some $\gamma > 0$. If $P$ is a polynomial of degree $1$ then, $$ \|P\|_{L^p(F)} \le C(\gamma) \|P\|_{L^p(G)}. $$ \end{lemma} \begin{lemma} \label{lemma1} Let $F=\{S_1,\cdots, S_m\}$ be a chain of cubes in $W_1$. Then, \begin{equation} \label{10} \|P_{S_1}(v) - P_{S_m}(v)\|_{L^p(S_1)^n} \le C c(m) \ell (S_1) \|\varepsilon(v)\|_{L^p(\cup_jS_j)^{n\times n}}, \end{equation} and \begin{equation} \label{11} \|P_{S_1}(v) - P_{S_m}(v)\|_{L^{\infty}(S_1)^n} \le C c(m) \ell (S_1) \|\nabla v\|_{L^{\infty}(\cup_jS_j)^{n\times n}}. \end{equation} \end{lemma} \begin{proof} We will use (\ref{7}) with $S$ being a cube or a union of two neighboring cubes. In view of (\ref{cubos3}) there are a finite number, depending only on the dimension $n$, of possible shapes for the union of two neighboring cubes, and so, we can take a uniform constant in (\ref{7}). Using Lemma \ref{equivalencia} we have \begin{align*} &\|P_{S_1}(v) - P_{S_m}(v)\|_{L^p(S_1)^n}\\ &\le \sum_{r=1}^{m-1} \|P_{S_r}(v) - P_{S_{r+1}}(v)\|_{L^p(S_1)^n} \\ &\le c(m) \sum_{r=1}^{m-1} \|P_{S_r}(v) - P_{S_{r+1}}(v)\|_{L^p(S_r)^n} \\ &\le c(m) \sum_{r=1}^{m-1}\{ \|P_{S_r}(v) - P_{S_r\cup S_{r+1}}(v)\|_{L^p(S_r)^n}\\ &\quad+ \|P_{S_r\cup S_{r+1}}(v) - P_{S_{r+1}}(v)\|_{L^p(S_{r+1})^n}\} \\ &\le c(m) \sum_{r=1}^{m-1}\{ \|v - P_{S_r}(v)\|_{L^p(S_r)^n} + \|v - P_{S_{r+1}}(v)\|_{L^p(S_{r+1})^n}\\ &\quad +\|v - P_{S_r \cup S_{r+1}}(v)\|_{L^p(S_r\cup S_{r+1})^n}\}\\ &\le C c(m) \ell(S_1) \|\varepsilon(v)\|_{L^p(\cup_j S_j)^{n\times n}} \end{align*} where we have used (\ref{7}). The proof of (\ref{11}) is analogous using now (\ref{9}). \end{proof} Now, for each $Q_j,Q_k\in W_3$ such that $ Q_j\cap Q_k\neq\emptyset$, we choose a chain $F_{j,k}$ connecting $Q^*_j$ to $Q^*_k$ and with $m\le C(\epsilon,\delta)$ and define $$ F(Q_j)= \bigcup_{Q_k\in W_3 ,\, Q_j\cap Q_k\neq\emptyset} F_{j,k} $$ then, \begin{equation} \label{estrella} \big\|\sum_{Q_k ,\, Q_j\cap Q_k\neq\emptyset} \chi_{\cup F_{j,k}}\big\|_{L^{\infty}(\mathbb{R}^n)} \le C \quad \forall Q_j\in W_3 \end{equation} The following lemmas will allow us to control the norms of $Ev$, $\varepsilon(Ev)$ and $\nabla(Ev)$ in $(\Omega^c)^0$. \begin{lemma} \label{lemma2} For $Q_0\in W_3$ we have \begin{gather} \label{12} \|Ev\|_{L^p(Q_0)^n} \le C \{ \|v\|_{L^p(Q^*_0)^n} + \ell(Q_0) \|\varepsilon(v)\|_{L^p(\cup F(Q_0))^{n\times n}} \}\,,\\ \label{13} \|\varepsilon(Ev)\|_{L^p(Q_0)^{n\times n}} \le C \|\varepsilon(v)\|_{L^p(\cup F(Q_0))^{n\times n}}\,,\\ \label{14} \|Ev\|_{L^{\infty}(Q_0)^n} \le C \{ \|v\|_{L^{\infty}(Q^*_0)^n} + \ell(Q_0) \|\nabla v\|_{L^{\infty}(\cup F(Q_0))^{n\times n}} \}\,,\\ \label{15} \|\nabla (Ev)\|_{L^{\infty}(Q_0)^{n\times n}} \le C \|\nabla v\|_{L^{\infty}(\cup F(Q_0))^{n\times n}} \}\,. \end{gather} \end{lemma} \begin{proof} On $Q_0$ we have $$ Ev=\sum_{Q_j\in W_3} P_j\phi_j $$ Now, since $\sum_{Q_j\in W_3} \phi_j \equiv 1$ on $\cup_{Q_j\in W_3} Q_j$, then $$ \| \sum_{Q_j\in W_3} P_j\phi_j \|_{L^p(Q_0)^n} \le \| P_0 \|_{L^p(Q_0)^n} + \| \sum_{Q_j\in W_3} (P_j- P_0) \phi_j \|_{L^p(Q_0)^n} =I + II $$ Using Lemma \ref{equivalencia} and (\ref{7}), we have \begin{align*} I&= \| P_0 \|_{L^p(Q_0)^n} \le C \| P_0 \|_{L^p(Q^*_0)^n}\\ &\le C \| P_0 - v \|_{L^p(Q^*_0)^n} + \| v \|_{L^p(Q^*_0)^n}\\ &\le C \{ \|v\|_{L^p(Q^*_0)^n} + \ell(Q_0) \|\varepsilon(v)\|_{L^p(Q^*_0)^{n\times n}} \} \end{align*} Now, since on $Q_0$ there are a finite number (depending only on $n$) of non-vanishing $\phi_j$ and $0\le \phi_j \le 1$, to bound $II$ it is enough to bound $\| (P_j - P_0) \|_{L^p(Q_0)^n}$. But, using (\ref{10}) and again Lemma \ref{equivalencia}, we have $$ \| (P_j - P_0) \|_{L^p(Q_0)^n} \le C \| (P_j - P_0) \|_{L^p(Q^*_0)^n} \le C \ell(Q_0) \|\varepsilon(v)\|_{L^p(\cup F_{0,j})^{n\times n}} $$ and therefore, summing up and using (\ref{estrella}) we obtain (\ref{12}). Analogously, we can prove (\ref{14}) using now (\ref{9}) and (\ref{11}). Now, calling $P_j^r$ the components of $P_j$ and recalling that $\varepsilon(P_j)=0$ we have \begin{equation} \label{epsilondelproducto} \varepsilon_{rs}(P_j\phi_j) = \frac12 P^r_j \frac{\partial\phi_j}{\partial x_s} + \frac12 P^s_j \frac{\partial\phi_j}{\partial x_r} \end{equation} On $Q_0$, $$ Ev=P_0 + \sum_{Q_j\in W_3} (P_j- P_0) \phi_j $$ and therefore, since $\varepsilon(P_0)=0$ we have, $$ \varepsilon(Ev)=\sum_{Q_j\in W_3}\varepsilon( (P_j- P_0) \phi_j) $$ but, there are at most $C$ cubes $Q_j$ such that $\phi_j$ does not vanishes in $Q_0$ and these $Q_j$ intersect $Q_0$ and therefore, $\ell(Q_j) \ge \frac14\ell(Q_0)$. Thus, $$ |\nabla \phi_j|\le C \ell(Q_0)^{-1} $$ whenever $\phi_j\neq 0$ for some $x\in Q_0$. Then, for these values of $j$, it follows from (\ref{epsilondelproducto}) and (\ref{10}) that \begin{align*} \|\varepsilon( &(P_j- P_0) \phi_j)\|_{L^p(Q_0)^{n\times n}} \le C \ell(Q_0)^{-1} \|P_j- P_0\|_{L^p(Q_0)^n}\\ &\le C \ell(Q_0)^{-1} \|P_j- P_0\|_{L^p(Q^*_0)^n} \le C \|\varepsilon(v)\|_{L^p(\cup F_{0,j})^{n\times n}} \end{align*} Summing up in $j$, we obtain (\ref{13}). The proof of (\ref{15}) is similar to that of (\ref{13}) but using now (\ref{7ymedio}). Indeed, we have $$ \nabla(Ev)= \nabla P_0 + \sum_{Q_j\in W_3}\nabla((P_j- P_0) \phi_j) $$ and we have to estimate also the terms $\nabla P_0$ and $\nabla(P_j-P_0)$. But, $$ \|\nabla P_0\|_{L^{\infty}(Q_0)^{n\times n}} \le C \|\nabla P_0\|_{L^{\infty}(Q^*_0)^{n\times n}} \le C \|\nabla v\|_{L^{\infty}(Q_0)^{n\times n}}, $$ and \begin{align*} \|\nabla (P_j- P_0)\|_{L^{\infty}(Q_0)^{n\times n}} &\le C \|\nabla (P_j-P_0)\|_{L^{\infty}(Q^*_0)^{n\times n}}\\ &\le C \|\nabla (P_j-P_0)\|_{L^{\infty}(Q^*_0 \cup Q^*_j)^{n\times n}} \le C \|\nabla v\|_{L^{\infty}(Q^*_0 \cup Q^*_j)^{n\times n}}\\ &\le C \|\nabla v\|_{L^{\infty}(\cup F_{j,0})^{n\times n}} \} \end{align*} and so (\ref{15}) holds. \end{proof} \begin{lemma} \label{lemma3} For $Q_0\in W_2 \setminus W_3$ we have \begin{gather} \label{16} \|Ev\|_{L^p(Q_0)^n} \le C \sum_{Q_j\in W_3 ,\, Q_j\cap Q_0 \neq\emptyset} \{ \|v\|_{L^p(Q^*_j)^n} + \|\varepsilon(v)\|_{L^p(Q^*_j)^{n\times n}} \}\,,\\ \label{17} \|\varepsilon(Ev)\|_{L^p(Q_0)^{n\times n}} \le C \sum_{Q_j\in W_3 ,\, Q_j\cap Q_0 \neq\emptyset} \{ \|v\|_{L^p(Q^*_j)^n} + \|\varepsilon(v)\|_{L^p(Q^*_j)^{n\times n}} \}\,,\\ \label{18} \|Ev\|_{L^{\infty}(Q_0)^n} \le C \sum_{Q_j\in W_3 ,\, Q_j\cap Q_0 \neq\emptyset} \{ \|v\|_{L^{\infty}(Q^*_j)^n} + \|\nabla v\|_{L^{\infty}(Q^*_j)^{n\times n}} \}\,,\\ \label{19} \|\nabla(Ev)\|_{L^{\infty}(Q_0)^{n\times n}} \le C \sum_{Q_j\in W_3 ,\, Q_j\cap Q_0 \neq\emptyset} \{ \|v\|_{L^{\infty}(Q^*_j)^n} + \|\nabla v\|_{L^{\infty}(Q^*_j)^{n\times n}} \} \end{gather} \end{lemma} \begin{proof} If $\phi_j$ does not vanish on $Q_0$ then $Q_j\cap Q_0 \neq\emptyset$ and so, $$ \ell(Q_j) \ge \frac14 \ell(Q_0) \ge \frac{\epsilon\delta}{64 n} $$ therefore, on $Q_0$, we have $$ |Ev|= |\sum_{Q_j\in W_3 ,\,Q_j\cap Q_0 \neq\emptyset} \phi_j P_j| \le C \sum_{Q_j\in W_3 ,\, Q_j\cap Q_0 \neq\emptyset} |P_j|, $$ but $$ \|P_j\|_{L^p(Q_0)^n} \le C \|P_j\|_{L^p(Q^*_0)^n} \le C \{ \|v-P_j\|_{L^p(Q^*_0)^n} + \|v\|_{L^p(Q^*_0)^n} \}. $$ Now, since $\Omega$ is bounded, $\ell(Q_j^*)$ is bounded by a constant depending only on $\Omega$ and therefore, using (\ref{7}), we obtain \begin{equation} \label{cotadePj} \|P_j\|_{L^p(Q_0)^n} \le C \{ \|\varepsilon(v)\|_{L^p(Q^*_0)^{n\times n}} + \|v\|_{L^p(Q^*_0)^n} \} \end{equation} and therefore, (\ref{16}) is proved. On the other hand, on $Q_0$ we have $$ \varepsilon_{rs}(Ev) = \frac12 \sum_{Q_j\in W_3,\,Q_j\cap Q_0 \neq\emptyset} \{ P^r_j \frac{\partial\phi_j}{\partial x_s} + P^s_j \frac{\partial\phi_j}{\partial x_r}\} $$ but, $\ell(Q_j) \ge \frac{\epsilon\delta}{64 n}$, and so $|\nabla\phi_j|\le C$ and (\ref{17}) follows using again (\ref{cotadePj}). Finally, (\ref{18}) and (\ref{19}) are obtained by similar arguments, using now (\ref{7ymedio}). \end{proof} \begin{corollary} If $v\in W^{1,\infty}(\Omega)$ then \begin{equation} \label{cota1} \|Ev\|_{L^p((\Omega^c)^0)^n} + \|\varepsilon(Ev)\|_{L^p((\Omega^c)^0 )^{n\times n}} \le C \{ \|v\|_{L^p(\Omega)^n} + \|\varepsilon(v)\|_{L^p(\Omega)^{n\times n}} \} \end{equation} and \begin{equation} \label{cota2} \|Ev\|_{W^{1,\infty}((\Omega^c)^0 )^n} \le C \|v\|_{W^{1,\infty}(\Omega )^n} \end{equation} \end{corollary} \begin{proof} It follows immediately from Lemmas \ref{lemma2} and \ref{lemma3} by summing up over all $Q_0\in W_2$. \end{proof} We can now state the main theorem which follows from the results above and arguments given in \cite{J}. \begin{theorem} \label{principal} If $\Omega$ is a bounded $(\epsilon,\delta)$ domain and $1