\documentclass[twoside]{article} \pagestyle{myheadings} \markboth{Lavrentiev Phenomenon in Microstructure Theory\hfil EJDE--1996/06}% {EJDE--1996/06\hfil Matthias Winter\hfil} \long\def\onefigure#1#2#3{% #1 picture, #2 caption #3 label \begin{figure*}[htb] \begin{center}#1\end{center} \caption{#2}\label{#3} \end{figure*} } %end onefigure def \def\Ipe#1{\def\IPEfile{#1}\input{\IPEfile}} \usepackage{amsmath} \usepackage{amsthm} \usepackage{amsfonts} \newcommand{\e}{\varepsilon} \newcommand{\epr}{$\qed$} \renewcommand{\theequation}{\arabic{section}.\arabic{equation}} \newcommand{\pf}{\paragraph{Proof.}} \newcommand{\pix}[1]{{\leavevmode\epsfbox{#1.ps}}} \newtheorem{lemma}{Lemma}[section] \newtheorem{proposition}[lemma]{Proposition} \newtheorem{theorem}[lemma]{Theorem} \newtheorem{corollary}[lemma]{Corollary} \newtheorem{remark}[lemma]{Remark} \newtheorem{definition}[lemma]{Definition} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol.\ {\bf 1996}(1996), No.\ 06, pp. 1--12. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113} \vspace{\bigskipamount} \\ Lavrentiev Phenomenon in Microstructure Theory \thanks{ {\em 1991 Mathematics Subject Classifications:} 49K40, 73S10.\newline\indent {\em Key words and phrases:} Singular perturbation, Lavrentiev phenomenon, \newline\indent martensitic phase transformation. \newline\indent \copyright 1996 Southwest Texas State University and University of North Texas.\newline\indent Submitted April 30, 1996. Published August 22, 1996.} } \date{} \author{Matthias Winter} \maketitle \begin{abstract} A variational problem arising as a model in martensitic phase transformation including surface energy is studied. It explains the complex, multi-dimensional pattern of twin branching which is often observed in a martensitic phase near the austenite interface. We prove that a Lavrentiev phenomenon can occur if the domain is a rectangle. We show that this phenomenon disappears under arbitrarily small shears of the domain. We also prove that other perturbations of the problem lead to an extinction of the Lavrentiev phenomenon. \end{abstract} \section{Introduction} \setcounter{equation}{0} Phase transitions in solids often involve structure on a microscale. In martensitic phase transformation for example this is quite well understood. A common approach is by elastic energy minimization (see Ball and James \cite{BJ87,BJ92} for a geometrically nonlinear theory or Khachaturyan, Shatalov and Roitburd \cite{Kh83,KS69,Ro78} for a geometrically linear theory). The stored energies are typically nonconvex (and not quasiconvex) and so the variational integrals involved are typically not lower semicontinuous. Therefore the minimum is not attained. However, there exist minimizing sequences, which involve finer and finer oscillations describing the microstructure in the solid. Considering elastic energy alone one is capable of predicting many properties of the microstructure, for example the layering directions in twinned patterns or the lattice orientation of the different phases. However, other features such as lengthscales are still arbitrary. If also interfacial energy is incorporated into the model these can be determined, too. We consider two ways to represent interfacial energy. The first is by adding a singular perturbation involving higher order gradients, the second is by essentially adding the surface area of the interfaces. In this paper we revisit a model which was introduced and analyzed by Kohn and M\"uller \cite{KM92,KM94,KM95}. The model is as follows. Minimize \begin{equation} E^{\epsilon}(u)=\int_{R_L}u_x^2+(u_y^2-1)^2+\epsilon^2 u_{yy}^2\,dx\,dy \label{prob} \end{equation} subject to \[ u=0 \mbox{ for } x=0 \] where $R_L=(0,L)\times(0,1)$. \onefigure{\Ipe{fig1.ps}}{A rectangular domain ($\Omega=R_L$)}{rect} The double-well potential $u_x^2+(u_y^2-1)^2$ represents elastic energy of the martensite, the preferred values $\nabla u=(0,\pm1)$ being the stress-free states of two different variants of martensite. The higher-order term $\epsilon^2 u_{yy}^2$ describes interfacial energy by singular perturbation. The boundary $x=0$ represents the austenite--twinned-martensite interface. The boundary condition $u=0$ for $x=0$ refers to elastic compatibility with the austenite phase in the extreme case of complete rigidity of the austenite. The variational problem (\ref{prob}) is closely related to the following one. Minimize \begin{equation} I^{\epsilon}(u)=\int_{R_L} u_x^2+\epsilon|u_{yy}|\,dx\,dy \label{probb} \end{equation} subject to \[ |u_y|=1 \mbox{ a.e.,} \hspace{1cm} u=0 \mbox{ for }x=0. \] (The precise class of admissible functions will be introduced in section 2.) Note that in both formulations (\ref{prob}) and (\ref{probb}) of the variational problem the surface terms consider only changes of $u$ in $y$-direction. To simplify the presentation other components are neglected since the transition zones or interfaces, respectively, between the two variants of martensite are expected to be essentially horizontal. Our results, in particular Theorem \ref{lav}, remain valid also without this approximation. There is no rigorous proof of a relationship between the two formulations of the problem. For a heuristic connection note that, following Modica \cite{Mo87}, \begin{eqnarray*} \int_{x=x_0} (u_y^2-1)^2+\epsilon^2u_{yy}^2\,dy &\geq& \int_{x=x_0} 2\epsilon |u_y^2-1|\,|u_{yy}|\,dy \\ &=&\int_{x=x_0} 2\epsilon |H(u_y)_y|\,dy \end{eqnarray*} where $H(t)$ is a primitive of $|t^2-1|$. The inequality becomes sharp if $\epsilon u_{yy}=\pm(u_y^2-1)$, i.e.~if in the layer where $u_y$ changes between $\pm1$ one has got the appropriate profile. Note that the unknowns of $I^{\epsilon}$ are the (sharp) interfaces where $u_y$ changes its value between $\pm1$, and $1/2 \int_0^1|u_{yy}|\,dy$ counts the number of these changes along the segment $x=\mbox{const}$, $0\leq y\leq1$. We will present a striking difference between the two formulations of the problem, namely that a Lavrentiev phenomenon holds for the ``sharp'' formulation (\ref{probb}) but not for the ``diffusional'' one (\ref{prob}). It was shown in \cite{KM92,KM94,KM95} that for energy minimization of elastic and interfacial energy it is not enough to consider only a one-dimensional twinned pattern. On the contrary, in this situation it is necessary to study complex, two-dimensional patterns which are asymptotically self-similar. A rigorous analysis is performed in the context of formulation (\ref{probb}) of the variational problem. See also Schreiber \cite{Sch94} who extended many of the results to the situation of (\ref{prob}). In this paper we show that for the variational problem (\ref{probb}) a ``{\it Lavrentiev phenomenon}'' occurs. Our main result is as follows. In the class $W^{1,\infty}(R_L)$ there is not even a function possessing finite energy in contrary to the class $H^1(R_L)$. On the other hand, this Lavrentiev phenomenon does not occur if $\Omega$ is a parallelogram. We prove this explicitly giving an example of a function in $W^{1,\infty}(\Omega)$ having finite energy. Note that a rectangle is mapped onto a parallelogram by an arbitrarily small shear. Thus the behavior observed here depends on changes of the domain in a highly singular way. To our knowledge this example is the first where such a highly singular behavior of the Lavrentiev phenomenon on changes of the domain has been observed. We show that this Lavrentiev phenomenon also vanishes if we consider the ``diffusional'' variational problem (\ref{prob}) instead of the ``sharp'' one (\ref{probb}).. Furthermore, we prove that if we omit the surface area term in (\ref{probb}) and study the energy functional \[ I^{\e}(u)=\int_{R_L}u_x^2 \,dx \,dy \] subject to \[ |u_y|=1 \mbox{ a.e.,} \hspace{1cm} u=0\mbox{ for }x=0 \] the Lavrentiev phenomenon also disappears. This shows that the introduction of surface energy into the model not only captures new physical features but also changes the problem in a fundamental way thus highlighting the importance of considering surface energy effects. A refinement of our results would be question: Is the minimal value the same for functions chosen in $H^1$ or in $W^{1,\infty}$? Our results clearly show that this not the case for a rectangular domain and the ``sharp'' formulation since the first is finite, the latter is infinite. We expect that in case the domain is a parallelogram and/or for the ``diffusional'' formulation the minimal values are the same. But to our knowledge these are open questions. In a general context the term Lavrentiev phenomenon is used to describe that the value of the minimum of a variational problem increases strictly if the admissibility class $W^{1,p}(\Omega)$ is replaced by $W^{1,q}(\Omega)$ where $\Omega$ is a bounded domain and $1\leq p0$ such that for $\e$ sufficiently small \begin{equation} c\e^{2/3}L^{1/3}\leq \min I^{\e} \leq C\e^{2/3}L^{1/3}. \end{equation} \label{scal} \end{theorem} We show that if we restrict the admissibility class to the set of Lipschitz functions \begin{eqnarray*} {\cal B}_0&=&{\cal A}_0 \cap W^{1,\infty}(R_L)\\ &=&\{u\in W^{1,\infty}(R_L):\,|u_y|=1 \mbox{ a.e., }u_{yy} \mbox{ is a Radon measure on }R_L \\ &&\mbox{ with finite mass, } u=0 \mbox { for } x=0\} \end{eqnarray*} this statement is no longer true. In fact, we prove the following \begin{theorem} If $\Omega=R_L$ then for all functions $u\in{\cal B}_0$ $I^{\e}(u)=\infty$. \label{lav} \end{theorem} \begin{remark} It is easy to see that for all $p\in [1,\infty)$ the class \begin{eqnarray*} &\{u\in W^{1,p}(R_L):\,|u_y|=1 \mbox{ a.e., }u_{yy} \mbox{ is a Radon measure on }R_L& \\ &\mbox{ with finite mass, } u=0 \mbox { for } x=0\}& \end{eqnarray*} contains a function $u$ such that $I^{\e}(u)<\infty.$ An example for this is obtained by modifying {\bf Example 3.1} below such that \[\begin{array}{ll} \theta\in \left(\frac{1}{4},\frac{1}{2}\right) & \mbox{if }1\leq p\leq 2, \\ \\ \theta\in \left(2^{p/(1-p)},\frac{1}{2}\right) & \mbox{if }22^{p/(1-p)}.\] Furthermore, note that \begin{eqnarray*}\int_{R_L}\e |u_{yy}|\,dy\,dx &=&\e\sum_{i=0}^{\infty}\int_{x_1}^L\int_0^{2^i} 2^{i}|u_{yy}|2^{-i}\theta^i\,dy\,dx\\ &=&\e\sum_{i=0}^{\infty} (2\theta)^{i}\int_{x_1}^L\int_0^1|u_{yy}|\,dy\,dx\end{eqnarray*} and the series is convergent if and only if \[\theta<\frac{1}{2}.\] \end{remark} \paragraph{Proof of Theorem \ref{lav}.} Assume that there is a constant $K>0$ such that \[ |\nabla u|\leq K \hspace{1cm}\mbox{ for a.e. }x\in R_{L}. \] Then we have by the Cauchy-Schwarz inequality \[u^2(l,y)=\left(\int_0^l 1\cdot u_x(x,y)\,dx\right)^2 \leq \int_0^l 1^2 \,dx \cdot\int_0^l u_x^2(x,y)\,dx. \] This implies the following Poincar\'e inequality \begin{equation} \int_0^1 u^2(l,y)dy\leq l \int_0^l\int_0^1 |\nabla u(x,y)|^2 \,dx \,dy\leq CK^2l^2. \hspace{1cm} \label{poin} \end{equation} for all $l\in (0,L]$. Next we use a ``zig-zag'' inequality which was proved by Kohn and M\"uller \cite{KM94}. \begin{lemma} Let $f\in W^{1,\infty}(0,1).$ Assume that $|f'|=1$ a.e. and that $f'$ changes sign $N$ times. Then \[ \int_0^1 f^2 \,dx \geq \frac{1}{12}(N+1)^{-2}= \frac{1}{12}\left(\frac{1}{2}\int_0^1|f''|dx+1\right)^{-2}. \] \label{zigzag} \end{lemma} Lemma \ref{zigzag} implies \begin{equation} \frac{1}{12}\left(\frac{1}{2}\int_0^1|u_{yy}(l,y)|dy +1\right)^{-2}\leq \int_0^1 u^2(l,y)dy. \label{zagzig} \end{equation} Combining (\ref{poin}) and (\ref{zagzig}) we get \[ \int_0^1 |u_{yy}(l,y)|dy\geq C K^{-1}l^{-1}-2 \] where $C$ is independent of $K$ and $l$. After integration we have \[ \int_0^L\int_0^1 \e |u_{yy}(l,y)|\, dy\, dl\geq C \int_0^L l^{-1}\, dl-2\e L=\infty. \] This implies Theorem \ref{lav}. \epr We now assume that the domain is a parallelogram. To simplify the presentation assume that the parallelogram has interior angles of $\pi/4$ and $3\pi/4$. But note that our method also works for other angles (except for $\pi/2$, of course). Set $\Omega=\{(x,y):\,y