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% Author Package file for use with AMS-LaTeX 1.2
\controldates{20-AUG-2001,20-AUG-2001,20-AUG-2001,20-AUG-2001}
 
\documentclass{era-l}
\issueinfo{7}{10}{}{2001}
\pagespan{72}{78}
\PII{S 1079-6762(01)00096-8}
\copyrightinfo{2001}{American Mathematical Society}

\newtheorem*{theorem*}{Theorem}

\numberwithin{equation}{section}



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\begin{document}

\title[Fully explicit quasiconvexification]{Fully explicit quasiconvexification of the
mean-square deviation of the gradient of the state in optimal design}

\author{Pablo Pedregal}
\address{Departamento de Matem\'aticas, ETSI Industriales, Universidad de
Castilla-La Mancha, 13071 Ciudad Real, Spain}
\email{ppedrega@ind-cr.uclm.es}
\thanks{I would like to acknowledge several stimulating
conversations with R. Lipton concerning the type of optimal design problems
considered here and to J. C. Bellido for carrying out various initial
computations.
 I also appreciate the criticism of several referees
which led to the improvement of several aspects of this note.}

\subjclass[2000]{Primary 49J45, 74P10}
\date{March 15, 2001}
\revdate{June 27, 2001}

\commby{Stuart Antman}

\begin{abstract}
We explicitly compute the quasiconvexification of the
resulting integrand associated with the mean-square deviation of the
gradient of the
state with respect to a given target field, when the underlying optimal design
problem in conductivity is reformulated as a purely 
variational problem. What is
remarkable, more than the formula itself, is the fact that it can be shown
to be the
full quasiconvexification.
\end{abstract}

\maketitle

\section{Introduction}
In a number of recent papers
(\cite{BellidoPedregalA}, \cite{PedregalN}), a variational approach to
optimal design and structural optimization has been proposed as an
alternative to the analysis of some of these problems. In particular, we
would like to focus in this note on one such typical situation in
conductivity, where two different conducting materials, with
conductivities $\ag$ and $\bg$,
$0<\ag<\bg$, are to be mixed, to fill out a simply-connected design domain
$\Og\subset\R^2$, so as to minimize the integral of the square of the
gradient of the
electric potential $u$ over
$\Og$. In more precise terms, given $\Og$, $u_0\in H^1(\Og)$, $f\in L^2(\Og)$, $\ag$,
$\bg$, as above, we want to
\[
\hbox{Minimize}\q  I(\jg)=\int_\Og\myab\nabla u(x)-f(x)\cb^2\,dx,
\]
where $\jg$ is the characteristic function of a subset of $\Og$ and $u\in
H^1(\Og)$ is
the unique solution of
\begin{equation}
\begin{split}
\mydiv\myap (\ag\jg(x)+\bg(1-\jg(x)))\nabla u(x)\mycp&=0\q\hbox{in
}\Og,\\
 u&=u_0\q\hbox{on }\partial\Og.
\end{split}
\nonumber
\end{equation}

Such problems are known to lack optimal solutions within the class of
characteristic functions since the pioneering work \cite{MuratE}. Lately,
this particular problem has been studied (\cite{Grabovsky},
\cite{LiptonVeloA},
\cite{TartarK}) by trying to extend the ideas of homogenization to tackle
the dependence on derivatives of states. We will further comment on this
in the last section. For a recent, very good introduction to
homogenization and its application to optimal design see \cite{TartarM}.
Our approach here is different and tries to avoid the nonlocal nature of
the equilibrium law by introducing a suitable potential as a new
independent field (see the references cited above). It is also interesting
to point out that in the particular case $f\equiv0$ the optimal solution
of the problem corresponds to the choice $\jg\equiv0$ or $\jg\equiv1$
(either pure phase) since in this case we obtain the harmonic function
with boundary values $u_0$. In a slightly different context the case
$f\equiv0$ has also been shown to have  an optimal solution by Lipton (see
\cite{Grabovsky}). For
$f$ not identically zero, it is plausible that optimal solutions do not exist.

Under the hypothesis of simple connectedness of $\Og$, there exists
a potential $v\in H^1(\Og)$ such that
\[
\mydiv\myap (\ag\jg(x)+\bg(1-\jg(x)))\nabla u(x)\mycp=0\q\hbox{in
}\Og
\]
is equivalent to
\[
(\ag\jg(x)+\bg(1-\jg(x)))\nabla u(x)+T\nabla v(x)=0,
\]
where $T$ is the counterclockwise $\pg/2$-rotation in the plane. If we
collect both
$u$ and $v$ in a single vector field $U=(u, v)$, it is not hard to realize
that our initial
optimal design problem is equivalent to
\[
\hbox{Minimize}\q I(U)=\int_\Og W(x, \nabla U(x))\,dx
\]
subject to
\[
U\in H^1(\Og),\q U^{(1)}=u_0\q\hbox{on }\partial\Og,
\]
where the density
\[
W:\Mdos\to\R^*=\R\cup\al+\infty\mycl
\]
is defined by
\[
W(x, A)=\begin{cases}
\myab A^{(1)}-f(x)\cb^2& \text{if
$A\in\Lg_\ag\cup\Lg_\bg$},\\
 +\infty& \text{otherwise.}
\end{cases}
\]
Here $U^{(i)}$, $i=1, 2$, denotes the $i$-th component of $U$, $A^{(i)}$,
$i=1, 2$,
denotes the $i$-th row of $A$, and $\Lg_{\cg}$ denotes the two-dimensional
subspace
of matrices defined by
\[
\Lg_\cg=\al A\in\Mdos: \cg A^{(1)}+TA^{(2)}=0\mycl.
\]

Since it is equivalent to our original optimal design problem, this new vector
variational problem does not admit optimal solutions. In such cases,
however, its
relaxation (\cite{DacorognaH}) usually provides all the information needed
to understand and approximate the optimal behavior. This relaxation amounts to
computing the quasiconvexification of $W$ and  determining optimal Young
measures (\cite{PedregalI}). This is the main accomplishment of this work.

\begin{theorem*}
Put
\begin{align}
g(A)&=\ag^2\bg^2\myab A^{(1)}\cb^4+\myab A^{(2)}\cb^4+
(\ag^2+6\ag\bg+\bg^2)\det A\ ^2\nonumber\\
&\quad-2\ag\bg\myab A^{(1)}\cb^2\myab A^{(2)}\cb^2
-2\ag\bg(\ag+\bg)\myab A^{(1)}\cb^2\det A-2(\ag+\bg)\myab A^{(2)}\cb^2\det
A,\nonumber\\
h(A)&=(\ag+\bg)\det A-\ag\bg\myab A^{(1)}\cb^2-\myab
A^{(2)}\cb^2,\nonumber
\end{align}
and consider the set of matrices
\[
\Gg=\al A\in\Mdos: h(A)\ge 0, g(A)\ge0\mycl.
\]
Then the quasiconvexification $QW$ of $W$ is given by
\begin{align*}
QW(x, A)&=\f1{2\ag\bg}\ac\ag\bg\myab A^{(1)}\cb^2-\myab
A^{(2)}\cb^2+(\ag+\bg)\det A-\myr{g(A)}\cc\\
&\quad-2A^{(1)}\cdot f(x)+\myab f(x)\cb^2
\end{align*}
if $A\in\Gg$, and
\[
QW(x, A)=+\infty
\]
otherwise. Moreover the unique optimal Young measure providing the value of this
quasiconvexification, when $A$ does not belong to $\Lg_\ag\cup\Lg_\bg$, is the
first-order laminate
\[
t\dg_{A^\ag}+(1-t)\dg_{A^\bg}
\]
where
\begin{align}
&t=\f12+\f1{2(\bg-\ag)\det A}\ac\ag\bg\myab A^{(1)}\cb^2-\myab
A^{(2)}\cb^2+\myr{g(A)}\cc,\nonumber\\
&A^\ag=
\begin{pmatrix}
z\\ \ag Tz\end{pmatrix},\q z=\f1t\f1{\bg-\ag}\myap\bg
A^{(1)}+TA^{(2)}\mycp,\nonumber\\
&A^\bg=\begin{pmatrix} w\\ \bg Tw\end{pmatrix},\q
w=\f1{(1-t)}\f1{\ag-\bg}\myap\ag A^{(1)}+TA^{(2)}\mycp.\nonumber
\end{align}
\end{theorem*}

\section{Sketch of proof}
We divide the proof in several steps.



Step 1. We know that in computing $QW(x, A)$ we must care about
(homogeneous) gradient Young measures supported in the set where $W$ is finite
and with first moment $A$ (\cite{PedregalI}). Therefore, let $\myng$ be such
a gradient Young measure supported in
$\Lg_\ag\cup\Lg_\bg$ with first moment $A$. Naturally we can decompose $\myng$
into two parts
\[
\myng=s\myng_\ag+(1-s)\myng_\bg,\q s\in[0, 1],
\]
where
\[
\supp(\myng_\ag)\subset\Lg_\ag, \q \supp(\myng_\bg)\subset\Lg_\bg.
\]
In the same way, set
\[
A_\ag=\int_{\Lg_\ag} F\,d\myng_\ag(F)\in\Lg_\ag,\q
A_\bg=\int_{\Lg_\bg} F\,d\myng_\bg(F)\in\Lg_\bg.
\]
It is also easy to check the following:
\begin{gather}
\det A=-A^{(1)}\cdot TA^{(2)},\nonumber\\
\det A=\ag\myab A^{(1)}\cb^2\q\hbox{if }A\in\Lg_\ag,\nonumber\\
\det A=\bg\myab A^{(1)}\cb^2\q\hbox{if }A\in\Lg_\bg,\nonumber\\
\det(sA_\ag+(1-s)A_\bg)=s\det A_\ag+(1-s)\det A_\bg-s(1-s)
\det(A_\ag-A_\bg).\nonumber
\end{gather}
This last formula for the determinant is only valid for $2\times2$ matrices.

We pretend to make use of the weak continuity of $\det$ and see what conclusion
we can reach. Indeed, we should have
\[
\int_\Mdos \det F\,d\myng(F)=\det\myap\int_\Mdos F\,d\myng(F)\mycp.
\]
By using all the decompositions and all the formulas written above, it is
elementary
to arrive at
\begin{gather}
\ag s\int_{\Lg_\ag}\myab F^{(1)}\cb^2\,d\myng_\ag(F)+\bg(1-s)\int_{\Lg_\bg}\myab
F^{(1)}\cb^2\,d\myng_\bg(F)\nonumber\\
=\ag s\myab \int_{\Lg_\ag}F^{(1)}\,d\myng_\ag(F)\cb^2+\bg(1-s)\myab\int_{\Lg_\bg}
F^{(1)}\,d\myng_\bg(F)\cb^2-s(1-s)\det(A_\ag-A_\bg).\nonumber
\end{gather}
By Jensen's inequality, we conclude that
\[
\det(A_\ag-A_\bg)\le0.
\]

Step 2. Consider the optimization problem
\[
\hbox{Minimize}\q t\myab A_\ag^{(1)}-f(x)\cb^2+(1-t)\myab A_\bg^{(1)}-f(x)\cb^2
\]
subject to
\begin{gather}
A=tA_\ag+(1-t)A_\bg,\q A_\ag\in\Lg_\ag, \q A_\bg\in\Lg_\bg,\q  t\in[0, 1]
\nonumber\\
\det(A_\ag-A_\bg)\le0.\nonumber
\end{gather}
Notice that the objective functional, under these constraints, can be written
as
\[
t\myab A_\ag^{(1)}\cb^2+(1-t)\myab A_\bg^{(1)}\cb^2-2
A^{(1)}\cdot f(x)+\myab f(x)\cb^2,
\]
so that we will be concerned about the optimization problem
\[
\hbox{Minimize}\q t\myab A_\ag^{(1)}\cb^2+(1-t)\myab A_\bg^{(1)}\cb^2
\]
subject to
\begin{gather}
A=tA_\ag+(1-t)A_\bg,\q A_\ag\in\Lg_\ag,\q A_\bg\in\Lg_\bg,\q t\in[0,
1]\nonumber\\
\det(A_\ag-A_\bg)\le0,\nonumber
\end{gather}
bearing in mind that the optimal value must be corrected in the end by the term
\[
-2A^{(1)}\cdot f(x)+\myab f(x)\cb^2.
\]
We drop the $x$-dependence of $QW$ in what follows.

By writing
\[
A_\ag=\begin{pmatrix} z\\ \ag Tz\end{pmatrix},\q
A_\bg=\begin{pmatrix} w\\ \bg Tw\end{pmatrix}
\]
for certain vectors $z$, $w$, it is elementary to find
\begin{gather}
z=\f1t\f1{\bg-\ag}\myap\bg A^{(1)}+TA^{(2)}\mycp,\nonumber\\
w=\f1{1-t}\f1{\ag-\bg}\myap\ag A^{(1)}+TA^{(2)}\mycp.\nonumber
\end{gather}
For simplicity, let us put
\[
A_1=\f1{\bg-\ag}\myap\bg A^{(1)}+TA^{(2)}\mycp,\q
A_2=\f1{\ag-\bg}\myap\ag A^{(1)}+TA^{(2)}\mycp.
\]
Then we can rewrite the above optimization problem in the form
\[
\hbox{Minimize}\q \myab A_1\cb^2\f1t+\myab A_2\cb^2\f1{1-t}
\]
subject to
\[
t\in(0, 1),\q \myab A_1\cb^2\f\ag{t^2}+\myab A_2\cb^2\f\bg{(1-t)^2}-A_1\cdot
A_2\f{\ag+\bg}{t(1-t)}\le0.
\]
Notice that this last expression is precisely $\det(A_\ag-A_\bg)$ and that the
vectors $A_1$ and $A_2$ are constant.
For future reference, put
\[
\vfg(t)=\myab A_1\cb^2\f\ag{t^2}+\myab A_2\cb^2\f\bg{(1-t)^2}-A_1\cdot
A_2\f{\ag+\bg}{t(1-t)}.
\]
Since the objective function for this new
formulation is convex in $t$, it tends to $+\infty$ as $t\to0^+$ and
$t\to1^-$, and
since the function determining the constraint is continuous, the minimum value
sought will correspond to equality in the restriction, provided that the
point of
absolute minimum of the objective function on the whole interval $(0, 1)$, $t_0$, is
such that
\[
\myab A_1\cb^2\f\ag{t_0^2}+\myab A_2\cb^2\f\bg{(1-t_0)^2}-A_1\cdot
A_2\f{\ag+\bg}{t_0(1-t_0)}\ge0.
\]
This is indeed an elementary calculus exercise. Since
\[
t_0=\f{\myab A_1\cb}{\myab A_1\cb+\myab A_2\cb},
\]
the previous expression simplifies to
\[
(\ag+\bg)\myap \myab A_1\cb+\myab A_2\cb\mycp^2\myap 1-\f{A_1}{\myab A_1\cb}\cdot
\f{A_2}{\myab A_2\cb}\mycp\ge0.
\]
Note that $A_i$ cannot vanish unless $A\in\Lg_\ag\cup\Lg_\bg$.



Step 3. By putting together steps 1 and 2, we conclude that if $\myng$ is a
gradient
Young measure supported in the set where $W$ is finite and having first moment
$A$, then, by decomposing $\myng$ as in step 1 and using Jensen's inequality
on each
part, we have
\begin{eqnarray}
\langle W, \myng\rangle&\ge&\min_t\al  \myab A_1\cb^2\f1t+\myab
A_2\cb^2\f1{1-t}:
\vfg(t)\le0\mycl\nonumber\\
&=&\min_t\al  \myab A_1\cb^2\f1t+\myab A_2\cb^2\f1{1-t}:
\vfg(t)=0\mycl\nonumber\\
&\ge&QW(A).\nonumber
\end{eqnarray}
The last inequality is correct because when $\vfg(t)=0$ we obtain a first-order
laminate. By taking the infimum in $\myng$ we get
\[
QW(A)=\min_t\al  \myab A_1\cb^2\f1t+\myab A_2\cb^2\f1{1-t}:
\vfg(t)=0\mycl.
\]
Notice how first order laminates are the only possibility for which we can have
equality.



Step 4. Computation of the previous minimum.
The equation $\vfg(t)=0$  is quadratic in $t$. Indeed, it can be rewritten as
\begin{equation}
\ag(1-t)^2\myab A_1\cb^2+\bg t^2\myab A_2\cb^2-(\ag+\bg)t(1-t)A_1\cdot A_2=0.
\end{equation}
The value of this parabola for $t=0$ and $t=1$ is positive if $A$ does not
belong to
either $\Lg_\ag$ or $\Lg_\bg$.
In order for this quadratic equation to have real roots
in the interval $(0, 1)$, we need to demand the discriminant to be
nonnegative, the
leading coefficient to be (strictly) positive and the vertex to belong to
$(0, 1)$.  After
a few computations we also have
\begin{align*}
&\myap\ag\myab A_1\cb^2+\bg\myab A_2\cb^2+(\ag+\bg) A_1\cdot A_2\mycp
t^2\\
&\qquad-\myap2\ag\myab A_1\cb^2+(\ag+\bg) A_1\cdot A_2\mycp t
+\ag \myab A_1\cb^2=0,
\end{align*}
or even further, bearing
in mind the expressions for the vectors $A_1$ and $A_2$,
\begin{align*}
\det A\  t^2&-\f1{\bg-\ag}\myap\ag\bg\myab A^{(1)}\cb^2-\myab
A^{(2)}\cb^2+(\bg-\ag)\det
A\mycp t\\ 
&+ \f1{(\bg-\ag)^2}\myap\ag\bg^2\myab A^{(1)}\cb^2+\ag\myab
A^{(2)}\cb^2-2\ag\bg\det A\mycp=0.
\end{align*}
Those three conditions mentioned above amount to
\[
g(A)\ge0,\q (\ag+\bg)\det A>\myab\ag\bg\myab A^{(1)}\cb^2-\myab A^{(2)}\cb^2\cb.
\]
After some algebra, it is elementary to show that these two conditions
together are
equivalent to those  defining the set
$\Gg$ in the statement of the theorem. For a matrix not belonging to this
set, the
quasiconvexification will be infinite.

Of the two roots of the above quadratic equation, by elementary continuity
arguments, it is easy to show that we are interested in the one with
positive root,
which, after some computations, has the form given in the statement of the
theorem. For the value of the minimum itself, we must calculate
\[
\myab A_1\cb^2\f1t+\myab A_2\cb^2\f1{1-t},
\]
precisely for this value of $t$. By exploiting the relationship of $t$ and
$1/t$ when
$t$ is the root of a quadratic equation, and realizing that (2.1) is
invariant when changing $t$ by $1-t$, $\ag$ to $\bg$ and $A_1$ to $A_2$,
the expression for the quasiconvexification is obtained after some
careful arithmetic. Finally the term
\[
-2A^{(1)}\cdot f(x)+\myab f(x)\cb^2
\]
must be added.

\section{Final comments}
It is worthwhile to notice that the function $QW$
given explicitly in the theorem is, thus, a quasiconvex function, and, in
particular,
the set where it is finite is a quasiconvex, nonconvex set, in the sense
that weak
limits of gradients taking values on it will also take values within the
same set. In
fact, our computations  amount to realizing that the polyconvexification and the
rank-one convexification coincide as is typical in explicit results  of
this nature (see,
for instance,
\cite{KohnA}, \cite{KohnStrang} among others).

It can actually be shown
that the set $\Gg$ can also be determined by using the $G$-closure 
of  $\al\ag\id,
\bg\id\mycl$ ($\id$ is the identity matrix). In fact, it is true that
\[
\Gg=\al A\in\Mdos: \hbox{there exists }\cg\in G[\ag\id, \bg\id]\hbox{ and }
\cg A^{(1)}+TA^{(2)}=0\mycl.
\]
This was noticed in \cite{SverakI} where the idea of introducing a
potential to replace
the differential constraint was indeed used and indicated. As far as we can
tell, this
is the main link between our approach and that of using relaxation based on
effective conductivities of generalized layouts as in \cite{LiptonVeloA},
\cite{TartarK}.
Notice however that here there is no restriction on the target field $f$.
Our main theorem
also ensures that optimal microstructures are always rank-one laminates and that
these are the only ones. This issue was also addressed in \cite{Grabovsky}  and
\cite{TartarK}. Further work may reveal a more profound connection between
both perspectives.

Our computations can also be extended to cover more general cases. I plan to
explore this in the near future (\cite{BellidoPedregalE}). On the other
hand, I believe
our theorem yields also the clue for the numerical approximation of optimal (or
quasi-optimal) designs  via relaxation of vector variational problems. I will
address this issue shortly too.

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\end{document}