PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ON
DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS
May 24 – 27, 2002, Wilmington, NC, USA
pp. 30-41
CONSTRAINED ENVELOPE FOR A GENERAL CLASS OF
DESIGN PROBLEMS
Ernesto Aranda and Pablo Pedregal
E.T.S. Ingenieros Industriales
Universidad de Castilla - La Mancha
13071 Ciudad Real
Abstract. We analyze the relaxation and computation of the relaxed density when
we reformulate a typical optimal design problem with volume constraint in two dimension as a fully vector variational problem. Our aim is to examine a general cost
functional depending explicitly on all variables and in particular in the gradient variable, and see how far computations and properties of the relaxed integrand can be
pushed.
1. Variational formulation in optimal design problems. In this paper we
would like to consider a variational approach to optimal design problem in conductivity. We want to mix two different conducting materials, with conductivities α
and β, 0 < α < β, in a design domain Ω ⊂ R2 (regular and simply-connected) in
order to minimize a general cost functional
Z
˜
I(χ)
=
W (x, χ(x), u(x), ∇u(x)) dx,
Ω
where χ(x), the design variable, is the characteristic function of a subset of Ω
occupied by the material with conductivity α, and u ∈ H 1 (Ω) is the unique solution
of the state equation:
− div [(αχ(x) + β(1 − χ(x))) ∇u(x)] = P
u = u0
in Ω,
on ∂Ω.
We assume W : Ω×R×R×R2 −→ R∗ = R∪{+∞} is continuous with respect χ(x),
u(x) and ∇u(x), and measurable in x; P ∈ H −1 (Ω) and u0 ∈ H 1 (Ω). Moreover,
there is a typical volume restriction
Z
1
χ(x) dx = t, for t ∈ (0, 1) fixed.
(1)
|Ω| Ω
Similar optimal design problems for particular cost functionals have been studied
in a number of papers ([1, 2, 3, 4, 8]). Our goal here is to see what can be done about
the analysis of a general cost functional in terms of bounds on the relaxed integrand
or even explicit computations in some cases. It is a well-known fact that these
problems fail to have optimal solutions within the class of characteristic functions.
Thus, some different techniques have been used to analyze such situations. Namely,
when the cost functional does not depend on derivatives of the state, homogenization
theory has been the main tool to understand such situations. In this paper we will
further explore a variational perspective that has already been used in [7], when
1991 Mathematics Subject Classification. 49J45, 74P10.
Key words and phrases. Optimal design, relaxation, constrained quasiconvexification.
30
CONSTRAINED ENVELOPE FOR DESIGN PROBLEMS
31
the cost functional consists of the square of the gradient of the underlying electric
potential. Our goal is to extend the ideas of these works for a general cost functional
W as presented here.
The variational version of this optimal design problem was developed in [6]. The
main idea consists in a reformulation of the equilibrium equation in the following
way. If p ∈ H01 (Ω) is the solution of the problem −∆p = P in H −1 (Ω), then
div [−αχ(x) − β(1 − χ(x))∇u(x) + ∇p(x)] = 0
implies that there exists a stream function v ∈ H 1 (Ω) such that
∇p(x) = (αχ(x) + β(1 − χ(x))) ∇u(x) + T ∇v(x),
where T is the counterclockwise π/2 rotation in the plane. Introducing the vectorvalued function U = (u, v), it is not difficult to realize that our original design
problem can be set up as a vector variational problem
Z
Minimize I(U ) =
ϕ(x, U (x), ∇U (x)) dx,
Ω
where
ϕ(x, U, F ) =

Wα (x, U (1) , F (1) )
if F ∈ Λ(α, ∇p(x))\Λ(β, ∇p(x)),



W (x, U (1) , F (1) )
if
F ∈ Λ(β, ∇p(x))\Λ(α, ∇p(x)),
β
©
ª
(1)
(1)

min Wα (x, U , 0), Wβ (x, U , 0)
if F ∈ Λ(α, ∇p(x)) ∩ Λ(β, ∇p(x)),



+∞
otherwise.
©
ª
and Λ(γ, B) = F ∈ M2×2 : γF (1) + T F (1) = B . Here, U (1) denotes the first component of U and F (1) stands for the first row of matrix F . We have also put
Wα (x, a, b) = W (x, 1, a, b) and Wβ (x, a, b) = W (x, 0, a, b).
On the other hand, the volume constraint (1) can be replaced by
Z
ψ(x, ∇U (x)) dx = t,
Ω
where

1


 |Ω| if F ∈ Λ(α, ∇p(x)),
ψ(x, F ) = 0
if F ∈ Λ(β, ∇p(x))\Λ(α, ∇p(x)),



+∞ else.
Therefore, we set up the variational problem
Z
Minimize I(U ) =
ϕ(x, U (x), ∇U (x)) dx, U ∈ H 1 (Ω), U (1) − u0 ∈ H01 (Ω),
Ω
Z
such that J(U ) =
ψ(x, ∇U (x)) dx = t.
Ω
The equivalence of these two optimization problems has been rigorously shown in
[6]. As equivalent to our original problem, this formulation does not admit optimal
solutions. However, this variational approach allows us to use all of the ideas and
techniques about non-convex vector variational problems. These include relaxation,
Young measures, microstructure, etc., that they provide the necessary information
to understand the optimal design problem.
32
ERNESTO ARANDA AND PABLO PEDREGAL
The paper is organized as follows. In Section 2 we rigorously prove a relaxation result which is relevant in our case because the integrands for both the cost
functional and the integral restriction are not Carathéodory functions. Section 3 is
devoted to the understanding of the set of admissible measures in computing the
relaxed integrand. In particular, we stress the role played by second moments of
such measures. Here we follow closely the analysis and computations of [7]. Bounds
on the relaxed integrand are contained in Section 4. These are valid for a general
cost functional. Finally, in Section 5 we show how in some cases these bounds are
in fact exact leading to a explicit form of the relaxed integrand.
2. Relaxation. Once we have reformulated our original problem as a typical problem in the Calculus of Variations we will proceed with relaxation. The main tool to
analyze relaxation under integral constraints is the constrained envelope proposed
in [6] in terms of gradient Young measures (YM),
½Z
¾
CQϕ(x, U, A, t) = inf
ϕ(x, U (x), F ) dν(F ) : ν ∈ A(x, A, t)
M
where
A(x, A, t) =
½
¾
Z
Z
1,∞
ν : ν homogeneous W
-YM,
F dν(F ) = A,
ψ(x, F ) dν(F ) = t
M
M
This is the appropriate integrand for the relaxed problem. The following result
establishes this fact.
Theorem 1. The two infima
½Z
m = inf
ϕ(x, U (x), ∇U (x)) dx : U ∈ H 1 (Ω), U (1) − u0 ∈ H01 (Ω),
Ω
¾
Z
ψ(x, ∇U (x)) dx = t
Ω
and
½Z
m̄ = inf
CQϕ(x, U (x), ∇U (x), λ(x)) dx : U ∈ H 1 (Ω), U (1) − u0 ∈ H01 (Ω),
Ω
¾
Z
0 ≤ λ(x) ≤ 1,
λ(x) dx = t|Ω|
Ω
coincide. Moreover, the second infimum is a minimum.
Note that when the integrand ϕ is a Carathéodory function, the above result is
true in general, but in our case, ϕ takes on the value +∞ and it is no longer such kind
of function. However, it is still possible to prove the relaxation in this case using
Young measures in the following way. Let us denote Λ(∇p(x)) = Λ(α, ∇p(x)) ∪
Λ(β, ∇p(x)) and define
©
A(u0 , t) = {νx }x∈Ω : ν is a H 1 -YM, supp(νx ) ⊂ Λ(∇p(x)), ∃U ∈ H 1 (Ω) :
¾
Z
Z Z
(1)
1
U − u0 ∈ H0 (Ω), ∇U (x) =
F dνx (F ),
ψ(x, F ) dνx (F ) dx = t
M
Ω
M
CONSTRAINED ENVELOPE FOR DESIGN PROBLEMS
and the problem
½Z Z
33
¾
m̃ = inf
ϕ(x, U (x), F ) dνx (F ) dx : ν ∈ A(u0 , t) .
Ω
M
We have the following result.
Theorem 2. Under above conditions,
m̃ = m̄ = m
Moreover, for each ν ∈ A(u0 , t) such that supp(νx ) ⊂ Λ(∇p(x)) a.e. x ∈ Ω there
exists a sequence {∇Uj } generating ν such that
Uj ∈ H 1 (Ω),
(1)
Uj
− u0 ∈ H01 (Ω),
{|∇Uj |2 } is equi-integrable,
Z
a.e. x ∈ Ω, ∀j,
ψ(x, ∇Uj (x)) dx = t, ∀j,
∇Uj (x) ∈ Λ(∇p(x)),
Ω
and
Z
lim
j→∞
Z Z
ϕ(x, Uj (x), ∇Uj (x)) dx =
Ω
ϕ(x, U (x), F ) dνx (F ) dx
Ω
M
where U is the weak limit of Uj .
Proof. The inequality m̃ ≤ m̄ ≤ m is standard (cf. [5]). Let us prove m ≤ m̃. Let
ν = {νx }x∈Ω ∈ A(u0 , t). Then there exists (cf. [5]) a sequence {∇Ūj } generating ν
which verifies
Ūj ∈ H 1 (Ω),
(1)
Ūj
− u0 ∈ H01 (Ω),
{|∇Ūj |2 } is equi-integrable.
Let us consider the following function,
½¯
¯2 ¯
¯2 ¾
¯
¯ ¯
¯
Φ(x, F ) = min ¯αF (1) + T F (2) − ∇p(x)¯ , ¯βF (1) + T F (2) − ∇p(x)¯ .
It is clear that Φ is a nonnegative Carathéodory function such that
Φ(x, ·)|Λ(∇p(x)) = 0.
Then
Z
lim
j→∞
Z Z
Φ(x, ∇Ūj (x)) dx =
Ω
Φ(x, F ) dνx (F ) dx = 0,
Ω
(2)
M
due to the fact that supp(νx ) ⊂ Λ(∇p(x)) and |∇Ūj |2 is equi-integrable.
Now we define a sequence of designs, σj (x) = αχj (x) + β(1 − χj (x)) such that
¯
¯2
¯
¯
(1)
(2)
Φ(x, ∇Ūj (x)) = ¯σj (x)∇Ūj (x) + T ∇Ūj (x) − ∇p(x)¯ .
Using that ν ∈ A(u0 , t),
Z Z
ψ(x, F ) dνx (F ) dx = t,
Ω
M
then, from ψ’s definition, it is clear that
Z
νx (Λ(α, ∇p(x))) dx = t|Ω|,
Ω
and therefore1
Z
χj (x) dx = t|Ω|.
lim |{x ∈ Ω : σj (x) = α}| = lim
j→∞
1 The
j→∞
Ω
proof of this fact uses a typical argument involving truncated functions.
34
ERNESTO ARANDA AND PABLO PEDREGAL
Then, it is possible to slightly modify the designs in order to construct a sequence
Uj such that ∇Uj (x) ∈ Λ(∇p(x)) a.e. x ∈ Ω. Let us put
Ωα,j = {x ∈ Ω : σj (x) = α}.
Without loss of generality we can assume that |Ωα,j | < t|Ω|. Using that Lebesgue
measure has no atoms, there exists a measurable subset Sα,j ⊂ Ω\Ωα,j with vanishing measure such that |Ωα,j ∪ Sα,j | = t|Ω|. Putting
(
α if x ∈ Ωα,j ∪ Sα,j ,
γj (x) =
β if x ∈ Ω\ (Ωα,j ∪ Sα,j ),
we can define the new sequence Uj (x) as the solution of
(1)
− div(γj (x)∇Uj (x)) = P (x)
u = u0
in Ω,
on ∂Ω.
¾
(3)
With this construction, it is clear that ∇Uj (x) ∈ Λ(∇p(x)) and
Z
ψ(x, ∇Uj (x)) dx = t, ∀j.
Ω
Moreover,
lim k∇Ūj − ∇Uj kL2 (Ω) = 0.
j→∞
(4)
As a consequence, ν is generated by ∇Uj . Let us prove (4). For simplicity we only
work with the first component of Uj = (uj , vj ). We have
Z
Z
2
α
|∇uj (x) − ∇ūj (x)| dx ≤
hγj (x) (∇uj (x) − ∇ūj (x)) , ∇uj (x) − ∇ūj (x)i dx
Ω
Ω
adding and subtracting T ∇vj (x), T ∇v̄j (x) and ∇p(x) in the first factor of the scalar
product
Z
=
h−γj (x)∇ūj (x) − T ∇v̄j (x) + ∇p(x)), ∇uj (x) − ∇ūj (x)i
Ω
Z
+
hγj (x)∇uj (x) + T ∇vj (x) − ∇p(x), ∇uj (x) − ∇ūj (x)i
Ω
Z
+
hT ∇v̄j (x) − T ∇vj (x), ∇uj (x) − ∇ūj (x)) .
Ω
The second term vanishes because of (3); the third one also vanishes because it is
the scalar product of a divergence-free function and ∇u − ∇ū has zero trace on ∂Ω.
Finally, using Hölder inequality
µZ
¶ 12
2
αk∇uj (x)−∇ūj (x)kL2 (Ω) dx ≤ lim
|γj (x)∇ūj (x) + T ∇v̄j (x) − ∇p(x)| dx
j→∞
Ω
The right hand side is equal to zero as a consequence of the definition of γj , Sα,j
and (2). This finishes the proof.
Once we have proved the relaxation result, our main achievement here is to address some properties of this envelope and the explicit computation in some special
cases.
Before that, in order to simplify the exposition we can assume, without loss of
generality, P ≡ 0 in the state equation. This is so because of Λ(γ, ∇p(x)) = V x +Λγ ,
where
Λγ = {F ∈ M2×2 : γF (1) + T F (2) = 0}
CONSTRAINED ENVELOPE FOR DESIGN PROBLEMS
and
µ
Vx =
0
−T ∇p(x)
35
¶
.
Therefore, we will compute CQϕ(x, U, A, t) by a simple evaluation of CQϕ] (x, U, A−
V x , t). Also, for simplicity, we will drop the dependence in (x, U ). Then

Wα (F (1) )
if F ∈ Λα \Λβ ,



W (F (1) )
if
F ∈ Λβ \Λα ,
β
ϕ] (F ) =

min{Wα (0), Wβ (0)} if F ∈ Λα ∩ Λβ ,



+∞
otherwise.
and
]
ψ (F ) =

1


 |Ω|
if F ∈ Λα \Λβ ,
if F ∈ Λβ \Λα ,
0


+∞ else.
It is clear now that the set of feasible measures is
½
¾
Z
Z
A(A, t) = ν : ν homog. W 1,∞ -YM,
F dν(F ) = A,
ψ ] (F ) dν(F ) = t
M
M
We will explore this set of measures in the following section.
3. Analysis of admisible measures. In order to compute CQϕ] for a fixed A and
t, we will firstly establish some general properties about measures ν ∈ A(A, t). By
definition of ϕ] and ψ ] , we can restrict our attention to those feasible ν’s supported
in Λα ∪ Λβ such that ν(Λα ) = t. Following the analysis in [7] we can decompose
ν = tνα + (1 − t)νβ ,
We set
supp(να ) ⊂ Λα , supp(νβ ) ⊂ Λβ .
Z
(5)
Z
Aα =
F dνα (F ),
Λα
Aβ =
F dνβ (F ),
Λβ
the first moments of να and νβ , respectively. Thus Aα ∈ Λα , Aβ ∈ Λβ , and
A = tAα + (1 − t)Aβ . Now, let z and w be such that
µ
¶
µ
¶
z
w
Aα =
, Aβ =
.
(6)
αT z
βT w
It is easy to check that
³
´
³
´
1
−1
z=
βA(1) + T A(2) , w =
αA(1) + T A(2) .
(7)
t(β − α)
(1 − t)(β − α)
Note that Aα and Aβ are uniquely determined by A and t, and they are independent
of ν itself. On the other hand, we can consider the second moments
Z
Z
xα =
|F (1) |2 dνα (F ), xβ =
|F (1) |2 dνβ (F ),
(8)
Λα
Λβ
Now, using that the determinant is a quasiaffine function, we have
Z
Z
Z
det A =
det F dν(F ) = t
det F dνα (F ) + (1 − t)
det F dνβ (F ),
M
M
and taking into account that if F ∈ Λα then det F = α|F
det F = β|F (1) |2 , (9) reads
det A = tαxα + (1 − t)βxβ .
(9)
M
(1) 2
| , and F ∈ Λβ implies
(10)
36
ERNESTO ARANDA AND PABLO PEDREGAL
From Jensen’s inequality follows
¯Z
Z
¯
xα =
|F (1) |2 dνα (F ) ≥ ¯¯
Λα
Λα
and similary
¯2
¯
2
2
F (1) dνα (F )¯¯ = |A(1)
α | = |z| ,
(11)
2
xβ ≥ |w| .
Let us define
©
ª
L(A, t) = (xα , xβ ) ∈ R2 : xα ≥ |z|2 , xβ ≥ |w|2 , det A = tαxα + (1 − t)βxβ
which describes a segment on the xα -xβ plane, and the second-moment function
S : A(A, t) −→ R2 ,
!
Z ¯
¯
¯
¯
¯ (1) ¯2
¯ (1) ¯2
¯F ¯ dνβ (F )
¯F ¯ dνα (F ),
ÃZ
S(ν) =
Λβ
Λα
We have the following result.
Lemma 1. Let A and t such that
g(A) ≥ 0,
h(A) ≥ 0,
r1 (A) ≤ t ≤ r2 (A);
(12)
where
¯
¯4 ¯
¯4
¯
¯
¯
¯
g(A) = α2 β 2 ¯A(1) ¯ + ¯A(2) ¯ + (α2 + 6αβ + β 2 )(det A)2
¯
¯2 ¯
¯2
¯
¯2
¯
¯2
¯
¯ ¯
¯
¯
¯
¯
¯
− 2αβ ¯A(1) ¯ ¯A(2) ¯ − 2αβ(α + β) ¯A(1) ¯ det A − 2(α + β) ¯A(2) ¯ det A,
¯
¯2 ¯
¯2
¯
¯
¯
¯
h(A) = (α + β) det A − αβ ¯A(1) ¯ − ¯A(2) ¯ ,
and
ri (A) =
1
1
+
2 2(β − α) det A
µ ¯
¶
¯2 ¯
¯2
p
¯
¯
¯
¯
αβ ¯A(1) ¯ − ¯A(2) ¯ + (−1)i g(A) .
(13)
(14)
(15)
Then,
L(A, t) = S(A(A, t))
Proof. The inclusion L(A, t) ⊂ S(A(A, t)) has already been done using Jensen’s
inequality and the determinant’s quasiaffinity (see [7]). Let us prove the right
inclusion. First, we have a restriction on A and t given by (10) and (11). That is,
det A ≥ αt|z|2 + β(1 − t)|w|2 .
(16)
This can be written as
µ ¯
¶
¯2
¯
¯2
¯ (1)
¯ (1)
2
2
(2) ¯
(2) ¯
2
(β − α) det At + t β ¯αA + T A ¯ − α ¯βA + T A ¯ − (β − α) det A
¯
¯2
¯
¯
+ α ¯βA(1) + T A(2) ¯ ≤ 0
If PA (t) stands for the second degree polynomial on the left hand side, it is easy to
check that PA (0) and PA (1) are positive2 if A 6∈ Λα ∪ Λβ . Then, we have to demand
that A and t are such that PA has solutions in [0, 1] and t is between both solutions.
That is to say, PA has to be an upward parabola with positive discriminant and
vertex in [0, 1], and t is between its roots. After some computations it results that
2 Note
that PA (0) = α|βA(1) + T A(2) |2 , PA (1) = β|αA(1) + T A(2) |2 .
CONSTRAINED ENVELOPE FOR DESIGN PROBLEMS
37
the discriminant is g(A) and the roots are ri (A). It is not difficult to realize that
the above conditions are exactly lemma’s hypotheses.
Then, given A and t satisfying (12), for each point in L we will associate an
element in S(A(A, t)). We will proceed in two steps.
1. The two extreme points of L:
¶
µ
det A − tα|z|2
,
sα = |z|2 ,
(1 − t)β
µ
sβ =
¶
det A − (1 − t)β|w|2
, |w|2 ,
tα
belong to the image of S.
This fact has already been proved in [7]. For the sake of completeness we
sketch the proof. We construct a pair of laminates associated to each extreme
point of L. For instance, for the second one we can consider
Z
(1)
xβ =
|F (1) |2 dνβ (F ) = |w|2 = |Aβ |2 ⇒ We can take νβ = δAβ .
Λβ
On the other hand, determining the rank one directions going through A with
extreme points on Λα and Λβ , we have A = rAα + (1 − r)Aβ , r ∈ [0, 1], with
Aα ∈ Λα , Aβ ∈ Λβ and det(Aα − Aβ ) = 0. After some computation, this
condition is equivalent to PA (r) = 0. Therefore, under (12), there exist two
first order laminates with barycenter A and support on matrices Aα,j , Aβ,j in
Λα and Λβ respectively, for j = 1, 2.
Now, considering Āα,j = Aβ + ζj (Aα,j − Aβ,j ) such that Āα,j ∈ Λα , then
the laminates
³
´
β
β
β
νi,j
= σi,j
δAα,i + (1 − σi,j
) ρβi,j δAβ + (1 − ρβi,j )δĀα,j , i, j = 1, 2, i 6= j,
where
β
σi,j
=
ri (rj − t)
,
rj − rj t − ri + ri rj
ρβi,j =
rj − rj t − ri + ri rj
,
rj − ri
β
satisfy S(νi,j
) = sβ .
In the same way, it is possible to construct the laminates
³
´
α
α
α
νi,j
= σi,j
δAβ,i + (1 − σi,j
) ρβi,j δAα + (1 − ρβi,j )δĀβ,j , i, j = 1, 2, i 6= j,
where Āβ,j = Aα + ξj (Aα,j − Aβ,j ), with ξj such that Āβ,j ∈ Λβ , and
α
σi,j
=
(rj − t)(ri − 1)
,
t − rj − rj t + ri rj
ρα
i,j =
t − rj − rj t + ri rj
.
ri − rj
α
In this case, S(νi,j
) = sα .
0
2. For every point (xα , x0β ) ∈ L(A, t) there exists a laminate µ such that S(µ) =
(x0α , x0β ). This is an immediate consequence of the convexity of the Young
measures set. Namely, if (x0α , x0β ) ∈ L(A, t), there exists λ0 ∈ [0, 1] such that
(x0α , x0β ) = λ0 sα + (1 − λ0 )sβ . Then, it is clear which laminates we can choose;
for i, j, k, l = 1, 2, j 6= i, k 6= l,
β
α
µ = λ0 νi,j
+ (1 − λ0 )νk,l
.
It is obvious that any of these laminates are in A(A, t) and S(µ) = (x0α , x0β ).
38
ERNESTO ARANDA AND PABLO PEDREGAL
We can define the set of admissible Young measures ν with barycenter A, ν(Λα ) =
t and fixed second moments s ∈ L(A, t) by
A(A, t, s) = S −1 (s).
Then the functional
Φ : L(A, t) −→ R,
½Z
¾
Φ(s) = min
ϕ] (F ) dν(F ) : ν ∈ A(A, t, s)
M
satisfies the following result.
Corollary 1.
½Z
¾
CQϕ] (A, t) = min
ϕ] (F ) dν(F ) : ν ∈ A(A, t)
=
M
min Φ(s) = min Φ(rsα + (1 − r)sβ ).
0≤r≤1
s∈L(A,t)
Unfortunately, we do not know how to find the optimal gradient Young measure
associated with Φ(s) for a general integrand ϕ] , except is some cases in which we
can obtain an explicit expression of Φ(s). We will analyze this situation in Section 5.
However, using Φ for the computation of CQϕ] allows us to give interesting bounds
on the semiconvex envelope.
4. Bounds. We start with the following properties on Φ.
Lemma 2. Φ is finite and convex on L(A, t).
Proof. The convexity of Φ is an immediately consequence of Lemma 1 and the
convexity of the Young measure set. Then,
Φ(rsα + (1 − r)sβ ) ≤ rΦ(sα ) + (1 − r)Φ(sβ ),
r ∈ [0, 1],
and it is clear that Φ(sα ) and Φ(sβ ) are finite.
As a consequence,
Φ(s) ≤ rΦ(sα ) + (1 − r)Φ(sβ )
and
Z
Z
α
Φ(s ) ≤
ϕ
M
]
α
(F ) dνi,j
(F ),
β
Φ(s ) ≤
M
β
ϕ] (F ) dνk,l
(F ).
Then we have the upper bound
½Z
¾
Z
β
α
(F ),
ϕ] (F ) dνk,l
CQϕ] (A, t) ≤ min
ϕ] (F ) dνi,j
(F ) .
M
M
On the other hand, if Wα and Wβ are convex functions on Λα and Λβ respectively,
using Jensen’s inequality we have a lower bound
CQϕ] (A, t) ≥ tWα (z) + (1 − t)Wβ (w),
where z and w were defined in (7). Moreover, this lower bound can only be attained
if the pair (A, t) is such that ri (A) = t for i = 1 or 2.
It is also possible to obtain another lower bound without convexity assumptions
on Wα and Wβ . If we define the convexification of a function ψ with fixed second
moment,
½Z
¾
Z
Z
(2)
2
C ψ(η, ζ) = inf
ψ(λ) dν(λ) :
λ dν(λ) = η,
|λ| dν(λ) = ζ
ν
CONSTRAINED ENVELOPE FOR DESIGN PROBLEMS
39
then we have the following result:
Lemma 3. For (A, t) verifying the hypotheses of Lemma 1, and s ∈ L(A, t),
Φ(s) ≥ tC (2) Wα (z, s1 ) + (1 − t)C (2) Wβ (w, s2 )
where s = (s1 , s2 ) and z and w were defined in (7).
Proof. Let be s ∈ L(A, t). By Lemma 1 there exists µ ∈ A(A, t) such that S(µ) = s.
Let now consider µ = tµα + (1 − t)µβ such that supp(µα ) ⊂ Λα and supp(µβ ) ⊂ Λβ
(as in (5)). Then, it is clear that µα and µβ are admissible for C (2) Wα and C (2) Wβ ,
respectively (remember that z and w are independent of µ ). Then,
Z
Z
C (2) Wα (z, s1 ) ≤
Wα (F (1) ) dµα (F ), C (2) Wβ (w, s2 ) ≤
Wβ (F (1) ) dµβ (F )
Λα
Λβ
and
Z
tC (2) Wα (z, s1 ) + (1 − t)C (2) Wβ (w, s2 ) ≤
ϕ] (F ) dµ(F ).
M
Due to last inequality is valid for all µ ∈ A(A, t, s) the result is followed.
From Corollary 1 and Lemma 3 we have:
Corollary 2 (Lower bound for CQϕ] (A, t)).
½
µ
¶
det A − β(1 − t)|w|2
]
(2)
2
CQϕ (A, t) ≥ min tC Wα z, r|z| + (1 − r)
0≤r≤1
tα
µ
¶¾
det A − αt|z|2
(2)
2
+(1 − t)C Wβ w, r
+ (1 − r)|w|
(1 − t)β
5. Exact computations. In this section we analyze two particular cases in which
the function Φ(s) can be explicitly written . Firstly, we will consider W not depending on the gradient variable. That is, W (x, χ(x), u(x), ∇u(x)) = W (x, χ, u(x)). In
this case,
Φ(s) = tWα (x, U (1) ) + (1 − t)Wβ (x, U (1) ),
]
∀s ∈ L(A, t)
(1)
and therefore CQϕ (x, U, A, t) = tWα (x, U ) + (1 − t)Wβ (x, U (1) ) for (A, t) ∈ B,
where
B = {(A, t) : g(A) ≥ 0, h(A) ≥ 0, r1 (A) ≤ t ≤ r2 (A)} .
A more interesting case appears when
W (x, χ(x), u(x), ∇u(x)) = f (x, χ(x), u(x))|∇u(x)|2 .
In this situation,
Φ(s1 , s2 ) = tfα (x, u(x))s1 + (1 − t)fβ (x, u(x))s2
where f (x, 1, a) = fα (x, a) and f (x, 0, a) = fβ (x, a). As it has been shown in
Corollary 1, CQϕ] coincides with the minimum of Φ(s) in L(A, t). It is obvious
that this problem attains the optimum at one of the extreme points of L(A, t).
Moreover, the minimum will be
µ
¶
2
fα (x, U (1) ) fβ (x, U (1) )
2 det A − tα|z|
|z| ,
, if
−
>0
(1 − t)β
α
β
or
µ
¶
det A − (1 − t)β|w|2
fα (x, U (1) ) fβ (x, U (1) )
2
, |w|
if
−
< 0.
tα
α
β
40
ERNESTO ARANDA AND PABLO PEDREGAL
For
fα (x,U (1) )
α
fβ (x,U (1) )
,
β
=
every point of the segment is optimal. Therefore,
CQϕ] (x, U, A, t) =

³ ¯
¯2 ¯
¯2
βfα (x,U (1) )−αfβ (x,U (1) )


β 2 ¯A(1) ¯ + ¯A(2) ¯

tβ(β−α)2
¡
¢
¢
+ t(β − α)2 fβ (x, U (1) ) − 2β det A


+∞
for
fα (x,U (1) )
α
−
fβ (x,U (1) )
β
> 0. In the same way,
CQϕ] (x, U, A, t) =

³ ¯
¯2 ¯
¯2
(1)
)−βfα (x,U (1) )

 αfβ (x,U
α2 ¯A(1) ¯ + ¯A(2) ¯

(1−t)α(β−α)2
¡
¢
¢
+ (1 − t)(β − α)2 fα (x, U (1) ) − 2α det A


+∞
for
fα (x,U (1) )
α
−
fβ (x,U (1) )
β
< 0. Finally, for
(
]
(A, t) ∈ B
otherwise,
CQϕ (x, U, A, t) =
fα (x,U (1) )
α
fβ (x,U (1) )
β
=
det A
+∞
(A, t) ∈ B
otherwise,
fβ (x,U (1) )
,
β
(A, t) ∈ B
otherwise,
Note that these values correspond to the upper and lower bounds given in the
previous section.
As a particular case, we can compute the exact relaxed integrand for a problem
described in [2]. The design problem is
Z
1
min
|∇u(x)|2 dx
Ω a(χ(x))
where a(χ) = αχ(x) + β(1 − χ(x)), satisfying the state equation
div (a(χ(x))∇u(x)) = 0
u = u0
in Ω,
on ∂Ω.
For this case,
f (x, χ(x), u(x)) =
1
αχ(x) + β(1 − χ(x))
so
CQϕ(x, U, A, t) =
³
(
α+β
tαβ 2 (β−α)
+∞
´
´
¯
¯2 ¯
¯2 ³
β 2 ¯A(1) ¯ + ¯A(2) ¯ + tα(β−α)
− 2β det A
α+β
(A, t) ∈ B
otherwise,
Acknowledgements. This work has been supported by Ministerio de Ciencia y
Tecnologı́a (Spain) through grant BMF2001-0738 and by Junta de Comunidades de
Castilla - La Mancha through grant GC-02-001.
CONSTRAINED ENVELOPE FOR DESIGN PROBLEMS
41
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Received July 2002.
E-mail address: [email protected]
[email protected]