Boletı́n de la Asociación Matemática Venezolana, Vol. XX, No. 1 (2013)
Bounds on the effective energy density of a
special class p-dielectric
Gaetano Tepedino Aranguren, Javier Quintero C,
Eribel Marquina, José Soto
Abstract. This work gives lower and upper bounds on the effective
f of a two phase composites material composed by
energy density W
a periodical mixed of two nonlinear homogeneous isotropic dielectric
materials in prescribed proportion. These bounds are given as a
function of θ, which is the volume fraction of the material with
lowest dielectric constant in the mixture. The dielectric constant
conductivity of the k−material are given respectively by
ν1 (z) = α1 |z|p−2 ,
ν2 (z) = α2 |z|p−2 ,
where 0 < α1 < α2 and 1 < p < ∞ .
For anisotropic composites the bounds are given in the form
Z
Φ(p, r, θ) ≤ ∼ W̃ ≤ Ψ(p, r, θ) ,
Sr
where the functions Φ, Ψ reduce smoothly to the optimal lower and
upper bound of the linear composite when p → 2.
The method to obtain this bounds, in the case p 6= 2, follows a
generalization of the Hashin–Shtrikman variational principles constructed from a comparison medium which is in general nonlinear
and reduces to linear when p = 2.
Resumen. Este trabajo da cotas inferiores y superiores para la
f de un material compuesto de dos fases,
densidad de energı́a eficaz W
constituido por una mezcla periódica de dos materiales dieléctricos
isotrópicos homogéneos no lineales en proporción prescrita. Estas
cotas son dadas como una función de θ, que es la fracción de volumen del material con constante dieléctrica más bajo en la mezcla.
La constante dieléctrica de conductividad del k-material se dan,
respectivamente, por
ν1 (z) = α1 |z|p−2 ,
ν2 (z) = α2 |z|p−2 ,
55
56
G. Tepedino, J. Quintero, E. Marquina, J. Soto.
donde 0 < α1 < α2 y 1 < p < ∞.
Para compuestos anisotrópicos los lı́mites se dan en la forma
Z
Φ(p, r, θ) ≤ ∼ W̃ ≤ Ψ(p, r, θ) ,
Sr
donde las funciones Φ, Ψ se reducen suavemente a la cota inferior y
superior del compuesto lineal cuando p → 2.
El método para obtener estas cotas, en el caso p 6= 2, sigue de una
generalización de los principios variacionales de Hashin–Shtrikman
construido a partir de un medio de comparación que es en general
no lineal y se reduce al lineal cuando p = 2.
1
Introduction
In this work we will follow the Y −periodic microstructure of the mixture, been
N
Y
Y the cell
(0, ai ), {a1 , a2 , . . . , aN } ⊂ (0, ∞), if θk , for k ∈ {1, 2}, is the
i=1
proportion of material type k in the mixed, Y = Y1 ∪ Y2 , Y1 ∩ Y2 = ∅ and χk is
the characteristic
function of the phase Yk which only contains material k, then
Z
θk = ∼ χk , 0 ≤ θk ≤ 1 and θ1 + θ2 = 1. Following the notation of [T.Q.M.S] ,
Y
the energy density of the composite is the Y −periodic extension of the function
W : RN × RN → R given by
W (x, z) = χ1 (x)W1 (z) + χ2 (x)W2 (z) , where
Wk (z) = αpk |z|p , and 0 < α1 < α2 , 1 < p < ∞ .
(1.1)
In [T.Q.M.S] has been proved that the effective energy density and its dual are
given respectively by the variational principles:
Z
Z
f (ξ) = inf ∼ W (x, v + ξ) dx , W
f ∗ (η) = inf ∼ W ∗ (x, σ + η) dx ,
W
(1.2)
v∈Vp
σ∈Sq
Y
Y
1
n
where Vp is the completion of Cper
(Y , R
Z ) under the Lp −norm and Sq is the
1
completion of N = {σ ∈ Cper
(Y, RN ) : ∼ σ = θ and div(σ) = 0 in Y } under
Y
the Lq −norm. Notice that v ∈ Vp ⇐⇒ v = ∇u for some u ∈ Kp . Here Kp is the

1/p
Z
Z
1
completion of Cper
(Y ) under the norm kuk1,p = ∼ |u|p dx + ∼ |∇u|p dx .
Y
Y
Bounds on the Effective ... Dielectric Composites...
57
Also we have Vp⊥ = Sq ⊕ CV , where CV are the constants vector fields. See
[T.Q.M.S] for a complete description of definitions and properties of these
spaces.
2
f , Γ−convergence and Homogenization
Existence of W
Lemma 1 The function W : Rn → RN → R defined by (1.1) satisfies:
(1)∀z ∈ RN : W (., z) is Y −periodic and measurable.
(2)∀x ∈ RN : W (x, .) is C 1 (Rn ) and strictly convex.
(3)∀x, z ∈ RN : 0 ≤ αp1 |z|p ≤ W (x, z) ≤ αp2 |z|p .
(4)∃L ≥ 0 such that ∀x, z1 , z2 ∈ RN : |W (x, z1 )1/p −W (x, z2 )1/p | ≤ L|z1 −z2 |
Proof The items (1) to (3) are direct consequences of 1.1. In other hand let
βk = (αk )1/p , then W (x, zk )1/p = β1 χ1 (x)|zk | + β2 χ2 (x)|zk | and W (x, z1 )1/p −
W (x, z2 )1/p = β1 χ1 (x)(|z1 | − |z2 |) + β2 χ2 (x)(|z1 | − |z2 |), therefore |W (x, z1 )1/p −
W (x, z2 )1/p | ≤ L(|z1 | − |z2 |) ≤ L|z1 − z2 |.
2
f and W
f ∗ are given by
Lemma 2 If W is the function defned by 1.1, then W
1.2.
Proof These are consequences of lema 1 and the articles [M.M] , [G.DA] .2
Lemma 3 (Elementary Bounds)
f is defined by 1.2 and p−1 + q −1 = 1, then
If W is defined by 1.1, W
−p/q
1
−q/p
−q/p
f (ξ) ≤ 1 (θ1 α1 + θ2 α2 )|ξ|p .
|ξ|p ≤ W
∀ξ ∈ RN :
θ1 α1
+ θ2 α2
p
p
(2.1)
f
These bounds are called The Elementary Lower and Upper Bounds on W .
Proof Since the null vector field belongs to Vp , then from 1.2 we obtain
Z
α2
1
f
W (ξ) ≤ ∼ W (x, ξ) dx = θ1 αp1 |ξ|p + θ2 |ξ|p = (θ1 α1 + θ1 α2 )|ξ|p .
(2.2)
p
p
Y
In other hand since the null vector field belongs to Sq , then from 1.2 we get
Z
f ∗ (η) ≤ ∼ W ∗ (x, η) dx = θ1 W1∗ (η) + θ2 W2∗ (η) = 1 θ1 α−q/p + θ2 α−q/p |η|q ,
W
1
2
q
Y
(2.3)
from this, using a typical result of convex analysis, see for example [E, T] , we
get
−p/q
f (ξ) ≥ 1 θ1 α−q/p + θ2 α−q/p
|ξ|p ,
W
p
1
2
from this last inequality and 2.2 we obtain 2.1.
2
58
G. Tepedino, J. Quintero, E. Marquina, J. Soto.
Lemma 4 Under the same hypothesis of lemma 3 we have:
∀x, ξ ∈ RN : W (x, ξ) − W1 (ξ) = χ2 (x)h(ξ), W2 (ξ) − W (x, ξ) = χ1 (x)h(ξ)
(2.4)
∀x, η ∈ RN : W1∗ (η) − W ∗ (x, η) = χ2 (x)g(η), W ∗ (x, η) − W2∗ (η) = χ1 (x)g(η)
(2.5)
β
p
∗
q
1
∀ξ, η ∈ RN : h(ξ) = α2 −α
|ξ|
,
h
(η)
=
|η|
,
p
q
(2.6)
κ
q
∗
p
2
g(η) = β1 −β
|η|
,
g
(ξ)
=
|ξ|
q
p
where βkp αkq = 1, β p (α2 − α1 )q = 1, and κq (β1 − β2 )p = 1.
(2.7)
Proof From 1.1 we get W −W1 = χ1 W1 +χ2 W2 −χ1 W1 −χ2 W1 = χ2 (W2 −W1 ),
1
|z|p which is a convex function, and h∗ (z) =
then h(z) = (W2 − W1 )(z) = α2 −α
p
β
q
q |z|
where β p (α2 − α1 )q = 1. Moreover W2 − W = χ1 W2 + χ2 W2 − χ1 W1 −
χ2 W2 = χ1 (W2 − W1 ).
In other hand W ∗ = χ1 Wi∗ + χ2 W2∗ and Wk∗ (z) = βqk |z|q where βkp αkq = 1.
Therefore W ∗ − W2∗ = χ2 (W2∗ − W1∗ ) and W1∗ − W ∗ = χ2 (W1∗ − W2∗ ), then
2
|z|q and g ∗ (z) = κp |z|p where κq (β1 − β2 )p = 1. 2
g(z) = (W1∗ − W2∗ )(z) = β1 −β
q
Lemma 5 Under the same hypothesis of lemma 1 we have:
Z
f (ξ) = inf inf ∼ [−h∇u + ξ, σi + χ1 (x)h∗ (σ) + W2 (∇u + ξ)] dx ,
∀ξ ∈ RN : W
q
σ∈Lper u∈Kp
Y
(2.8)
Z
inf ∼ [−hσ + η, ζi + χ2 (x)g ∗ (ζ) + W1∗ (σ + η)] dx .
f ∗ (η) = inf
∀η ∈ RN : W
p
ζ∈Lper σ∈Sq
Y
(2.9)
Proof These variational principles are consequences of theorem 3 of the article
[T.Q.M.S] and the lemma 4.
2
Lemma 6 Let ϕ is a Y −periodic solution of ∆ϕ = χk − θk in Y , H its Hessian
matrix, δ ∈ R, r > 0, η ∈ RN and u = δh∇ϕ, ηi, then
Z
Z
Z
Z
u ∈ Kp , ∇u = δHη, ∼ Hχk = ∼ H 2 , ∼ h∇u, ηiχk = δ∼ |Hη|2 ,
(2.10)
Y
Y
Y
ZZ
r2
∼ ∼ |Hη|2 = θ1 θ2 .
N
Y
(2.11)
Sr Y
Proof See for example the theorem 4 of [T.Q.M.S] .
2
Bounds on the Effective ... Dielectric Composites...
Lemma 7 If 2 ≤ p < ∞, r > 0 and H as in the lemma 6, then
ZZ
p+1
r2
G(p, r) = ∼ ∼ |Hη|p ≤ θ1 θ2 + (p − 2)N 2 θ1 θ2 Z(p)(r2 + rp+1 ) ,
N
59
(2.12)
Sr Y
where Z(p) = C
in [T.Q.M] .
p+1
(p + 1) and
C is the Calderon-Zygmund-Stein constant given
2
Proof Using the corollary 3 of [T.Q.M] there is t ∈ [2, p] such that
G(p, r) ≤ G(2, r) + (p − 2)N
t+1
t+1
2
rt+1 C
t+1
(t + 1)θ1 θ2 (θ1t + θ2t ) .
(2.13)
p+1
Clearly C (t + 1) ≤ C
(p + 1) = Z(p), θkt ≤ θk2 ≤ θk then θ1t + θ2t ≤ 1 and
(t+1)/2
(p+1)/2
N
≤N
, then using 2.11 and 2.13 we get 2.12.
2
Lemma 8 Given N ∈ N, 2 ≤ p < ∞, p−1 + q −1 = 1, 0 < < 1 and s ≥ 0, then
∀x, y ∈ RN :
1
1
1
|x+y|p ≤
1 + (p − 1)(p − 2)2p−2 |x|p +hx, yi|x|p−2 + (p−1)2p−2 −sp/2 |y|p .
p
p
q
(2.14)
Proof If p > 2 the function F : RN → R given as F (x) = p1 |x|p is of class
C 2 (RN ), then given x, y ∈ RN there is t ∈ [0, 1] such that F (x + y) = F (x) +
h∇F (x), yi+ 12 hA(z)y, yi where z = x+ty and A(z) is the Hessian matrix of F at
z. We have ∇F (x) = x|x|p−2 and A(z) = I|z|p−2 + (p − 2)B(z)|z|p−4 , where I is
the identity matrix and B(z) is the matrix ((zi , zj )), clearly the greatest eigenvalue
of B(z) is |z|2 , the greatest eigenvalue of A(z) is (p−1)|z|p−2 , then 12 hA(z)y, yi ≤
2
p−2
1
. Using a standard inequality gives |y|2 |z|p−2 = (−s |y|2 )(s |z|p−2 ) ≤
2 |y| |z|
2 −sp/2
p
sp/p−2
sp/p−2 p−1
|y| + p−2
|z|p ≤ p2 −sp/2 |y|p + p−2
2 (|x|p + |y|p ) =
p
p p p−2 sp/p−2 p−1
2 |x|p + p2 (−sp/2 + (p − 2)sp/p−2 2p−2 )|y|p , then
p (p − 1)
(p − 2)sp/p−2 2p−2 +
p
(p − 1) −sp/2
+
+ (p − 2)sp/p−2 2p−2 |y|p ,
p
1
hA(z)y, yi ≤
2
then
1
|x + y|p
p
≤
1
1 + (p − 1)(p − 2)sp/p−2 2p−2 |x|p +
p
1 −sp/2
+
+ (p − 2)sp/p−2 2p−2 |y|p +
q
+hx, yi|x|p−2 .
60
G. Tepedino, J. Quintero, E. Marquina, J. Soto.
Since 0 < < 1, s > 0 and p > 2, then sp/p−2 ≤ 1, 2p−2 > 1, 1 ≤ −sp/2
and −sp/2 + (p − 2)sp/p−2 2p−2 2p−2 < −sp/2 + (p − 2)2p−2 < −sp/2 2p−2 (p −
2)−sp/2 2p−2 = (p − 1)−sp/2 2p−2 . From this we obtain 2.14. We notice that in
the special case p = 2 we have 12 |x + y|2 = 12 + hx, yi + 12 |y|2 ≤ 12 |x|2 + hx, yi +
1 −s
|y|2 , therefore the inequality 2.14 is also true when p = 2.
2
2
N
Lemma 9 Given
Z h γ1 ∈ R, γ2 > 0, η ∈ R i and T2 : K2 → R defined as
γ2
T2 (u) = ∼ γ1 h∇u, ηiχk + |∇u|2 dx, then inf = T2 (b
u), where u
b =
u∈K2
2
Y
− γγ12 h∇ϕ, ηi and ϕ is the Y −periodic solution of ∆ϕ = χk − θk in Y .
Proof Clearly T2 is a proper strictly convex function and G1 −differentiable
on the reflexive Banach space K2 . By the used of the Poincarè inequality
we can prove that lim T2 (u) = +∞, therefore there is an unique mini||u||→∞
mizer u
b ∈ K2 which satisfies ∀u Z∈ K2 : DT2 (b
u, u) = 0. Since
u, u) =
Z
Z DT2 (b
∼ [γ1 h∇u, ηiχk + γ2 h∇u, ∇b
ui] = ∼ h∇u, γ1 ηχk + γ2 ∇b
ui, then ∼ div (γ1 ηχk +
Y
Y
Y
γ2 ∇b
u)u dx = 0, hence γ2 δb
u = −div (γ1 ηχk ) in Y , from here we get the expected
result.
2
Lemma 10 ZGiven γ1 ∈ R, γ2 > 0, ξ ∈ RN and M2 : S2 → R defines as
M2 (σ) = ∼ γ1 hσ, ξiχk + γ22 |σ|2 dx, then inf M2 (σ) = M2 (b
σ ), where σ
b=
σ∈S2
γ1
γ2 (Hξ
Y
− (χk − θk )ξ) and H is the Hessian matrix of the Y −periodic solution of
∆ϕ = χk − θk in Y .
Proof Clearly M2 is a proper strictly convex function which is G1 -differentiable
and coercive (Poincarè inequality) on the reflexive Banach space S2 , therefore
there is an unique Zminimizer σ
b which satisfies ∀σ
σ , σ) = 0,
Z ∈ S2 : DM2 (b
since DM2 (b
σ , σ) = ∼ [γ1 hσ, ξiχk + γ2 hσ, σ
bi] dx = ∼ hσ, γ1 ξχk + γ2 σ
bi dx, then
Y
Y
(γ1 ξχk + γ2 σ
b) ∈ S2⊥ = V2 ⊕ CV , then there is u ∈ K2 and c ∈ RN such that
γ1 ξχk + γ2 σ
b = ∇u + c. Since div(b
σ ) = 0 in Y we have ∆u = div(γ1 ξχk ), that is
u = γ1 h∇ϕ, ξi where ϕ is the Y −periodic solution of ∆ϕ = χZk − θk in Y . Since
∇u = γ1 Hξ, then γ1 ξχk + γ2 σ
b = γ1 Hξ + c. From the fact ∼ σ
b = θ, we obtain
γ1 ξθk = c and σ
b=
γ1
γ2 (Hξ
Y
− (χk − θk )ξ).
2
Bounds on the Effective ... Dielectric Composites...
3
61
f when 2 ≤ p < ∞
An Upper Bound on W
f as 1.1 and 1.2, then if 2 ≤ p < ∞ and p−1 +q −1 =
Theorem 1 Given W and W
1 we have
Z
f )∗ (η) ≤ F1 (p, θ, r, t) = arq −br2 t−cr2 tp +d(r2 +rp+1 )tp ,
∀r > 0, t > 0 : ∼ (W 0 −W
Sr
(3.1)
0
where W (ξ) =
and a =
β
,
qθ1p−1
p
C0
p |ξ| ,
2−2/p
b=
θ2
2/p
N α2 θ1
, c=
0
p−2
C = α2 [1 + (p − 1)(p − 2)2
(p−1)2p−2 θ2
,
N qθ1p−1 αp−1
2
d=
]
(3.2)
(p−1)(p−2)2p−2 θ1 θ2 N (p+1)/2 Z(p)
θ1p−1 αp−1
2
(3.3)
Proof We will use the variational principle 2.8 where h is given by 2.6. Given
η ∈ RN and σ(x) = χ1 (x)η we have
Z
f (ξ) ≤ θ1 hξ, ηi+θ1 h∗ (η)+ inf ∼ [−h∇u, ηiχ1 (x) + W2 (∇u + ξ)] dx .
∀ξη ∈ RN : W
u∈Kp
Y
(3.4)
Using lemma 8 with x = ξ, y = ∇u and s = 2 − 4/p we get
W2 (∇u + ξ) ≤
α2 1 + (p − 1)(p − 2)2p−2 +
p
α2
+ α2 hξ, ∇ui|ξ|p−2 +
(p − 1)2p−2 2−p |∇u|p ,
q
substituting this into 3.4 we obtain
f (ξ) − W 0 (ξ) ≤ −θ1 hξ, ηi + θ1 h∗ (η) + inf Tp (u) ,
∀ξ, η ∈ RN : W
u∈Kp
(3.5)
Z
α2
where Tp (u) = ∼ [−h∇u, ηiχ1 +
(p − 1)2p−2 2−p |∇u|p ] dx.
q
Y
It is known that inf T2 (u) = T2 (ũ) where ũ =
u∈K2
1
α2 h∇ϕ, ηi
being ϕ the
Y −periodic solution of ∆ϕ = χ1 − θ1 , then ∀t > 0:
inf Tp (u) ≤ Tp (t(θ1 θ2 )1−p/2 ũ) =
Z α2
≤ ∼ −t(θ1 θ2 )1−2/p h∇ũ, ηiχ1 +
(p − 1)2p−2 2−p tp (θ1 θ2 )p−2 |∇ũ|p dx .
q
u∈Kp
Y
.
62
G. Tepedino, J. Quintero, E. Marquina, J. Soto.
If H is the Hessian matrix of ϕ, then using the lemma 6 and replacing 2.10 into
the last inequality we obtain that ∀t > 0:
#
Z "
t
p − 1 p−2 2−p p (θ1 θ2 )p−2
p
1−2/p
2
inf Tp (u) ≤ ∼ − (θ1 θ2 )
|Hη| dx ,
|Hη| +
2 t
u∈Kp
α2
q
α2p−1
Y
(3.6)
choosing = θ1 θ2 and replacing 3.6 into 3.5 we get ∀ξ, η ∈ RN , ∀t > 0:
f − W 0 )(ξ) ≤ − θ1 hξ, ηi + θ1 h∗ (η) − t
(W
Z
(θ1 θ2 )1−2/p
∼ |Hη|2 dx+
α2
Y
Z
p − 1 p−2 tp
2
∼ |Hη|p dx ,
+
q
α2p−1
Y
replacing η
η/θ1 , adding hξ, ηi to both sides of the last result and taking sup
over ξ ∈ RN we obtain ∀η ∈ RN , ∀t > 0:
Z
1−2/p Z
tθ2
p − 1 p−2 tp
∗
∗
2
f
(W −W ) (η) ≤ θ1 h (η/θ1 )−
∼ |Hη| dx+
∼ |Hη|p dx .
2
1+2/p
q
α2p−1 θ1p
α2 θ1
0
Y
Y
(3.7)
Given r > 0, integrating both sides of 3.7 over Sr and using 2.11 of lemma 6
and 2.12 of lemma 7 we obtain 3.1, 3.2 and 3.3.
2
Theorem 2 Under the same hypothesis of theorem 1, given F1 and C 0 by 3.1
and 3.2, then:
Z
h
i
f ≤ 1 C 0 (p) − (qA1 (p))1−p rp , where
∀r > 0 : ∼ W
(3.8)
p
Sr
A1 (p) = inf inf r−q F1 (p, r, t) .
r>0 t>0
(3.9)
f (rξ) = rp W
f (ξ), then
Proof Since ∀ξ ∈ RN , ∀r > 0 : W 0 (rξ) = rp W 0 (p) and W
p
0
∗
q
0
∗
0
0
f )(rξ) = r (W − W
f )ξ) and (W − W
f (rη) = r (W − W
f ) (η), hence
(W − W
∗
−q
0
∗
0
f
f
(W − W ) (η) = r (W − W ) (η) and by 3.1 we obtain
Z
Z
f )∗ (η) ds(η) = rq ∼ (W 0 − W
f )∗ (rη) ds(η)
∼ (W 0 − W
S1
S1
Z
f )∗ ≤ r−q F1 (p, r, t) ,
= r−q ∼ (W 0 − W
Sr
Bounds on the Effective ... Dielectric Composites...
63
Z
f )∗ ≤ A1 (p), where A1 is given by 3.9.
therefore ∼ (W 0 − W
S1
f ∗ (η) ≥ hξ, ηi − W 0 (ξ) + W
f (ξ). Let η =
In other hand, ∀ξ, η ∈ RN : (W 0 − W
6
∗
0
f ) (η) ≥ r|η|−W (r|η/|η|)+ W
f (rη/|η|) =
θ, r > 0 and ξ = rη/|η|, we get (W 0 − W
0
f (η/|η|), integrating over S1 we obtain A1 (p) ≥ r − C 0 rp +
r|η| − Cp rp + rp W
p
Z
Z
Z
1−p
−p
C0
C0
p f
f
f
r ∼ W , from this ∼ W ≤ p − r
+ r A1 (p), then ∼ W ≤ p inf {−r1−p +
r>0
S1
−p
S1
S1
C0
p
+ rb1−p + A1 (p)b
r−p where rb = qA1 (p), the substitution gives the
Z
Z
f
f (rξ) ds(ξ) =
estimation 3.9 with r = 1, then ∼ W (ξ) ds(ξ) = ∼ W
r
A1 (p)} =
Sr
Z
f (ξ) ds(ξ) ≤
r ∼W
p
1
p
h
S1
0
C − (qA1 (p))
1−p
i
2
p
r .
S1
f is isotropic, then
Corollary 1 Under the same hypothesis of theorem 1, if W
h
i
f (ξ) ≤ 1 C 0 − (qA1 (p))1−p |ξ|p ,
(3.10)
∀ξ ∈ RN : W
p
where C 0 and A1 are given by 3.2 and 3.9.
β 2
2θ1 r −
(α2 − α1 )−1 ,
Observation-(1): In the limit case p = 2 we have F1 (2, r, t) =
2
= β2 θ1−1 r2 + ( t2 − t)α2−1 θ1−1 θ2 N −1 r2 , where β =
then A1 (2) = inf inf r−2 F1 (2, r, t) = 2θ11 (β − αθ22N ), therefore
θ2
θ2
2
α2 N θ1 r t + 2θ1 N α2
r>0 t>0
1
2
h
i
−1
C 0 − (2A1 (2))
=
"
1
2
α2 − θ1
θ2
1
−
α2 − α1
α2 N
−1 #
,
which is the optimal upper bound of the linear composite.
4
f when 1 < p ≤ 2
An Upper Bound On W
f as 1.1 and 1.2, then if 1 < p ≤ 2 and p−1 +q −1 = 1
Theorem 3 Given W and W
we have
Z
f )∗ ≤ F2 (p, r, t) = arq − btr2 + ctp rp ,
∀r > 0, ∀t > 0 : ∼ (W2 − W
(4.1)
Sr
where a =
β
,
θ1q−1
b=
θ2
(θ1 θ2 )p/2 N −p/2
.
, c=
α2 N θ1
pα2p−1 θ1p
(4.2)
64
G. Tepedino, J. Quintero, E. Marquina, J. Soto.
Z
Proof Given u ∈ Kp and ξ ∈ RN , since ∼ h∇u, ξi = 0, using the Jensen
Y
inequality and the inequality (|a| + |b|)p/2 ≤ |a|p/2 + |b|p/2 when 1 < p ≤ 2, we
have

p/2 
p/2

p/2
Z
Z
Z
Z
∼ |∇u+ξ|p ≤ ∼ |∇u + ξ|2 
= |ξ|2 + ∼ |∇u|2 
≤ |ξ|p +∼ |∇u|2 
,
Y
Y
Y
Y
N
substituting this result into 3.4 we obtain ∀ξ, η ∈ R :
f − W2 )(ξ) ≤ −θ1 hξ, ηi + θ1 h∗ (η) + inf Mp (u),
(W
u∈Kp

p/2
Z
Z
where Mp (u) = αp2 ∼ |∇u|2  − ∼ h∇u, ηiχ1 .
Y
(4.3)
Y
It is known that inf M2 (u) = M2 (b
u) where u
b =
u∈K2
1
α2 h∇ϕ, ηi
being ϕ the
Y −periodic solution of ∇ϕ = χ1 − θ1 in Y. Therefore ∀t > 0 : inf Mp (u) ≤
u∈Kp
Mp (tb
u) and by the same arguments used in the proof of theorem 1 we obtain

p/2
Z
Z
p
t
∼ |Hη|2  − t ∼ |Hη|2 ,
inf ≤
u∈Kp
α2
pα2p−1
Y
Y
substituting this into 4.3 we get
∀ξ, η ∈ RN , ∀t > 0 :

p/2
Z
Z
p
t
f − W2 )(ξ) ≤ −θ1 hξ, ηi + θ1 h∗ (η) +
∼ |Hη|2  − t ∼ |Hη|2 .
(W
α2
pα2p−1
Y
Y
(4.4)
Replacing η
η/θ1 in 4.4, adding hξ, ηi to both sides of that result and taking
sup over ξ ∈ RN we obtain
∀ξ ∈ RN , ∀t > 0 :
f )∗ (η) ≤ θ1 h∗ (η/θ1 ) +
(W2 − W
t
p
pα2p−1 θ1p

p/2
Z
∼ |Hη|2  −
Y
Z
t
∼ |Hη|2 .
α2 θ12
Y
(4.5)

p/2

p/2
Z Z
ZZ
Given r > 0 the Jensen inequality gives ∼ ∼ |Hη|2 
≤ ∼ ∼ |Hη|2  .
Sr
Y
Sr Y
Hence integrating 4.5 over Sr and using 2.11 we get 4.1 and 4.2.
2
Bounds on the Effective ... Dielectric Composites...
65
Theorem 4 Under the same hypothesis of theorem 3, given F2 by 4.1and 4.2,
then:
Z
h
i
f ≤ 1 α2 − (qA2 (p))1−p rp ,
∼W
(4.6)
p
Sr
where A2 (p) = inf inf r−q F2 (p, r, t) .
(4.7)
r>0 t>0
2
Proof Similar to the proof of theorem 2.
f is isotropic, then
Corollary 2 Under the same hypothesis of theorem 3, if W
h
i
f (ξ) ≤ 1 α2 − (qA2 (p))1−p rp ,
(4.8)
∀ξ ∈ RN : W
p
where A2 is given by 4.6.
Observation-(2): In the limit case p = 2 the formula 4.8 gives the optimal
upper bound of the linear composite.
5
f when 2 ≤ p < ∞
A Lower Bound On W
f as 1.1 and 1.2, then if 2 ≤ p < ∞ and p−1 +q −1 =
Theorem 5 Given W and W
1 we have
Z
f )∗ ≤ F3 (p, r, t) = arp − btr2 + dtq rq ,
∀r > 0, ∀t > 0 : ∼ (W1 − W
(5.1)
Sr
1−q
κ 1−p
q/2 −q/2 β1
θ2 , b = (1 − N −1 )θ1 θ2−1 β1−1 , d = (1 − N −1 )q/2 θ1 θ2
,
p
q
(5.2)
and κ, β1 are given by 2.7
where a =
Proof We choose the variational principal 2.9, given ξ ∈ RN we take ζ = χ2 ξ,
then ∀ξ, η ∈ RN :
Z f ∗ (η) ≤ −hη, ξiθ2 + θ2 g ∗ (ξ) + inf ∼ −hσ, ξiχ2 + β1 |σ + η|q dx . (5.3)
W
σ∈Sq
q
Y

q/2
Z
Z
Z
Since 1 < q ≤ 2 and ∼ hσ, ηi = 0, then ∼ |σ + η|q ≤ ∼ |σ + η|2 
=
Y
Y
Y
q/2

q/2
Z
Z
|η|2 + ∼ |σ|2 
≤ |η|q + ∼ |σ|2  , hence

Y
∀η, ξ ∈ R
Y
N
f∗
: W (η) ≤
W1∗ (η)
− hη, ξiθ2 + θ2 g ∗ (ξ) + inf Mq (σ) ,
σ∈Sq
(5.4)
66
G. Tepedino, J. Quintero, E. Marquina, J. Soto.
where Mq (σ) =
β1
q

q/2
Z
Z
∼ |σ|2  −∼ hσ, ξiχ2 . We know that inf M2 (σ) = M2 (b
σ ),
σ∈S2
Y
Y
where σ
b = − β11 (Hξ − (χ2 − θ2 )ξ), being H the Hessian matrix of the Y −periodic
solution of ∆ϕ = χ2 − θ2 , then

q/2
Z
Z
β1 q 
σ |2  − t∼ hb
σ , ξiχ2 .
t ∼ |b
inf Mq (σ) ≤ Mq (tb
σ) =
σ∈Sq
q
Y
(5.5)
Y
Z
Z
2
1
1
1
We have hb
σ , ξi = − β1 hHξ, ξiχ2 + β1 (χ2 −θ2 )|ξ| χ2 and ∼ hb
σ , ξiχ2 = − β1 ∼ |Hξ|2 +
Y
Y
2
1
β1 θ1 θ2 |ξ| .
Z
∼ |b
σ |2 =
Y
1
β12
Also |b
σ |2 = β12 |Hξ|2 − 2(χ2 − θ2 )hHξ, ξi + (χ2 − θ2 )2 |ξ|2
1


Z
θ1 θ2 |ξ|2 − ∼ |Hξ|2 . Hence
and
Y

q/2
Z
Z
β11−q q 
t
t
2
2
inf M2 (σ) ≤
t θ1 θ2 |ξ| − ∼ |Hξ|
+ ∼ |Hξ|2 − θ1 θ2 |ξ|2 .
σ∈Sq
q
β1
β1
Y
Y
(5.6)
Substituting 5.5 5.6 into 5.4 we get

q/2
Z
1−q
β
f ∗ (η) − W1∗ (η) ≤ −hη, ξiθ2 + θ2 g ∗ (ξ) + 1 tq θ1 θ2 |ξ|2 − ∼ |Hξ|2 
W
q
Y
Z
+ βt1 ∼ |Hξ|2 − βt1 θ1 θ2 |ξ|2 .
Y
(5.7)
Replacing ξ
ξ/θ2 in 5.7, adding hξ, ηi to both side of the last result and taking
sup over η ∈ RN we get

q/2
Z
1−p
1−q q
κθ
β
t
f ∗ )∗ (ξ) ≤ 2 |ξ|p + 1
θ1 θ2−1 |ξ|2 − θ2−2 ∼ |Hξ|2 
(W1∗ − W
p
q
(5.8)
Y
Z
t −2
t
2
2
+ β1 θ2 ∼ |Hξ| − β1 θ1 θ2 |ξ| ,
Y
integrating over Sr we finally get 5.1 and 5.2.
2
Bounds on the Effective ... Dielectric Composites...
Theorem 6 Under the same hypothesis of theorem 5 we have
Z
h
i Z
h
i1−p
f ∗ ≤ 1 β1 − (pA3 (p))1−q , ∼ W
f ≥ 1 β1 − (pA3 (p))1−q
∼W
,
q
p
S1
67
(5.9)
S1
where A3 (p) = inf inf r−p F( p, r, t) being F3 given by 5.1.
r>0 t>0
Proof By the same homogeneity properties using in the proof of the theorem 4
we have
Z
Z
∗ ∗
−p
∗
f
f ∗ )∗ ≤ A3 (p). Since (W ∗ −
∼ (W2 − W ) ≤ r F3 (p, r, t), then ∼ (W2∗ − W
1
S1
S1
f ∗ )∗ ≥ hξ, ηi − W1∗ (η) + W
f ∗ (η), fixing ξ 6= θ and r > 0, let η = rξ/|ξ|,
W
∗ ∗
q
∗
q f∗
∗
f
we obtain (W1 − W ) (ξ) ≥ r|ξ|
Z − r W1Z(ξ/|ξ|) + r W (ξ/|ξ|), then A3 (p) ≥
Z
f ∗ )∗ ≥ r − r q β 1 + r q ∼ W
f ∗ and ∼ W
f ∗ ≤ r−q A3 (p) − r1−q + β1 , hence
∼ (W ∗ − W
1
q
q
S1
SZ1
f ∗ ≤ inf r−q A3 (p) − r1−q +
∼W
r>0
S1
β1
q ,
that is
S1
Z
h
i
f ∗ ≤ 1 β1 − (pA3 (p))1−q = B3 (p) .
∼W
q
(5.10)
S1
f , taking η =
f ∗ (η) ≥ hξ, ηi − W
In other hand, since W
6 θ, r > 0 and ξ = rη/|η|,
∗
pf
f
we get W (η) ≥
Z r|η| − r ZW (η/|η|), integrating over S1 and using 5.10 we obtain
p f
f ≥ r1−p − r−p B(p), then sup r1−p − r−p B(p) ≤
B(p) ≥ r − r ∼ W and ∼ W
r>0
S1
S1
Z
f
∼ W . From here and 5.10 we obtain 5.9.
2
S1
Corollary 3 Under the same hypothesis of theorem 5 we have
Z
Z
h
i
h
i1−p
f ∗ ≤ 1 β1 − (pA3 (p))1−q rq , ∼ W
f ≥ 1 β1 − (pA3 (p))1−q
∀r > 0 : ∼ W
rp ,
q
p
Sr
Sr
(5.11)
f is isotropic, then
Corollary 4 Under the same hypothesis of theorem 5, if W
∀ξ, η ∈ RN :
h
i
h
i1−p
f ∗ (η) ≤ 1 β1 − (pA3 (p))1−q |η|q , W
f (ξ) ≥ 1 β1 − (pA3 (p))1−q
W
|ξ|p .
q
p
(5.12)
68
G. Tepedino, J. Quintero, E. Marquina, J. Soto.
f , when 1 < p ≤ 2
Lower Bound on W
6
f as 1.1 and 1.2, then if 1 ≤ p ≤ 2 and p−1 +q −1 = 1
Theorem 7 Given W and W
we have
Z
f ∗ )∗ ≤ F4 (p, r, t) = arp − br2 t + dr2 tq + ctq (r2 + rq+1 )
∀r > 0, t > 0 : ∼ (W0∗ − W
Sr
(6.1)
where
W0∗ (η)
=
C0∗
q
q |η| ,
C0∗
q−2
= β1 1 + (q − 1)(q − 2)2
and
(6.2)
1 2−2/q −2/q
βq
θ21−p κ
1
, b = (1 − )β1
θ2
, c = 1 (q − 1)22q−2 θ2−1 (1 + ),
p
N
q
N
a=
β −q
d = 1q (q − 1)(q − 2)22q−2 θ2−q N (q+1)/2 Z(q) and Z(p) =
Stein-constants.
C (q + 1)q+1
(6.3)
is the
We choose the variational principle 4.3, then ∀η ∈ RN :
Z f (η) ≤ −hη, ξiθ2 + g ∗ (ξ)θ2 + inf ∼ −hσ, ξiχ2 + β1 |σ + η|q dx .
W
σ∈Sq
q
(6.4)
Y
Since q ≥ 2 using 2.14 with x = η, y = σ and s = 2 − 4/q we get
1
1
1
|σ +η|q ≤
1 + (q − 1)(q − 2)2q−2 |η|q +hη, σi|η|q−2 + (q −1)2q−2 2−q |σ|q ,
q
q
q
substituting this inequality into 6.4 we get
f (η) − W0 (η) ≤ −hη, ξiθ2 + g ∗ (η)θ2 + inf Mq (σ) ,
∀ξ, η ∈ RN : W
(6.5)
σinSq
Z 1
where Mq (σ) = ∼ −hσ, ξiχ2 + (q − 1)(q − 2)2q−2 2−q |σ|q dx. Since
q
Y
inf M2 (σ) = M2 (b
σ ), where σ
b = − β11 [Hξ − (χ2 − θ2 )], being H the Hessian
σ∈S2
matrix of the Y −periodic solution of ∆ϕ = χ2 − θ2 , then ∀t > 0:
inf Mq (σ) ≤ Mq (t(θ1 θ2 )1−2/q σ
b)
Z
Z
β1
= −t(θ1 θ2 )1−2/q ∼ hb
σ , ξiχ2 + (q − 1)2q−2 2−q tq (θ1 θ1 )q−2 ∼ |b
σ |q .
q
σ∈Sq
Y
Y
(6.6)
69
Bounds on the Effective ... Dielectric Composites...
Z
Z
In other hand hb
σ , ξi = − β11 hHξ, ξi − (χ2 − θ2 )|ξ|2 and ∼ hb
σ , ξiχ2 = − β11 ∼ |Hξ|2 .
Y
Y
Therefore from here, 6.6, taking = θ1 θ1 and using an standard inequality, we
get
inf Mq (σ) ≤
σ∈Sq
−tβ1−1 (θ1 θ2 )2−2/q |ξ|2
Z
+
t(θ1 θ2 )1−2/q β1−1 ∼ |Hξ|2
+
Y
Z
−q
−1
q
q
q
q β1
2q−3
2q−3
q β1
+t
2
(θ1 θ2 + θ1 θ2 )|ξ| + t
(q − 1)2
∼ |Hξ|q ,
q
q
Y
substituting this inequality into 6.4 we obtain that ∀ξ, η ∈ RN :
f ∗ (η) − W0∗ (η) ≤ − hξ, ηiθ2 + g ∗ (ξ)θ2 − tβ −1 (θ1 θ2 )2−2/q |ξ|2
W
1
Z
tq β1−1
+ t(θ1 θ2 )1−2/q β1−1 ∼ |Hξ|2 +
(q − 1)22q−2 |ξ|2 +
q
Y
Z
tq β1−q
+
(q − 1)22q−3 ∼ |Hξ|q ,
q
(6.7)
Y
replacing ξ
ξ/θ2 , adding hξ, ηi to both side of the result and taking sup over
η ∈ RN , we finally get 6.1, 6.2 and 6.3.
2
Theorem 8 Under the same hypothesis of theorem 7 we have
Z
Z
i
i1−p
h
h
f ∗ ≤ 1 C0∗ − (pA4 (p))1−q rq , ∼ W
f ≥ 1 C0∗ − [pA4 (p))1−q
rp ,
∀r > 0 : ∼ W
q
p
Sr
Sr
(6.8)
where A4 = inf inf F4 (p, r, t).
r>0 t>0
f is isotropic we
Corollary 5 Under the same hypothesis of theorem 7, when W
N
have ∀ξ, η ∈ R :
f (ξ) ≥
W
f ∗ (η) ≤
W
1−p p
1 ∗
C − (pA4 (p))1−q
|ξ|
p 0
1 ∗
C0 − (pA4 (p))1−q |η|q .
q
70
7
G. Tepedino, J. Quintero, E. Marquina, J. Soto.
Summary and Conclusions
Given 1 < p < ∞, p−1 + q −1 = 1, 1 < θk < 1, θ1 + θ2 = 1, θ = θ1 , we obtain
C 0 (p) > 0 and C0 (p) > 0 such that
Z
i1−p
1h
q(p−1)
1−q
p
f
C0 (p) − θ1
(pA(p, θ))
r ≤ ∼W
p
Sr
i
1h 0
p(q−1)
1−p
C (p) − θ2
(qB(p.θ))
rp
≤
p
C0 (p) =
α2 [1 + (p − 1)(p − 2)2p−1 ] if
α2
if
A(θ, p) =
C 0 (p) =
inf
+
(x,y)∈R+
0 ×R0
x−p F(θ, p, x, y)
β1
if
β1 [1 + (q − 1)(q − 2)2q−1 ] if
B(θ, p) =
inf
+
(x,y)∈R+
0 ×R0
p≥2
,
1<p<2
p≥2
,
1<p<2
x−q (θ, p, x, y)
where the functions F, are given in this work.
In the isotropic case we have
i1−p
1h
q(p−1)
1−q
f (ξ)
|ξ|p ≤ W
C0 (p) − θ1
(pA(p, θ))
p
i
1h 0
p(q−1)
1−p
≤
C (p) − θ2
(qB(p, θ))
|ξ|p .
p
Here we have θ1 = θ and θ2 = 1 − θ, where 1 ≤ θ ≤ 1 is the volume fraction
of the first material.
In the limit case p = 2, the left and right hand sides of the last inequality
give the optimal lower and upper bound of the linear composite.
Acknowledgement: The authors; want to thank Vicenzo Constenzo Alvarez
of the Universidad Simón Bolı́var, Department of Physics, for having revised this
paper and Oswaldo Araujo of the University of the Andes, Faculty of Science,
Department of Mathematics who helped this paper to be published in these Bulletin.
Likewise, we thank Mr. Antonio Vizcaya P. for transcription it.
Bounds on the Effective ... Dielectric Composites...
71
References
[1] J. Ash and R. Jones. Calderon-Zygmund and Beyond. Conference in
Honor of Stephen Vagi Retirement. De Paul University, Chicago, Illinois,
2006.
[2] B.Bergman. Bulk Physical Properties of a Composite Media. lectures
Notes. L’Ecole d’ete’ de Analyse Numerique, 1983.
[3] A. Braides. Omogeneizazione di Integrali non Coercive. Estratto de
Ricerche di Matematica, Volume XXXII, pp.347, 1983.
[4] R.Burridge, S.Childress, G.Papanicolaou. Macroscopic Properties of
Disordered Media. Spronger-Verlag, New York, 1982.
[5] I.Ekland,R.Temam. Convex Analysis and variational Problems. North
Holland, 1976.
[6] G.Dell’Antonio. Non-linear Electrostatic in Inhomogeneous Media.
Preprint, Dipartamento di Matematica, Universita’ di Roma, La Sapienza,
Italia, 1987.
[7] K.Lurie, A.Cherkaev. Exact Estimates of Conductivity of Composites
Formed by two Isotropic Conducting Media Taken in Prescribed Proportion.
Proc. Royal Soc. Edimburg, 29A, pp.71-87,1984.
[8] Hashin, Shtrikman. A Variational Approach to the Theory
of the Effective Magnetic Permeability of Multi-phase Materials.
J.Appl.Phys,33,pp.3125-3131, 1962.
[9] M.Milkis. Effective Dielectric Constants of a Non-Linear Composites
Material. SIAM, J.Appl Math, Vol 43, N.3, Oct,1983.
[10] Yves-Patricck Pellegrini. Self-consistent Effective-medium Approximation for Strongly Nonlinear Media. Phis.Rev B. Vol. 64, 2001.
[11] Sundaram Thangavelu . Harmonic Analysis on the Heisemberg Group.
Springer, 1998.
[12] Sundaram Thangavelu . An Introduction to the Uncertainty Principle.
Springer, 2004.
[13] Dong-Hau Kuo, Wun-Ku Wang. Dielectric Properties of three Ceramic
Epoxy Composites. Materials Chemistry and Physics. Volume 85. Issue 1,
2004, pages 201-206.
[14] S.Talbot, J.Willis. Variational Principles for Inhomogeneous Non-linear
Media. IMA, Journal of Applied Mathematics, 35,pp. 39-54, 1985.
72
G. Tepedino, J. Quintero, E. Marquina, J. Soto.
[15] G.Tepedino, J. Quintero, E. Marquina. An application of the Theory
of Calderon-Zygmund to the Sciences of the Materials. Journal of Mathematical Control Science and Applications (JMCSA). Vol.5, N.1, June 2012,
pp, 11-18.)
[16] G.Tepedino. Bounds on the Effective Energy Density of Nonlinear Composites. Doctoral Thesis, Courant Institute of Mathematical Sciences, 1988.
[17] G.Tepedino, J. Quintero, E. Marquina. Bounds on the Effective Energy Density of a More General Class Willis Dielectric Composites. To
appear.
[18] Hung T. Vo and Frank G. Shi Towards Model-Based Engineering of
Optoelectronic Packing Materials: Dielectric Constant Modeling. Microelectronics Journal. Volume 33. Issues 5-6, 2002, pages 409-415.
[19] Tom Wolf. Lectures in Harmonic Analysis. AMS University Lecture Series,
Vol 29, 2003.
[20] J.Willis. Variational and Related Methods for the Overall Properties of
Composites. Advanced in Applied Mechanics. Academic Press, Vol. 21, 1981.
[21] Zhou C., Neese B., Zhang Q., Bauur F. Relaxor Ferroelectric
Poly(Vinylidene fluoride-trifluororthylene-chlofluoroethylene) Terpolymer
for High Energy Density Storage Capacitors. Dielectrics and Electrical
Insulation, IEEE Transaction . Vol 13, Issue 5, 2006.
Gaetano Tepedino Aranguren,
Departamento de Matemáticas
Facultad de Ciencias,
Universidad de los Andes
Mérida, Venezuela.
Javier Quintero C.
Área de Matemática,
Universidad Nacional Abierta de
Mérida.
Eribel Marquina.
Área de Matemática,
UNEFA, Mérida.
José Soto,
Departamento de Matemáticas
Facultad de Ciencias,
Universidad de los Andes
Mérida, Venezuela.