Bounds on the effective energy density of a special class p-dielectric

Bounds on the effective energy density of a special class p-dielectric

Gaetano Tepedino Aranguren, Javier Quintero C, Eribel Marquina, Jos´ e Soto

Abstract. This work gives lower and upper bounds on the effective energy densityWf of a two phase composites material composed by 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 thek−material are given respectively by

ν1(z) =α1|z|^p−2, ν2(z) =α2|z|^p−2, where 0< α1< α2and 1< p <∞.

For anisotropic composites the bounds are given in the form Φ(p, r, θ)≤ ∼

Z

S_r

W˜ ≤Ψ(p, r, θ),

where the functions Φ,Ψ reduce smoothly to the optimal lower and upper bound of the linear composite whenp→2.

The method to obtain this bounds, in the case p6= 2, follows a generalization of the Hashin–Shtrikman variational principles con- structed from a comparison medium which is in general nonlinear and reduces to linear whenp= 2.

Resumen. Este trabajo da cotas inferiores y superiores para la densidad de energ´ıa eficazfW de un material compuesto de dos fases, constituido por una mezcla peri´odica de dos materiales diel´ectricos isotr´opicos homog´eneos no lineales en proporci´on prescrita. Estas cotas son dadas como una funci´on deθ, que es la fracci´on de volu- men del material con constante diel´ectrica m´as bajo en la mezcla.

La constante diel´ectrica de conductividad del k-material se dan, respectivamente, por

ν₁(z) =α₁|z|^p−2, ν₂(z) =α₂|z|^p−2,

donde 0< α1< α2 y 1< p <∞.

Para compuestos anisotr´opicos los l´ımites se dan en la forma Φ(p, r, θ)≤ ∼

Z

Sr

W˜ ≤Ψ(p, r, θ),

donde las funciones Φ,Ψ se reducen suavemente a la cota inferior y superior del compuesto lineal cuandop→2.

El m´etodo para obtener estas cotas, en el casop6= 2, sigue de una generalizaci´on de los principios variacionales de Hashin–Shtrikman construido a partir de un medio de comparaci´on que es en general no lineal y se reduce al lineal cuandop= 2.

1 Introduction

In this work we will follow theY−periodic microstructure of the mixture, been Y the cell

N

Y

i=1

(0, a_i), {a1, a₂, . . . , a_N} ⊂ (0,∞), if θ_k, for k ∈ {1,2}, is the proportion of material typekin the mixed,Y =Y₁∪Y₂,Y₁∩Y₂=∅ andχ_k is the characteristic function of the phaseYk which only contains materialk, then θk=∼

Z

Y

χk, 0≤θk≤1 andθ1+θ2= 1. Following the notation of [T.Q.M.S] , theenergy densityof the composite is theY−periodic extension of the function W :R^N ×R^N →Rgiven by

W(x, z) =χ₁(x)W₁(z) +χ₂(x)W₂(z), where

W_k(z) =^α_p^k|z|^p, and 0< α₁< α₂,1< p <∞. (1.1) In [T.Q.M.S] has been proved that theeffective energy densityand its dual are given respectively by the variational principles:

fW(ξ) = inf

v∈V_p∼ Z

Y

W(x, v+ξ)dx , Wf^∗(η) = inf

σ∈S_q∼ Z

Y

W^∗(x, σ+η)dx , (1.2)

whereVp is the completion ofC_per¹ (Y ,Rⁿ) under the Lp−norm andSq is the completion ofN ={σ∈C_per¹ (Y,R^N) :∼

Z

Y

σ=θ and div(σ) = 0 in Y} under theLq−norm. Notice thatv∈Vp ⇐⇒ v=∇ufor someu∈Kp. HereKpis the completion ofC_per¹ (Y) under the normkuk1,p=



∼

Z

Y

|u|^pdx+∼ Z

Y

|∇u|^pdx





1/p

.

Also we haveV_p^⊥ =Sq⊕CV, whereCV are the constants vector fields. See [T.Q.M.S] for a complete description of definitions and properties of these spaces.

2 Existence of W f , Γ−convergence and Homogenization

Lemma 1 The functionW :Rⁿ →R^N →Rdefined by (1.1) satisfies:

(1)∀z∈R^N :W(., z)is Y−periodic and measurable.

(2)∀x∈R^N :W(x, .)isC¹(Rⁿ)and strictly convex.

(3)∀x, z∈R^N : 0≤^α_p¹|z|^p≤W(x, z)≤ ^α_p²|z|^p.

(4)∃L≥0such that∀x, z1, z2∈R^N :|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, thenW(x, zk)^1/p=β1χ1(x)|zk|+β2χ2(x)|zk|andW(x, z1)^1/p− W(x, z2)^1/p=β1χ1(x)(|z1| − |z2|) +β2χ2(x)(|z1| − |z2|), therefore|W(x, z1)^1/p− W(x, z2)^1/p| ≤L(|z1| − |z2|)≤L|z1−z2|. 2 Lemma 2 IfW is the function defned by 1.1, then fW and Wf^∗ are given by 1.2.

Proof These are consequences of lema 1 and the articles [M.M],[G.DA].2 Lemma 3 (Elementary Bounds)

IfW is defined by 1.1, fW is defined by 1.2 andp⁻¹+q⁻¹= 1, then

∀ξ∈R^N : 1 p

θ₁α^−q/p₁ +θ₂α^−q/p₂ ^−p/q

|ξ|^p≤Wf(ξ)≤1

p(θ₁α₁+θ₂α₂)|ξ|^p. (2.1) These bounds are called The Elementary Lower and Upper Bounds on Wf.

Proof Since the null vector field belongs to Vp, then from 1.2 we obtain fW(ξ)≤ ∼

Z

Y

W(x, ξ)dx=θ1α1

p|ξ|^p+θ2

α2

p |ξ|^p= 1

p(θ1α1+θ1α2)|ξ|^p. (2.2) In other hand since the null vector field belongs toSq, then from 1.2 we get

fW^∗(η)≤ ∼ Z

Y

W^∗(x, η)dx=θ₁W₁^∗(η) +θ₂W₂^∗(η) =1 q

θ₁α^−q/p₁ +θ₂α^−q/p₂

|η|^q, (2.3) from this, using a typical result of convex analysis, see for example[E,T], we get

fW(ξ)≥_p¹

θ1α^−q/p₁ +θ2α^−q/p₂ −p/q

|ξ|^p,

from this last inequality and 2.2 we obtain 2.1. 2

Lemma 4 Under the same hypothesis of lemma 3 we have:

∀x, ξ∈R^N :W(x, ξ)−W₁(ξ) =χ₂(x)h(ξ), W₂(ξ)−W(x, ξ) =χ₁(x)h(ξ) (2.4)

∀x, η∈R^N :W₁^∗(η)−W^∗(x, η) =χ2(x)g(η), W^∗(x, η)−W₂^∗(η) =χ1(x)g(η) (2.5)

∀ξ, η∈R^N :h(ξ) =^α2−α_p ¹|ξ|^p, h^∗(η) = ^β_q|η|^q,

g(η) = ^β1−β_q ²|η|^q, g^∗(ξ) =^κ_p|ξ|^p (2.6) where β_k^pα^q_k = 1, β^p(α2−α1)^q= 1, and κ^q(β1−β2)^p= 1. (2.7) Proof From 1.1 we getW−W1=χ₁W₁+χ₂W₂−χ1W₁−χ2W₁=χ₂(W₂−W1), then h(z) = (W₂−W₁)(z) = ^α2−α_p ¹|z|^p which is a convex function, and h^∗(z) =

β

q|z|^q where β^p(α2−α1)^q = 1. MoreoverW2−W =χ1W2+χ2W2−χ1W1− χ2W2=χ1(W2−W1).

In other hand W^∗=χ1W_i^∗+χ2W₂^∗ and W_k^∗(z) = ^β_q^k|z|^q whereβ_k^pα^q_k = 1.

Therefore W^∗ −W₂^∗ = χ2(W₂^∗−W₁^∗) and W₁^∗ −W^∗ = χ2(W₁^∗−W₂^∗), then g(z) = (W₁^∗−W₂^∗)(z) = ^β1−β_q ²|z|^q andg^∗(z) = ^κ_p|z|^p whereκ^q(β1−β2)^p= 1. 2 Lemma 5 Under the same hypothesis of lemma 1 we have:

∀ξ∈R^N :Wf(ξ) = inf

σ∈L^q_per inf

u∈Kp

∼ Z

Y

[−h∇u+ξ, σi+χ1(x)h^∗(σ) +W2(∇u+ξ)]dx , (2.8)

∀η∈R^N :fW^∗(η) = inf

ζ∈L^p_per inf

σ∈Sq

∼ Z

Y

[−hσ+η, ζi+χ2(x)g^∗(ζ) +W₁^∗(σ+η)]dx . (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 Letϕis aY−periodic solution of∆ϕ=χ_k−θ_k inY, H its Hessian matrix,δ∈R,r >0,η∈R^N andu=δh∇ϕ, ηi, then

u∈K_p, ∇u=δHη, ∼ Z

Y

Hχ_k =∼ Z

Y

H², ∼ Z

Y

h∇u, ηiχ_k=δ∼

Z

Y

|Hη|², (2.10)

∼ Z

Sr

∼ Z

Y

|Hη|²= r²

Nθ₁θ₂. (2.11)

Proof See for example the theorem 4 of [T.Q.M.S]. 2

Lemma 7 If2≤p <∞,r >0 andH as in the lemma 6, then G(p, r) =∼

Z

S_r

∼ Z

Y

|Hη|^p≤ r²

Nθ1θ2+ (p−2)N^p+12 θ1θ2Z(p)(r²+r^p+1), (2.12) whereZ(p) =

C

^{p+1(p+ 1)and}

C

is the Calderon-Zygmund-Stein constant given

in[T.Q.M]. 2

Proof Using the corollary 3 of [T.Q.M] there ist∈[2, p] such that

G(p, r)≤ G(2, r) + (p−2)N^t+12 r^t+1

C

^t+1(t+ 1)θ1θ2(θ^t₁+θ^t₂). (2.13) Clearly

C

^t+1(t+ 1)≤

C

^p+1(p+ 1) =Z(p),θ^t_k ≤θ_k²≤θk thenθ₁^t+θ^t₂≤1and N^(t+1)/2≤N^(p+1)/2, then using 2.11 and 2.13 we get 2.12. 2 Lemma 8 GivenN∈N,2≤p <∞, p⁻¹+q⁻¹= 1,0< <1ands≥0, then

∀x, y∈R^N: 1

p|x+y|^p≤ 1 p

1 + (p−1)(p−2)2^p−2

|x|^p+hx, yi|x|^p−2+1

q(p−1)2^p−2−sp/2|y|^p. (2.14) Proof If p >2 the function F : R^N →R given as F(x) = ¹_p|x|^p is of class C²(R^N), then given x, y∈R^N there is t∈[0,1] such thatF(x+y) =F(x) + h∇F(x), yi+¹₂hA(z)y, yiwherez=x+tyandA(z)is the Hessian matrix ofF at z. We have∇F(x) =x|x|^p−2andA(z) =I|z|^p−2+ (p−2)B(z)|z|^p−4, whereI is the identity matrix andB(z)is the matrix((zi, zj)), clearly the greatest eigenvalue ofB(z)is|z|², the greatest eigenvalue ofA(z)is(p−1)|z|^p−2, then ¹₂hA(z)y, yi ≤

1

2|y|²|z|^p−2. Using a standard inequality gives|y|²|z|^p−2= (^−s|y|²)(^s|z|^p−2)≤

2

p^−sp/2|y|^p + ^p−2_p ^sp/p−2|z|^p ≤ ²_p^−sp/2|y|^p + ^p−2_p ^sp/p−22^p−1(|x|^p +|y|^p) =

p−2

p ^sp/p−22^p−1|x|^p+²_p(^−sp/2+ (p−2)^sp/p−22^p−2)|y|^p, then 1

2hA(z)y, yi ≤ (p−1)

p (p−2)^sp/p−22^p−2+ +(p−1)

p

^−sp/2+ (p−2)^sp/p−22^p−2

|y|^p, then

1

p|x+y|^p ≤ 1 p

1 + (p−1)(p−2)^sp/p−22^p−2

|x|^p+ +1

q

^−sp/2+ (p−2)^sp/p−22^p−2

|y|^p+ +hx, yi|x|^p−2.

Since 0 < < 1, s > 0 and p > 2, then ^sp/p−2 ≤ 1,2^p−2 > 1,1 ≤ ^−sp/2 and ^−sp/2+ (p−2)^sp/p−22^p−22^p−2 < ^−sp/2+ (p−2)2^p−2< ^−sp/22^p−2(p− 2)^−sp/22^p−2= (p−1)^−sp/22^p−2. From this we obtain 2.14. We notice that in the special casep= 2we have ¹₂|x+y|²= ¹₂+hx, yi+¹₂|y|²≤¹₂|x|²+hx, yi+

1

2^−s|y|², therefore the inequality 2.14 is also true whenp= 2. 2 Lemma 9 Givenγ₁∈R, γ₂>0, η∈R^N andT₂:K₂→Rdefined as

T2(u) = ∼ Z

Y

h

γ1h∇u, ηiχk+γ2

2 |∇u|²i

dx, then inf

u∈K2

= T2(u), whereb ub =

−^γ_γ¹

2h∇ϕ, ηiandϕis theY−periodic solution of∆ϕ=χk−θk inY.

Proof Clearly T2 is a proper strictly convex function and G¹−differentiable on the reflexive Banach space K2. By the used of the Poincar`e inequality we can prove that lim

||u||→∞T2(u) = +∞, therefore there is an unique mini- mizer ub ∈ K2 which satisfies ∀u ∈ K2 : DT2(bu, u) = 0. Since DT2(u, u) =b

∼ Z

Y

[γ₁h∇u, ηiχ_k+γ₂h∇u,∇bui] = ∼ Z

Y

h∇u, γ₁ηχ_k +γ₂∇ui, thenb ∼ Z

Y

div(γ₁ηχ_k + γ2∇u)u dxb = 0, henceγ2δbu=−div(γ1ηχk)inY, from here we get the expected

result. 2

Lemma 10 Given γ1∈R, γ2>0, ξ∈R^N andM2:S2→Rdefines as M2(σ) =∼

Z

Y

γ1hσ, ξiχk+^γ₂²|σ|²

dx, then inf

σ∈S2

M2(σ) =M2(σ), whereb bσ=

γ1

γ2(Hξ−(χ_k−θ_k)ξ)andH is the Hessian matrix of theY−periodic solution of

∆ϕ=χ_k−θ_k inY.

Proof ClearlyM₂is a proper strictly convex function which isG¹-differentiable and coercive (Poincar`e inequality) on the reflexive Banach spaceS₂, therefore there is an unique minimizer bσ which satisfies ∀σ ∈ S₂ : DM₂(σ, σ) = 0,b since DM2(bσ, σ) = ∼

Z

Y

[γ1hσ, ξiχk+γ2hσ,σi]b dx = ∼ Z

Y

hσ, γ1ξχk +γ2bσidx, then (γ1ξχk+γ2bσ)∈ S₂^⊥ =V2⊕CV, then there isu∈K2 and c ∈R^N such that γ1ξχk+γ2σb=∇u+c. Since div(bσ) = 0inY we have∆u=div(γ1ξχk), that is u=γ1h∇ϕ, ξiwhere ϕis theY−periodic solution of ∆ϕ=χk−θk inY. Since

∇u=γ₁Hξ, then γ₁ξχ_k+γ₂bσ=γ₁Hξ+c. From the fact ∼ Z

Y

σb=θ, we obtain γ1ξθk =candσb=^γ_γ¹

2(Hξ−(χk−θk)ξ). 2

3 An Upper Bound on W f when 2 ≤ p < ∞

Theorem 1 GivenW andfW as 1.1 and 1.2, then if2≤p <∞andp⁻¹+q⁻¹= 1 we have

∀r >0, t >0 :∼ Z

S_r

(W⁰−fW)^∗(η)≤ F1(p, θ, r, t) =ar^q−br²t−cr²t^p+d(r²+r^p+1)t^p, (3.1) where W⁰(ξ) = ^C_p⁰|ξ|^p, C⁰=α₂[1 + (p−1)(p−2)2^p−2] (3.2) and a= ^β

qθ^p−1₁ , b= ^θ

2−2/p 2

N α₂θ^2/p₁ , c= ^{(p−1)2p−2θ2}

N qθ^p−1₁ α^p−1₂ , d=(p−1)(p−2)2^p−2θ1θ2N^(p+1)/2Z(p) θ₁^p−1α^p−1₂ .

(3.3) Proof We will use the variational principle 2.8 wherehis given by 2.6. Given η∈R^N andσ(x) =χ₁(x)η we have

∀ξη∈R^N :fW(ξ)≤θ₁hξ, ηi+θ₁h^∗(η)+ inf

u∈Kp

∼ Z

Y

[−h∇u, ηiχ₁(x) +W₂(∇u+ξ)]dx . (3.4) Using lemma 8 withx=ξ, y=∇uands= 2−4/pwe get

W2(∇u+ξ)≤α2

p

1 + (p−1)(p−2)2^p−2 + +α₂hξ,∇ui|ξ|^p−2+α₂

q (p−1)2^p−22−p|∇u|^p, substituting this into 3.4 we obtain

∀ξ, η∈R^N :Wf(ξ)−W⁰(ξ)≤ −θ1hξ, ηi+θ1h^∗(η) + inf

u∈Kp

Tp(u), (3.5)

whereTp(u) =∼ Z

Y

[−h∇u, ηiχ1+α2

q (p−1)2^p−22−p|∇u|^p]dx.

It is known that inf

u∈K₂T₂(u) = T₂(˜u) where u˜ = _α¹

2h∇ϕ, ηi being ϕ the Y−periodic solution of∆ϕ=χ1−θ1, then ∀t >0:

u∈Kinfp

Tp(u)≤Tp(t(θ1θ2)^1−p/2u) =˜

≤ ∼ Z

Y

−t(θ1θ2)^1−2/ph∇˜u, ηiχ1+α₂

q (p−1)2^p−22−pt^p(θ1θ2)^p−2|∇˜u|^p

dx .

IfH is the Hessian matrix ofϕ, then using the lemma 6 and replacing 2.10 into the last inequality we obtain that ∀t >0:

u∈Kinfp

Tp(u)≤ ∼ Z

Y

"

− t α2

(θ1θ2)^1−2/p|Hη|²+p−1

q 2^p−22−pt^p(θ1θ2)^p−2 α^p−1₂ |Hη|^p

# dx , (3.6) choosing =θ₁θ₂ and replacing 3.6 into 3.5 we get ∀ξ, η∈R^N,∀t >0:

(fW−W⁰)(ξ)≤ −θ₁hξ, ηi+θ₁h^∗(η)−t(θ₁θ₂)^1−2/p α2

∼ Z

Y

|Hη|²dx+

+p−1

q 2^p−2 t^p α^p−1₂ ∼

Z

Y

|Hη|^pdx ,

replacingη η/θ1, addinghξ, ηito both sides of the last result and taking sup overξ∈R^N we obtain ∀η∈R^N,∀t >0:

(W⁰−fW)^∗(η)≤θ₁h^∗(η/θ₁)− tθ₂^1−2/p α₂θ^1+2/p₁ ∼

Z

Y

|Hη|²dx+p−1

q 2^p−2 t^p α^p−1₂ θ₁^p∼

Z

Y

|Hη|^pdx . (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, givenF1 andC⁰ by 3.1 and 3.2, then:

∀r >0 : ∼ Z

S_r

Wf≤ 1 p h

C⁰(p)−(qA1(p))^1−pi

r^p, where (3.8)

A1(p) = inf

r>0inf

t>0r^−qF1(p, r, t). (3.9) Proof Since∀ξ∈R^N,∀r >0 :W⁰(rξ) =r^pW⁰(p)andWf(rξ) =r^pfW(ξ), then (W⁰−fW)(rξ) =r^p(W⁰−fW)ξ)and(W⁰−fW^∗(rη) =r^q(W⁰−Wf)^∗(η), hence (W⁰−fW)^∗(η) =r^−q(W⁰−Wf)^∗(η)and by 3.1 we obtain

∼ Z

S₁

(W⁰−fW)^∗(η)ds(η) =r^q∼ Z

S₁

(W⁰−Wf)^∗(rη)ds(η)

=r^−q∼ Z

S_r

(W⁰−fW)^∗≤r^−qF1(p, r, t),

therefore∼ Z

S1

(W⁰−Wf)^∗≤A₁(p), whereA₁ is given by 3.9.

In other hand,∀ξ, η∈R^N : (W⁰−fW^∗(η)≥ hξ, ηi −W⁰(ξ) +Wf(ξ). Let η6=

θ, r >0andξ=rη/|η|, we get(W⁰−fW)^∗(η)≥r|η|−W⁰(r|η/|η|)+fW(rη/|η|) = r|η| −^C_p⁰r^p+r^pfW(η/|η|), integrating over S1 we obtain A1(p)≥r−^C_p⁰r^p+ r^p∼

Z

S₁

Wf, from this∼ Z

S₁

Wf≤ ^C_p⁰ −r^1−p+r^−pA1(p), then∼ Z

S₁

Wf≤ ^C_p⁰ inf

r>0{−r^1−p+ r^−pA1(p)}= ^C_p⁰+br^1−p+A1(p)br^−p where br=qA1(p), the substitution gives the estimation 3.9 withr= 1, then∼

Z

Sr

fW(ξ)ds(ξ) =∼ Z

S1

fW(rξ)ds(ξ) = r^p∼

Z

S₁

Wf(ξ)ds(ξ)≤¹_ph

C⁰−(qA₁(p))^1−pi

r^p. 2

Corollary 1 Under the same hypothesis of theorem 1, if fW is isotropic, then

∀ξ∈R^N :Wf(ξ)≤1 p

hC⁰−(qA₁(p))^1−pi

|ξ|^p, (3.10) whereC⁰ andA1 are given by 3.2 and 3.9.

Observation-(1): In the limit case p = 2 we have F1(2, r, t) = _2θ^β

1r²−

θ₂

α2N θ1r²t+_2θ^θ2

1N α2 = ^β₂θ⁻¹₁ r²+ (^t₂²−t)α⁻¹₂ θ⁻¹₁ θ₂N⁻¹r², whereβ= (α₂−α₁)⁻¹, thenA1(2) = inf

r>0inf

t>0r⁻²F1(2, r, t) = _2θ¹

1(β−_α^θ2

2N), therefore

1 2

h

C⁰−(2A1(2))⁻¹i

=¹₂

"

α2−θ1

1 α2−α1

− θ₂ α2N

−1# , which is theoptimal upper bound of the linear composite.

4 An Upper Bound On W f when 1 < p ≤ 2

Theorem 3 GivenW andWfas 1.1 and 1.2, then if1< p≤2andp⁻¹+q⁻¹= 1 we have

∀r >0,∀t >0 : ∼ Z

Sr

(W2−fW)^∗≤ F2(p, r, t) =ar^q−btr²+ct^pr^p, (4.1)

where a= β

θ₁^q−1, b= θ2

α₂N θ₁, c= (θ1θ2)^p/2N^−p/2

pα^p−1₂ θ₁^p . (4.2)

Proof Given u ∈ K_p and ξ ∈ R^N, since ∼ Z

Y

h∇u, ξi = 0, using the Jensen inequality and the inequality(|a|+|b|)^p/2≤ |a|^p/2+|b|^p/2 when 1< p≤2, we have

∼ Z

Y

|∇u+ξ|^p≤



∼

Z

Y

|∇u+ξ|²





p/2

=



|ξ|²+∼ Z

Y

|∇u|²





p/2

≤ |ξ|^p+



∼

Z

Y

|∇u|²





p/2

,

substituting this result into 3.4 we obtain∀ξ, η∈R^N: (fW−W2)(ξ)≤ −θ1hξ, ηi+θ1h^∗(η) + inf

u∈Kp

Mp(u), where Mp(u) =^α_p²



∼

Z

Y

|∇u|²





p/2

− ∼ Z

Y

h∇u, ηiχ1.

(4.3)

It is known that inf

u∈K2

M2(u) = M2(bu) where bu = _α¹

2h∇ϕ, ηi being ϕ the Y−periodic solution of ∇ϕ=χ1−θ1 in Y. Therefore ∀t >0 : inf

u∈Kp

Mp(u)≤ Mp(tu)b and by the same arguments used in the proof of theorem 1 we obtain

u∈Kinfp

≤ t^p pα^p−1₂



∼

Z

Y

|Hη|²





p/2

− t α₂∼

Z

Y

|Hη|²,

substituting this into 4.3 we get

∀ξ, η∈R^N,∀t >0 :

(fW−W2)(ξ)≤ −θ1hξ, ηi+θ1h^∗(η) + t^p pα^p−1₂



∼

Z

Y

|Hη|²





p/2

− t α2

∼ Z

Y

|Hη|². (4.4) Replacingη η/θ₁ in 4.4, addinghξ, ηito both sides of that result and taking sup overξ∈R^N we obtain

∀ξ∈R^N,∀t >0 :

(W2−fW)^∗(η)≤θ1h^∗(η/θ1) + t^p pα^p−1₂ θ₁^p



∼

Z

Y

|Hη|²





p/2

− t α₂θ²₁∼

Z

Y

|Hη|². (4.5) Given r >0the Jensen inequality gives ∼

Z

S_r



∼

Z

Y

|Hη|²





p/2

≤



∼

Z

S_r

∼ Z

Y

|Hη|²





p/2

. Hence integrating 4.5 overS_r and using 2.11 we get 4.1 and 4.2. 2

Theorem 4 Under the same hypothesis of theorem 3, given F2 by 4.1and 4.2, then:

∼ Z

S_r

fW ≤ 1 p h

α2−(qA2(p))^1−pi

r^p, (4.6)

where A2(p) = inf

r>0inf

t>0r^−qF2(p, r, t). (4.7) Proof Similar to the proof of theorem 2. 2 Corollary 2 Under the same hypothesis of theorem 3, if fW is isotropic, then

∀ξ∈R^N : Wf(ξ)≤ 1 p h

α2−(qA2(p))^1−pi

r^p, (4.8)

whereA₂ is given by 4.6.

Observation-(2): In the limit casep= 2the formula 4.8 gives the optimal upper bound of the linear composite.

5 A Lower Bound On W f when 2 ≤ p < ∞

Theorem 5 GivenW andfW as 1.1 and 1.2, then if2≤p <∞andp⁻¹+q⁻¹= 1 we have

∀r >0,∀t >0 :∼ Z

S_r

(W₁−Wf)^∗≤ F3(p, r, t) =ar^p−btr²+dt^qr^q, (5.1)

where a= κ

pθ₂^1−p, b= (1−N⁻¹)θ₁θ₂⁻¹β₁⁻¹, d= (1−N⁻¹)^q/2θ₁^q/2θ₂^−q/2β₁^1−q q , (5.2) andκ, β1 are given by 2.7

Proof We choose the variational principal 2.9, given ξ∈R^N we takeζ=χ2ξ, then ∀ξ, η∈R^N:

Wf^∗(η)≤ −hη, ξiθ2+θ2g^∗(ξ) + inf

σ∈Sq

∼ Z

Y

−hσ, ξiχ2+β1

q |σ+η|^q

dx . (5.3)

Since 1 < q ≤ 2 and ∼ Z

Y

hσ, ηi = 0, then ∼ Z

Y

|σ+η|^q ≤



∼

Z

Y

|σ+η|²





q/2

=



|η|²+∼ Z

Y

|σ|²





q/2

≤ |η|^q+



∼

Z

Y

|σ|²





q/2

, hence

∀η, ξ∈R^N :Wf^∗(η)≤W₁^∗(η)− hη, ξiθ2+θ2g^∗(ξ) + inf

σ∈Sq

Mq(σ), (5.4)

whereMq(σ) = ^β_q¹



∼

Z

Y

|σ|²





q/2

−∼

Z

Y

hσ, ξiχ2. We know that inf

σ∈S2

M2(σ) =M2(bσ), wherebσ=−_β¹

1(Hξ−(χ2−θ2)ξ), beingHthe Hessian matrix of theY−periodic solution of∆ϕ=χ2−θ2, then

inf

σ∈Sq

M_q(σ)≤M_q(tbσ) =β₁ q t^q



∼

Z

Y

|bσ|²





q/2

−t∼

Z

Y

hbσ, ξiχ2. (5.5)

We havehbσ, ξi=−_β¹

1hHξ, ξiχ2+_β¹

1(χ2−θ2)|ξ|²χ2and∼ Z

Y

hbσ, ξiχ2=−_β¹

1∼ Z

Y

|Hξ|²+

1

β1θ₁θ₂|ξ|². Also |σ|b² = _β¹2 1

|Hξ|²−2(χ₂−θ₂)hHξ, ξi+ (χ₂−θ₂)²|ξ|² and

∼ Z

Y

|σ|b²= _β¹2 1



θ₁θ₂|ξ|²− ∼ Z

Y

|Hξ|²



. Hence

σ∈Sinfq

M2(σ)≤β₁^1−q q t^q



θ1θ2|ξ|²− ∼ Z

Y

|Hξ|²





q/2

+ t β₁∼

Z

Y

|Hξ|²− t

β₁θ1θ2|ξ|². (5.6) Substituting 5.5 5.6 into 5.4 we get

fW^∗(η)−W₁^∗(η)≤ −hη, ξiθ2+θ₂g^∗(ξ) +β₁^1−q q t^q



θ₁θ₂|ξ|²− ∼ Z

Y

|Hξ|²





q/2

+_β^t

1∼ Z

Y

|Hξ|²−_β^t

1θ1θ2|ξ|².

(5.7) Replacingξξ/θ₂ in 5.7, addinghξ, ηito both side of the last result and taking sup overη∈R^N we get

(W₁^∗−fW^∗)^∗(ξ)≤ κθ^1−p₂

p |ξ|^p+β₁^1−qt^q q



θ1θ⁻¹₂ |ξ|²−θ⁻²₂ ∼ Z

Y

|Hξ|²





q/2

+_β^t

1θ⁻²₂ ∼ Z

Y

|Hξ|²−_β^t

1θ₁θ₂|ξ|²,

(5.8)

integrating overSr we finally get 5.1 and 5.2. 2

Theorem 6 Under the same hypothesis of theorem 5 we have

∼ Z

S1

Wf^∗≤ 1 q h

β1−(pA3(p))^1−qi , ∼

Z

S1

fW ≥ 1 p h

β1−(pA3(p))^1−qi1−p

, (5.9)

whereA₃(p) = inf

r>0inf

t>0r^−pF₍p, r, t)beingF₃ given by 5.1.

Proof By the same homogeneity properties using in the proof of the theorem 4 we have

∼ Z

S₁

(W₂^∗−fW^∗)^∗≤r^−pF3(p, r, t), then∼ Z

S₁

(W₂^∗−Wf^∗)^∗≤A3(p). Since (W₁^∗− fW^∗)^∗ ≥ hξ, ηi −W₁^∗(η) +Wf^∗(η), fixing ξ 6= θ and r > 0, let η = rξ/|ξ|, we obtain (W₁^∗−Wf^∗)^∗(ξ) ≥r|ξ| −r^qW₁^∗(ξ/|ξ|) +r^qWf^∗(ξ/|ξ|), then A3(p)≥

∼ Z

S₁

(W₁^∗−fW^∗)^∗≥r−r^{q β}_q¹+r^q∼ Z

S₁

Wf^∗ and∼ Z

S₁

fW^∗≤r^−qA3(p)−r^1−q+ ^β_q¹, hence

∼ Z

S1

Wf^∗≤ inf

r>0 r^−qA3(p)−r^1−q

+^β_q¹, that is

∼ Z

S₁

fW^∗≤1 q h

β1−(pA3(p))^1−qi

=B3(p). (5.10)

In other hand, sinceWf^∗(η)≥ hξ, ηi −fW, taking η6=θ, r >0 andξ=rη/|η|, we getWf^∗(η)≥r|η| −r^pWf(η/|η|), integrating over S1 and using 5.10 we obtain B(p)≥r−r^p∼

Z

S1

Wf and ∼ Z

S1

fW ≥r^1−p−r^−pB(p), then sup

r>0

r^1−p−r^−pB(p)

≤

∼ Z

S₁

Wf. From here and 5.10 we obtain 5.9. 2

Corollary 3 Under the same hypothesis of theorem 5 we have

∀r >0 : ∼ Z

S_r

Wf^∗≤ 1 q h

β1−(pA3(p))^1−qi r^q, ∼

Z

S_r

Wf≥1 p h

β1−(pA3(p))^1−qi1−p

r^p, (5.11) Corollary 4 Under the same hypothesis of theorem 5, if fW is isotropic, then

∀ξ, η∈R^N : fW^∗(η)≤ 1

q h

β1−(pA3(p))^1−qi

|η|^q, Wf(ξ)≥ 1 p h

β1−(pA3(p))^1−qi1−p

|ξ|^p. (5.12)

6 Lower Bound on W f , when 1 < p ≤ 2

Theorem 7 GivenW andWfas 1.1 and 1.2, then if1≤p≤2andp⁻¹+q⁻¹= 1 we have

∀r >0, t >0 : ∼ Z

S_r

(W₀^∗−fW^∗)^∗≤ F4(p, r, t) =ar^p−br²t+dr²t^q+ct^q(r²+r^q+1) (6.1) where W₀^∗(η) =^C_q^0∗|η|^q, C₀^∗=β₁ 1 + (q−1)(q−2)2^q−2

and (6.2) a= θ^1−p₂ κ

p , b= (1− 1

N)β₁^2−2/qθ^−2/q₂ , c=β₁^q

q (q−1)2^2q−2θ⁻¹₂ (1 + 1

N), (6.3) d = ^β

−q 1

q (q−1)(q−2)2^2q−2θ₂^−qN^(q+1)/2Z(q) and Z(p) =

C

(q+ 1)^q+1 is the Stein-constants.

We choose the variational principle 4.3, then ∀η∈R^N:

fW(η)≤ −hη, ξiθ2+g^∗(ξ)θ₂+ inf

σ∈Sq

∼ Z

Y

−hσ, ξiχ2+β₁

q |σ+η|^q

dx . (6.4)

Sinceq≥2using 2.14 with x=η,y=σ ands= 2−4/q we get 1

q|σ+η|^q ≤ 1 q

1 + (q−1)(q−2)2^q−2

|η|^q+hη, σi|η|^q−2+1

q(q−1)2^q−22−q|σ|^q, substituting this inequality into 6.4 we get

∀ξ, η∈R^N :fW(η)−W0(η)≤ −hη, ξiθ2+g^∗(η)θ2+ inf

σinSq

Mq(σ), (6.5)

whereMq(σ) =∼ Z

Y

−hσ, ξiχ2+1

q(q−1)(q−2)2^q−22−q|σ|^q

dx. Since

σ∈Sinf2

M2(σ) = M2(bσ), where σb = −_β¹

1 [Hξ−(χ2−θ2)], being H the Hessian matrix of theY−periodic solution of∆ϕ=χ2−θ2, then∀t >0:

σ∈Sinfq

Mq(σ)≤Mq(t(θ1θ2)^1−2/qσ)b

=−t(θ1θ₂)^1−2/q∼ Z

Y

hσ, ξiχb 2+β₁

q (q−1)2^q−22−qt^q(θ₁θ₁)^q−2∼ Z

Y

|bσ|^q. (6.6)

In other handhσ, ξib =−_β¹

1

hHξ, ξi −(χ₂−θ₂)|ξ|² and∼

Z

Y

hσ, ξiχb ₂=−_β¹

1∼ Z

Y

|Hξ|². Therefore from here, 6.6, taking=θ1θ1 and using an standard inequality, we

get

σ∈Sinfq

Mq(σ)≤ −tβ₁⁻¹(θ1θ2)^2−2/q|ξ|²+t(θ1θ2)^1−2/qβ⁻¹₁ ∼ Z

Y

|Hξ|²+

+t^qβ₁^−q

q 2^2q−3(θ₁θ₂^q+θ₁^qθ₂)|ξ|^q+t^qβ⁻¹₁

q (q−1)2^2q−3∼ Z

Y

|Hξ|^q,

substituting this inequality into 6.4 we obtain that∀ξ, η∈R^N: Wf^∗(η)−W₀^∗(η)≤ − hξ, ηiθ₂+g^∗(ξ)θ₂−tβ₁⁻¹(θ₁θ₂)^2−2/q|ξ|²

+t(θ1θ2)^1−2/qβ₁⁻¹∼ Z

Y

|Hξ|²+t^qβ₁⁻¹

q (q−1)2^2q−2|ξ|²+ +t^qβ₁^−q

q (q−1)2^2q−3∼ Z

Y

|Hξ|^q, (6.7)

replacingξξ/θ₂, addinghξ, ηito both side of the result and taking sup over

η∈R^N, we finally get 6.1, 6.2 and 6.3. 2

Theorem 8 Under the same hypothesis of theorem 7 we have

∀r >0 :∼ Z

Sr

fW^∗≤1 q h

C₀^∗−(pA4(p))^1−qi r^q, ∼

Z

Sr

Wf≥1 p h

C₀^∗−[pA4(p))^1−qi1−p

r^p, (6.8) whereA₄= inf

r>0inf

t>0F₄(p, r, t).

Corollary 5 Under the same hypothesis of theorem 7, when fW is isotropic we have∀ξ, η∈R^N:

Wf(ξ) ≥ 1 p

C₀^∗−(pA₄(p))^1−q1−p

|ξ|^p

fW^∗(η) ≤ 1 q

C₀^∗−(pA4(p))^1−q

|η|^q.

7 Summary and Conclusions

Given 1< p < ∞, p⁻¹+q⁻¹ = 1,1< θ_k <1, θ₁+θ₂ = 1, θ =θ₁, we obtain C⁰(p)>0 andC₀(p)>0 such that

1 p h

C0(p)−θ^q(p−1)₁ (pA(p, θ))^1−qi1−p

r^p ≤ ∼ Z

S_r

Wf

≤ 1 p h

C⁰(p)−θ₂^p(q−1)(qB(p.θ))^1−pi r^p

C0(p) =

α₂[1 + (p−1)(p−2)2^p−1] if p≥2

α₂ if 1< p <2 ,

A(θ, p) = inf

(x,y)∈R⁺0×R⁺0

x^−pF(θ, p, x, y)

C⁰(p) =

β₁ if p≥2

β₁[1 + (q−1)(q−2)2^q−1] if 1< p <2 , B(θ, p) = inf

(x,y)∈R⁺₀×R⁺₀

x^−q (θ, p, x, y) where the functionsF,are given in this work.

In the isotropic case we have 1

p h

C0(p)−θ^q(p−1)₁ (pA(p, θ))^1−qi1−p

|ξ|^p ≤ Wf(ξ)

≤ 1 p h

C⁰(p)−θ₂^p(q−1)(qB(p, θ))^1−pi

|ξ|^p.

Here we haveθ1=θ andθ2= 1−θ, where1≤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 Sim´on Bol´ıvar, 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.

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Gaetano Tepedino Aranguren, Departamento de Matem´aticas Facultad de Ciencias,

Universidad de los Andes M´erida, Venezuela.

Javier Quintero C.

Area de Matem´´ atica,

Universidad Nacional Abierta de M´erida.

Eribel Marquina.

Area de Matem´´ atica, UNEFA, M´erida.

Jos´e Soto,

Departamento de Matem´aticas Facultad de Ciencias,

Universidad de los Andes M´erida, Venezuela.