Introverted algebras with mean value

arXiv:1303.7355v1 [math.FA] 29 Mar 2013

INTROVERTED ALGEBRAS WITH MEAN VALUE AND APPLICATIONS JEAN LOUIS WOUKENG Abstract. Let A be an introverted algebra with mean value. We prove that its spectrum ∆(A) is a compact topological semigroup, and that the kernel K(∆(A)) of ∆(A) is a compact topological group over which the mean value on A can be identified as the Haar integral. Based on these facts and also on the fact that K(∆(A)) is an ideal of ∆(A), we define the convolution over ∆(A). We then use it to derive some new convergence results involving the convolution product of sequences. These convergence results provide us with an efficient method for studying the asymptotics of nonlocal problems. The obtained results systematically establish the connection between the abstract harmonic analysis and the homogenization theory. To illustrate this, we work out some homogenization problems in connection with nonlocal partial differential equations.

1. Introduction and the main results Let A be an algebra with mean value on RN , that is, a closed subalgebra of the commutative Banach algebra BUC(RN ) (of bounded uniformly continuous functions on RN ) that contains the constants, is closed under complex conjugation, is translation invariant and has an invariant mean value. Thus A is a commutative Banach algebra with spectrum denoted by ∆(A). We consider each element of ∆(A) as a multiplicative linear functional on A. The usual (or Gelfand) topology of ∆(A) is the relative weak∗ topology induced on ∆(A) by σ(A′ , A). The properties of the Gelfand space ∆(A) are well known and can be found in any text book about Banach algebras, see for instance [15]. The commonly known property is that ∆(A) is a compact topological space. It is also known that it becomes metrizable provided that A is separable. In some special cases, ∆(A) is well characterized. For example, when A is the algebra of almost periodic functions, ∆(A) is a compact topological abelian group, and in particular if A is the algebra of periodic functions, ∆(A) is the N -dimensional torus TN . On the other hand, it seems that almost nothing is known about ∆(A) for general algebras with mean value A. In this paper our aim is to characterize the space ∆(A) for some general algebras A and present some applications. The relevance and importance of this characterization is due, among other things, to the fact that if A is introverted [19, p. 121] (see also [7, p. 540] for the general concept of introversion) then ∆(A) is a compact topological semigroup (see Theorem 2 below). We are particularly interested in the following questions: Date: February, 2013. 2000 Mathematics Subject Classification. Primary 46J10, 45K05; Secondary 35B40. Key words and phrases. Introverted algebras with mean value, semigroups, sigma-convergence, convolution, homogenization. 1

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JEAN LOUIS WOUKENG

(a) When is the space ∆(A) a topological semigroup? (b) Under which conditions is ∆(A) a topological group? (c) In the case ∆(A) is a topological semigroup, what are the properties of its kernel? (d) Is it possible to define the convolution on ∆(A)? (e) In the case the convolution can be defined on ∆(A), what is the connection between it and the Σ-convergence method? In this paper we try to answer these questions and present some applications of the obtained results, and the notions introduced to study them. In particular we test these questions on the algebra of almost periodic functions, the algebra of functions that converge at infinity, and in general on any closed subalgebra of the algebra of weakly almost periodic functions. The content of the paper is summarized as follows. Section 2 deals with the fundamental notions about algebras with mean value, the generalized Besicovitch spaces and the derivation theory both associated to them. In Section 3 we study the introverted algebras with mean value. We give the answer to the questions (a), (b) and (c) raised above, but this in the special setting of introverted algebras with mean value. The main results of this section are as follows. (1) If the algebra A is introverted, then ∆(A) is a compact topological semigroup. This result is known in the general theory of Banach algebras of uniformly continuous functions; see e.g. [16, 19]. Our proof relies only on the compactness of ∆(A). We also show that, if further the multiplication defined on ∆(A) is jointly continuous, then ∆(A) is a compact topological group. The second main result of Section 3 reads as (2) If A is introverted, then A is a subalgebra of the weakly almost periodic functions, and moreover, if the multiplication in ∆(A) is jointly continuous, then A is a subalgebra of the almost periodic functions. The third main result of Section 3 is the answer to the question (c) raised above, and is this (3) If A is introverted, then the kernel K(∆(A)) of ∆(A) is a compact topological group, and the mean value on A can be identified as the Haar integral over K(∆(A)). In all the previous works dealing with algebras with mean value, the mean value were identified as the integral over the spectrum only, see for instance [21, 22, 27]. Here we go further and we will see that this result is of first importance when defining the convolution over the spectrum of such algebras. The last main result of Section 3 is the basic tool that enables us to establish the connection between the convolution and the Σ-convergence method. It is new and constitutes the point of departure of all the results dealing with convergence of sequences involving delay. It reads as (4) Let A be an introverted algebra with mean value on RN . Then if δ y denotes the Dirac mass at y, we have δ y ∈ K(∆(A)) for almost all y ∈ RN . We end Section 3 with the answer to question (d). We define the convolution on the spectrum ∆(A) in terms of its kernel K(∆(A)). In Section 4, in order to deal with some applications in homogenization theory, we state and prove a De Rham type result. More precisely, the main result of this section is this (5) If A is an algebra with mean value on RN and L is a bounded linear func1,p′ N tional on (BA ) which vanishes on the kernel of the divergence, then there exists

INTROVERTED ALGEBRAS WITH MEAN VALUE

3

p a function f ∈ BA such that L = ∇y f . Although classically known in the general framework of the distribution theory and in the special setting of periodic functions, the above result is new in the framework of general algebras with mean value. In Section 5 we gather the notation and basic facts we need about the Σconvergence method. Section 6 deals with the answer to question (e). We study therein the connection between the convolution and the Σ-convergence method. The main result of this section is the following. (6) Let (uε )ε>0 ⊂ Lp (Ω) and (vε )ε>0 ⊂ Lq (RN ) be two sequences with p ≥ 1, 1 q ≥ 1 and 1p + q1 = 1 + m . Assume that, as ε → 0, uε → u0 in Lp (Ω)-weak Σ and p q vε → v0 in Lq (RN )-strong Σ, where u0 and v0 are in Lp (Ω; BA ) and Lq (RN ; BA ) respectively. Then, as ε → 0, uε ∗ vε → u0 ∗ ∗v0 in Lm (Ω)-weak Σ where ∗∗ stands for the double convolution. This result is also new. One of the main motivation of the present study arises from the importance and applications of the phenomena with delay in the real life. The results of Section 6 have applications in mathematical neuroscience, engineering sciences etc. In order to show how it works, we present in Sections 7 and 8 two applications of the results of the previous sections to homogenization theory. In particular in Section 8, we give the answer to a question raised by Attouch and Damlamian [2] about the homogenization of nonlinear operators involving convolution. This is a true advance as far as the comprehension of the spectrum of an algebra with mean value as well as the homogenization theory in general, are concerned. We end this section with some preliminary notions. A directed set is a set E equipped with a binary relation . such that

• α . α for all α ∈ E; • if α . β and β . γ then α . γ; • for any α, β ∈ E there exists γ ∈ E such that α . γ and β . γ.

A net in a set X is a mapping α 7→ xα from a directed set E into X. We usually denote such a mapping by (xα )α∈E , or just by (xα ) if E is understood, and we say that (xα ) is indexed by E. Here below are some examples of directed sets: (i) The set of positive integers N, with j . k if and only if j ≤ k. (ii) The set R\{a} (a ∈ R), with x . y if and only if |x − a| ≥ |y − a|. Unless otherwise stated, vector spaces throughout are assumed to be real vector spaces, and scalar functions are assumed to take real values. The results obtained here easily carry over mutatis mutandis to the complex setting. The results of Section 6 were announced in [31]. 2. Fundamentals of algebras with mean value We refer the reader to [38] for details regarding some of the results of this section. A closed subalgebra A of the C*-algebra of bounded uniformly continuous functions BUC(RN ) is an algebra with mean value on RN [14, 27, 41] if it contains the constants, is translation invariant (u(· + a) ∈ A for any u ∈ A and each a ∈ RN ) and is such that any of its elements possesses a mean value, that is, for any u ∈ A, the sequence (uε )ε>0 (defined by uε (x) = u(x/ε), x ∈ RN ) weakly∗-converges in L∞ (RN ) to some constant real function M (u) as ε → 0. It is known that A (endowed with the sup norm topology) is a commutative C*-algebra with identity. We denote by ∆(A) the spectrum of A and by G the Gelfand transformation on A. We recall that ∆(A) (a subset of the topological

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JEAN LOUIS WOUKENG

dual A′ of A) is the set of all nonzero multiplicative linear functionals on A, and G is the mapping of A into C(∆(A)) such that G(u)(s) = hs, ui (s ∈ ∆(A)), where h, i denotes the duality pairing between A′ and A. When equipped with the relative weak∗ topology on A′ (the topological dual of A), ∆(A) is a compact topological space, and the Gelfand transformation G is an isometric ∗-isomorphism identifying A with C(∆(A)) as C*-algebras. Moreover the mean value M defined on A is a nonnegative continuous linear functional that can be expressed in terms of a Radon measure β (of total mass 1) R in ∆(A) (called the M -measure for A [21]) satisfying the property that M (u) = ∆(A) G(u)dβ for u ∈ A. To any algebra with mean value A we associate the following subspaces: Am = {ψ ∈ C m (RN ) : Dyα ψ ∈ A ∀α = (α1 , ..., αN ) ∈ NN with |α| ≤ m} (where Dyα ψ =

αN ∂ |α| ψ/∂y1α1 · · · ∂yN ). Under the norm k|u|km = sup|α|≤m Dyα ψ ∞ , Am is a Banach space. We also define the space A∞ = {ψ ∈ C ∞ (RN ) : Dyα ψ ∈ A ∀α = (α1 , ..., αN ) ∈ NN }, a Fr´echet space when endowed with the locally convex topology defined by the family of norms k|·|km . p Next, let BA (1 ≤ p < ∞) denote the Besicovitch space associated to A, that is the closure of A with respect to the Besicovitch seminorm  1/p Z 1 p kukp = lim sup |u(y)| dy r→+∞ |Br | Br where Br is the open ball of RN centered at the origin and of radius r > 0. It p q p is known that BA is a complete seminormed vector space verifying BA ⊂ BA for ∞ 1 ≤ p ≤ q < ∞. From this last property one may naturally define the space BA as follows: p ∞ BA = {f ∈ ∩1≤p 0, and choose v ∈ A such that ku − vkp < For 1 ≤ p < ∞, let u ∈ BA η/(kϕkL1 (RN ) + 1). Using Young’s inequality, we have ku ∗ ϕ − v ∗ ϕkp ≤ kϕkL1 (RN ) ku − vk2 < η,

p hence u ∗ ϕ ∈ BA since v ∗ ϕ ∈ A. We may therefore define the convolution between p p p ∞ N BA and C0 (R ). Indeed, for u = u+N ∈ BA (RN ) (with u ∈ BA ) and ϕ ∈ C0∞ (RN ), we define u ⊛ ϕ as follows

u ⊛ ϕ := u ∗ ϕ + N ≡ ̺(u ∗ ϕ).

(4.1)

INTROVERTED ALGEBRAS WITH MEAN VALUE

15

p Indeed, this is well defined as justified below by (4.5). Thus, for u ∈ BA and p ∞ N ϕ ∈ C0 (R ) we have u ⊛ ϕ ∈ BA with α

Dy (u ⊛ ϕ) = ̺(u ∗ Dyα ϕ), all α ∈ NN . N

(4.2)

∞

We deduce from (4.2) that u ⊛ ϕ ∈ DA (R ) since u ∗ ϕ ∈ A . Moreover, we have 1

ku ⊛ ϕkp ≤ |Suppϕ| p kϕkLp′ (RN ) kukp ,

(4.3)

where Suppϕ stands for the support of ϕ and |Suppϕ| its Lebesgue measure. Indeed, we have   p1 Z , ku ⊛ ϕkp = k̺(u ∗ ϕ)kp = lim sup |Br |−1 |(u ∗ ϕ)(y)|p dy r→+∞

and

Z

Br

p

|(u ∗ ϕ)(y)| dy

Z

Br

p Z

 p |u(y)| dy Br Br Z p |u(y)|p dy, ≤ |Br ∩ Suppϕ| kϕkLp′ (B ) ≤

|ϕ| dy

r

Br

hence (4.3). For u ∈ A and ϕ ∈ C0∞ (RN ) the convolution u b ⊛ ϕ is defined as follows Z τd (s ∈ ∆(A)), (4.4) (b u ⊛ ϕ) (s) = y u(s)ϕ(y)dy RN

where u b = G(u) and τ y u = u(· + y). It is easily seen that u b ⊛ ϕ ∈ C(∆(A)). We have u[ ∗ϕ=u b ⊛ ϕ for all u ∈ A and ϕ ∈ C0∞ (RN ). (4.5) Indeed, for x ∈ RN , we have Z Z (b u ⊛ ϕ) (δ x ) = τd τ y u(x)ϕ(y)dy y u(δ x )ϕ(y)dy = RN

=

RN

(u ∗ ϕ) (x) = u[ ∗ ϕ(δ x ),

and (4.5) follows by the continuity of both u b ⊛ ϕ and u[ ∗ ϕ, and the denseness of {δ x : x ∈ RN } in ∆(A). As claimed above, (4.5) justifies that u ⊛ ϕ is wellp defined by (4.1) for u ∈ BA (RN ) and ϕ ∈ C0∞ (RN ). Indeed, for u, v ∈ u, we have p b = vb and so u = u + N = v + N . It emerges from (4.5) that u, v ∈ BA with u u[ ∗ϕ=u b ∗ ϕ = vb ∗ ϕ = v[ ∗ ϕ, hence u ∗ ϕ + N = v ∗ ϕ + N . We also have the obvious equality ∂ϕ for all 1 ≤ i ≤ N . (4.6) ∂i (b u ⊛ ϕ) = u b⊛ ∂yi The following De Rham type result holds. ′

1,p N Theorem 6. Let 1 < p < ∞. Let L be a bounded linear functional on (BA ) p which vanishes on the kernel of the divergence. Then there exists a function f ∈ BA such that L = ∇y f , i.e., Z ′ c v dβ for all v ∈ (B 1,p )N . fb divb L(v) = − A ∆(A)

p p Moreover f is unique modulo IA , that is, up to an additive function g ∈ BA verifying ∇y g = 0.

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Proof. Let u ∈ A∞ (hence ̺(u) ∈ DA (RN )). Define Lu : D(RN )N → R by Lu (ϕ) = L(̺(u ∗ ϕ)) for ϕ = (ϕi ) ∈ D(RN )N where u ∗ ϕ = (u ∗ ϕi )i ∈ (A∞ )N . Then Lu defines a distribution on D(RN )N . Moreover if divy ϕ = 0 then divy (̺(u ∗ ϕ)) = ̺(u ∗ divy ϕ) = 0, hence Lu (ϕ) = 0, that is, Lu vanishes on the kernel of the divergence in D(RN )N . By the De Rham theorem, there exists a distribution S(u) ∈ D′ (RN ) such that Lu = ∇y S(u). This defines an operator S : A∞ → D′ (RN ); u 7→ S(u) satisfying the following properties: (i) S(τ y u) = τ y S(u) for all y ∈ RN and all u ∈ A∞ ; ′ (ii) S maps linearly and continuously A∞ into Lploc (RN ); (iii) There is a positive constant Cr (that is locally bounded as a function of r) such that 1

kS(u)kLp′ (Br ) ≤ Cr kLk |Br | p′ k̺(u)kp′ . The property (i) easily comes from the obvious equality Lτ y u (ϕ) = Lu (τ y ϕ) ∀y ∈ RN . Let us check (ii) and (iii). For that, let ϕ ∈ D(RN )N with Suppϕi ⊂ Br for all 1 ≤ i ≤ N . Then |Lu (ϕ)| =

≤ ≤

|L(̺(u ∗ ϕ))|

kLk k̺(u) ⊛ ϕk(B1,p′ )N A

1

max |Suppϕi | p′ kLk k̺(u)kp′ kϕkW 1,p (Br )N ,

1≤i≤N

the last inequality being due to (4.3). Hence, as Suppϕi ⊂ Br (1 ≤ i ≤ N ), 1

kLu kW −1,p′ (Br )N ≤ kLk |Br | p′ k̺(u)kp′ . (4.7) R Now, let g ∈ C0∞ (Br ) with Br gdy = 0; then by [24, Lemma 3.15] there exists ϕ ∈ C0∞ (Br )N such that div ϕ = g and kϕkW 1,p (Br )N ≤ C(p, Br ) kgkLp (Br ) . We have |hS(u), gi| = |− h∇y S(u), ϕi| = |hLu , ϕi| ≤ kLu kW −1,p′ (Br )N kϕkW 1,p (Br )N 1

≤ C(p, Br ) kLk |Br | p′ k̺(u)kp′ kgkLp (Br ) , ′

p ′ p and by a density argument, we get that S(u) R ∈ (L (Br )/R) = L (Br )/R for any p′ p′ r > 0, where L (Br )/R = {ψ ∈ L (Br ) : Br ψdy = 0}. The properties (ii) and (iii) therefore follow from the above series of inequalities. Taking (ii) as granted it comes that Z S(u) divy ϕdy for all ϕ ∈ D(RN )N . (4.8) Lu (ϕ) = − RN

INTROVERTED ALGEBRAS WITH MEAN VALUE

17

We claim that S(u) ∈ C ∞ (RN ) for all u ∈ A∞ . Indeed let ei = (δ ij )1≤j≤N (δ ij the Kronecker delta). Then owing to (i) and (iii) above, we have



  



−1

S t−1 (τ tei u − u) − ∂u

t (τ tei S(u) − S(u)) − S ∂u =

∂yi Lp′ (Br ) ∂yi Lp′ (Br )

 

−1 ∂u

. ≤ c t (̺(τ tei u − u)) − ̺ ∂yi ′ p

Hence, passing to the limit as t → 0 above leads us to   ∂u ∂ for all 1 ≤ i ≤ N. S(u) = S ∂yi ∂yi

Repeating the same process we end up with Dyα S(u) = S(Dyα u) for all α ∈ NN . ′

So all the weak derivative of S(u) of any order belong to Lploc (RN ). Our claim is therefore a consequence of [28, Theorem XIX, p. 191]. This being so, we derive from the mean value theorem the existence of ξ ∈ Br such that Z S(u)(ξ) = |Br |

−1

S(u)dy.

Br

On the other hand, the map u 7→ S(u)(0) is a linear functional on A∞ , and by the above equality we get Z |S(u)(0)| ≤ lim sup |Br |−1 |S(u)| dy r→0

≤ ≤

Br

lim sup |Br |

− p1′

r→0

c kLk k̺(u)kp′ .

Z

p′

Br

|S(u)| dy

 1′ p

e Hence, defining Se : DA (RN ) → R by S(v) = S(u)(0) for v = ̺(u) with u ∈ A∞ , we N get that Se is a linear functional on DA (R ) satisfying e (4.9) S(v) ≤ c kLk kvkp′ ∀v ∈ DA (RN ). ′

p We infer from both the density of DA (RN ) in BA and (4.9) the existence of a p function f ∈ BA with kf kp ≤ c kLk such that Z p′ e fbvbdβ for all v ∈ BA . S(v) = ∆(A)

In particular

S(u)(0) =

Z

∆(A)

fbu bdβ ∀u ∈ A∞

where u b = G(u) = G1 (̺(u)). Now, let u ∈ A∞ and let y ∈ RN . By (i) we have Z b S(u)(y) = S(τ y u)(0) = τd y uf dβ. ∆(A)

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JEAN LOUIS WOUKENG

Thus Lu (ϕ) = = = = = = = = ′

L(̺(u ∗ ϕ)) = − − − − − − −

Z

RN

Z

∆(A)

Z

∆(A)

Z

∆(A)

Z

∆(A)

Z

∆(A)

Z

∆(A)

Z

RN

Z

RN

S(u)(y) divy ϕdy (by (4.8)) !

b τd y uf dβ

divy ϕdy

 b τd y u(s) div y ϕdy f dβ

fb(b u ⊛ divy ϕ)dβ (by (4.4))

fb G (u ∗ divy ϕ) dβ (by (4.5)) fb G (divy (u ∗ ϕ)) dβ


fb G1 divy (̺(u ∗ ϕ)) dβ

 ∇y f, ̺(u ∗ ϕ) .

1,p N Finally let v ∈ (BA ) and let (ϕn )n ⊂ D(RN ) be a mollifier. Then v ⊛ ϕn → v in ′ 1,p N (BA ) as n → ∞, where v ⊛ ϕn = (vi ⊛ ϕn )i . We have v ⊛ ϕn ∈ DA (RN )N and L(v ⊛ ϕn ) → L(v) by the continuity of L. On the other hand Z Z
c v dβ. fb divb fb G1 divy (v ⊛ ϕn ) dβ → ∆(A)

∆(A)

1,p′ N on (BA ) , p f2 in BA be

We deduce that L and ∇y f agree i.e., L = ∇y f . For the uniqueness, let f1 and such that L = ∇y f1 = ∇y f2 , then p ∇y (f1 − f2 ) = 0, which means that f1 − f2 ∈ IA . 

The preceding result together with its proof are still valid mutatis mutandis when the function spaces are complex-valued. In that case, we only require the algebra A to be closed under complex conjugation (u ∈ A whenever u ∈ A). As a result of the preceding theorem, we have the p N Corollary 4. Let f ∈ (BA ) be such that Z fb · gbdβ = 0 ∀g ∈ DA (RN )N with divy g = 0. ∆(A)

1,p p Then there exists a function u ∈ BA , uniquely determined modulo IA , such that f = ∇y u. i′ h R 1,p′ N 1,p′ N v dβ. Then L lies in (BA Proof. Define L : (BA ) , ) → R by L(v) = ∆(A) fb· b

p and it follows from Theorem 6 the existence of u ∈ BA such that f = ∇y u. This 1,p shows at once that u ∈ BA . The uniqueness is shown as in Theorem 6. 

p p Remark 4. Let u ∈ BA be such that ∇y u = 0; then u ∈ IA . Thus, for the uniqueness argument, we may choose the function u in Corollary 4 to belong to 1,p p BA /IA , which space we shall henceforth equip with the norm gradient norm as above; see (2.3).

INTROVERTED ALGEBRAS WITH MEAN VALUE

19

5. Sigma convergence method Throughout this section, Ω is an open subset of RN , and unless otherwise specified, A is an algebra with mean value on RN . Definition 1. (1) A sequence (uε )ε>0 ⊂ Lp (Ω) (1 ≤ p < ∞) is said to weakly p Σ-converge in Lp (Ω) to some u0 ∈ Lp (Ω; BA ) if as ε → 0, Z ZZ b (x, s) dxdβ (s) uε (x) ψ ε (x) dx → u b0 (x, s) ψ (5.1) Ω

Ω×∆(A)

b (x, ·) = for all ψ ∈ Lp (Ω; A) (1/p′ = 1 − 1/p) where ψ ε (x) = ψ (x, x/ε) and ψ p G(ψ (x, ·)) a.e. in x ∈ Ω. We denote this by uε → u0 in L (Ω)-weak Σ. (2) A sequence (uε )ε>0 ⊂ Lp (Ω) (1 ≤ p < ∞) is said to strongly Σ-converge in p Lp (Ω) to some u0 ∈ Lp (Ω; BA ) if it is weakly Σ-convergent and further satisfies the following condition: kuε kLp (Ω) → kb u0 kLp (Ω×∆(A)) . ′

We denote this by uε → u0 in Lp (Ω)-strong Σ.

b = G◦ψ, G1 being the isometric isomorphism We recall here that u b0 = G1 ◦u0 and ψ p p sending BA onto L (∆(A)) and G, the Gelfand transformation on A. In the sequel the letter E will throughout denote a fundamental sequence, that is, any ordinary sequence E = (εn )n (integers n ≥ 0) with 0 < εn ≤ 1 and εn → 0 as n → ∞. The following result holds. Theorem 7. (i) Any bounded sequence (uε )ε∈E in Lp (Ω) (where 1 < p < ∞) admits a subsequence which is weakly Σ-convergent in Lp (Ω). (ii) Any uniformly integrable sequence (uε )ε∈E in L1 (Ω) admits a subsequence which is weakly Σ-convergent in L1 (Ω). Below is one fundamental result involving the gradient of sequences. Theorem 8. Let 1 < p < ∞. Let (uε )ε∈E be a bounded sequence in W 1,p (Ω). Then p 1,p there exist a subsequence E ′ of E, and a couple (u0 , u1 ) ∈ W 1,p (Ω; IA )×Lp (Ω; BA ) such that, as E ′ ∋ ε → 0, uε → u0 in Lp (Ω)-weak Σ;

∂u0 ∂uε ∂u1 → + in Lp (Ω)-weak Σ, 1 ≤ i ≤ N. ∂xi ∂xi ∂yi Proof. Since the sequences (uε )ε∈E and (∇uε )ε∈E are bounded respectively in p Lp (Ω) and in Lp (Ω)N , there exist a subsequence E ′ of E and u0 ∈ Lp (Ω; BA ), p ∂uε p N p v = (vj )j ∈ L (Ω; BA ) such that uε → u0 in L (Ω)-weak Σ and ∂xj → vj in Lp (Ω)-weak Σ. For Φ ∈ (C0∞ (Ω) ⊗ A∞ )N we have Z Z ε∇uε · Φε dx = − (uε (divy Φ)ε + εuε (div Φ)ε ) dx. Ω

Ω

Letting E ′ ∋ ε → 0 we get

−

ZZ

Ω×∆(A)

c Φdxdβ b = 0. u b0 div

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JEAN LOUIS WOUKENG

p (see (2.2)), that is, This shows that ∇y u0 = 0, which means that u0 (x, ·) ∈ IA p p u0 ∈ L (Ω; IA ). Next let Φε (x) = ϕ(x)Ψ(x/ε) (x ∈ Ω) with ϕ ∈ C0∞ (Ω) and Ψ = (ψ j )1≤j≤N ∈ (A∞ )N with divy Ψ = 0. Clearly N Z N Z X X ∂uε ∂ϕ uε ψ εj ϕψ εj dx = − dx ∂xj ∂xj j=1 Ω j=1 Ω

where ψ εj (x) = ψ j (x/ε). Passing to the limit in the above equation when E ′ ∋ ε → 0 we get N ZZ N ZZ X X b ∂ϕ dxdβ. b u b0 ψ vbj ϕψ j dxdβ = − (5.2) j ∂xj Ω×∆(A) Ω×∆(A) j=1 j=1 First, taking Φ = (ϕδ ij )1≤i≤N with ϕ ∈ C0∞ (Ω) (for each fixed 1 ≤ j ≤ N ) in (5.2) we obtain Z Z ∂ϕ dx (5.3) M (u0 ) M (vj )ϕdx = − ∂xj Ω Ω

and reminding that M (vj ) ∈ Lp (Ω) we have by (5.3) that

∂u0 ∂xj

∂u0 ∂xj is 1,p

W

p ∈ Lp (Ω; IA ), where

the distributional derivative of u0 with respect to xj . We deduce that u0 ∈ p (Ω; IA ). Coming back to (5.2) we get ZZ b (b v − ∇b u0 ) · Ψϕdxdβ = 0, Ω×∆(A)

and so, as ϕ is arbitrarily fixed, Z b (b v(x, s) − ∇b u0 (x, s)) · Ψ(s)dβ =0 ∆(A)

for all Ψ as above and for a.e. x. Therefore we infer from Corollary 4 the existence 1,p of a function u1 (x, ·) ∈ BA such that v(x, ·) − ∇u0 (x, ·) = ∇y u1 (x, ·)

for a.e. x. From which the existence of a function u1 : x 7→ u1 (x, ·) with values in 1,p BA such that v = ∇u0 + ∇y u1 . 

p Remark 5. If we assume the algebra A to be ergodic, then IA consists of constant functions, so that the function u0 in Theorem 8 does not depend on y, that is, u0 ∈ W 1,p (Ω). We thus recover the already known result proved in [27] in the case of ergodic algebras.

6. On the interplay between Σ-convergence and convolution In order to take full advantage the results of Section 3, we assume throughout this section that the algebra A is introverted. Then its spectrum is a compact topological semigroup. With this in mind, let a ∈ RN be fixed. Appealing to Theorem 5, we may assume without lost of generality that (δ a/ε )ε>0 is a net in the compact group K(∆(A)), hence it possesses a weak∗ cluster point r ∈ K(∆(A)). In the sequel we shall consider a subnet still denoted by (δ a/ε )ε>0 (if there is no danger of confusion) that converges to r in the relative topology of ∆(A), i.e. δ aε → r in K(∆(A))-weak ∗ as ε → 0.

(6.1)

INTROVERTED ALGEBRAS WITH MEAN VALUE

21

Finally let Ω be an open subset of RN , and let (uε )ε>0 be a sequence in Lp (Ω) p (1 ≤ p < ∞) which is weakly Σ-convergent to u0 ∈ Lp (Ω; BA ). We will see that a micro-translation of the sequence uε induces a micro-translation on its limit, while a macro-translation of uε induces both micro- and macro-translations on the limit u0 . This is the main goal of the following result. Theorem 9. Let (uε )ε>0 be a sequence in Lp (Ω) (1 ≤ p < ∞) and let the sequences (vε )ε>0 and (wε )ε>0 be defined by vε (x) = uε (x + a) (x ∈ Ω − a) and wε (x) = uε (x + εa) (x ∈ Ω). p Finally, let u0 ∈ Lp (Ω; BA ) and assume that uε → u0 in Lp (Ω)-weak Σ (resp. -strong Σ).

Proposition 3.

(i) If p > 1 then wε → w0 in Lp (Ω)-weak Σ (resp. -strong Σ)

(6.2)

where w0 is defined by w0 (x, y) = u0 (x, y + a) for (x, y) ∈ Ω × RN . Moreover if p = 1 then (6.2) is still valid provided that the sequence (uε )ε>0 is uniformly integrable. (ii) If (6.1) holds true then vε → v0 in Lp (Ω − a)-weak Σ (resp. -strong Σ)

(6.3)

p where v0 ∈ Lp (Ω − a, BA ) is defined by vb0 (x, s) = u b0 (x + a, sr) for (x, s) ∈ (Ω − a) × ∆(A).

Proof. We split the proof into two parts. Part 1. We consider first the case of weak Σ-convergence. Let us first check (6.2). Prior to this, let us note that the proof of (6.2) in the special case p = 1 has been done in [27]. In the general situation when p > 1, the proof is very similar to the one of the previous case, and for that reason, we just sketch it here. Let ϕ ∈ C0∞ (Ω) and let ψ ∈ A; we have Z

wε ϕψ ε dx = Ω

= =

=

Z

ZΩ

uε (x + εa)ϕ(x)ψ

x ε

uε (x)ϕ(x − εa)ψ

dx

x

 − a dx

ε ZΩ+εa  x − a dx uε (x)ϕ(x − εa)ψ ε Ω Z x  − uε (x)ϕ(x − εa)ψ − a dx ε Ω\(Ω+εa) Z x  uε (x)ϕ(x − εa)ψ + − a dx ε (Ω+εa)\Ω

(I) − (II) + (III).

22

JEAN LOUIS WOUKENG

As in [27], we have that

ZZ

→

(I)

Ω×K(∆(A))

ZZ

=

Ω×K(∆(A))

ZZ

=

Ω×K(∆(A))

u b0 (x, s)ϕ(x)τ\ −a ψ(s)dxdβ

b u b0 (x, s)ϕ(x)τ δ −a ψ(s)dxdβ

b τ δa u b0 (x, s)ϕ(x)ψ(s)dxdβ.

We infer from the inequality Z x  1 − a dx ≤ kϕk∞ kψk∞ (meas[(Ω + εa)∆Ω]) p′ kuε kLp (Ω) |uε (x)| |ϕ(x − εa)| ψ ε (Ω+εa)∆Ω ((Ω + εa)∆Ω being denoting the symmetric difference between Ω + εa and Ω) that (II) and (III) tend to 0 as ε → 0. This proves (6.2). The proof of (6.3) is more involved. It has just been done in [31] (in the case of weak Σ-convergence of course!) under the restricted assumption that Ω is bounded and ∆(A) is a group. We present here a general proof in which all these assumptions are relaxed. Let ϕ ∈ C0∞ (Ω − a) and ψ ∈ A. Let (yn )n be a net in RN (independent of ε) such that δ yn ∈ K(∆(A)) and δ yn → r in K(∆(A))-weak ∗ with n.

We have

Z

x

dx ε ZΩ−a x a dx − uε (x)ϕ(x − a)ψ = ε ε ZΩ h x a x i uε (x)ϕ(x − a) ψ = −ψ − − yn dx ε ε ε Ω Z x  uε (x)ϕ(x − a)ψ + − yn dx ε Ω = (I) + (II)
 R
 R where (I) =
Ω uε (x)ϕ(x−a) ψ xε − aε − ψ xε − yn dx and (II) = Ω uε (x)ϕ(x− a)ψ xε − yn dx. We first consider (II). For ε → 0, it holds that ZZ b ψ(s)dxdβ(s) u b0 (x, s)ϕ(x − a)τ δ (II) → uε (x + a)ϕ(x)ψ

−yn

Ω×K(∆(A))

ZZ

=

and

=

lim n ZZ

ZZ

(Ω−a)×K(∆(A))

b b0 (x + a, s)ϕ(x)ψ(s)dxdβ(s), τ δyn u

(Ω−a)×K(∆(A))

b b0 (x + a, s)ϕ(x)ψ(s)dxdβ(s) τ δ yn u

(Ω−a)×K(∆(A))

=

ZZ

(Ω−a)×K(∆(A))

b τ ru b0 (x + a, s)ϕ(x)ψ(s)dxdβ(s)

b u b0 (x + a, sr)ϕ(x)ψ(s)dxdβ(s).

INTROVERTED ALGEBRAS WITH MEAN VALUE

23

As for (I), we easily verify (using the fact that G is an isometry) that

b b − τδ ψ |(I)| ≤ c τ δ− a ψ

. −yn ∞

ε

Since the mapping s 7→ s−1 is continuous in K(∆(A)), we get that δ − aε = δ −1 → a ε

−1 r−1 in K(∆(A)) weak∗ and δ −yn = δ −1 in K(∆(A)) weak∗. Invoking the yn → r b uniform continuity of ψ, we are led to

b b − τδ ψ

τ δ− a ψ

→ 0 as ε → 0 and next n → ∞. −yn ∞

ε

(6.3) follows thereby.

Part 2. We now assume strong Σ-convergence of (uε )ε . Then from Part 1, we have (6.2) and (6.3) in the weak sense. Moreover we deduce from both the equality kuε kLp (Ω) = kuε (· + a)kLp (Ω−a) and the translation invariance of the measure β that kuε (· + a)kLp (Ω−a) → kb u0 (· + a, ·r)kLp ((Ω−a)×K(∆(A))) . This concludes the proof of (6.3) in both cases (weak and strong Σ-convergence). The same also holds for (6.2) in the case of strong Σ-convergence. The proof is complete.  The next important result deals with the convergence of convolution sequences. 1 . Let (uε )ε>0 ⊂ Lp (Ω) Let p, q, m ≥ 1 be real numbers such that p1 + q1 = 1 + m and (vε )ε>0 ⊂ Lq (RN ) be two sequences. One may view uε as defined in the whole RN by taking its extension by zero outside Ω. Define Z (uε ∗ vε )(x) = uε (t)vε (x − t)dt (x ∈ RN ), RN

which lies in Lm (RN ) and satisfies the Young’s inequality kuε ∗ vε kLm (Ω) ≤ kuε kLp (Ω) kvε kLq (Ω) .

(6.4)

We have the following result. Theorem 10. Let (uε )ε>0 and (vε )ε>0 be as above. Assume that, as ε → 0, uε → u0 in Lp (Ω)-weak Σ and vε → v0 in Lq (RN )-strong Σ, where u0 and v0 are p q in Lp (Ω; BA ) and Lq (RN ; BA ) respectively. Then, as ε → 0, uε ∗ vε → u0 ∗ ∗v0 in Lm (Ω)-weak Σ. Proof. In view of (6.4), the sequence (uε ∗ vε )ε>0 is bounded in Lm (Ω). Now, let N η > 0 and let ψ 0 ∈ K(RN ; A) (the space of continuous

functions from R into

b A with compact support in RN ) be such that b v0 − ψ ≤ η . Since 0

q

N

vε → v0 in L (R )-strong Σ we have that vε −

b Σ, hence kvε − ψ ε0 kLq (RN ) → b v0 − ψ 0 q N

ψ ε0

L (R ×∆(A))

ε → 0. So, there is α > 0 such that

Lq (RN ×∆(A))

2

→ v0 − ψ 0 in Lq (RN )-strong

b = b v0 − ψ as 0 q N

kvε − ψ ε0 kLq (RN ) ≤ η for 0 < ε ≤ α.

L (R ×K(∆(A)))

(6.5)

24

JEAN LOUIS WOUKENG

For f ∈ K(Ω; A), we have (by still denoting by uε the zero extension of uε off Ω)   Z Z Z  x x uε (t)vε (x − t)dt f x, dx = dx (uε ∗ vε )(x)f x, ε ε RN Ω Ω Z  Z  x uε (t) = vε (x − t)f x, dx dt ε N RN   ZR  Z t x dx dt = uε (t) vε (x)f x + t, + ε ε N RN ZR  Z ε ε uε (t) = (vε (x) − ψ 0 (x))f (x + t)dx dt RN RN Z  Z ε ε + uε (t) ψ 0 (x)f (x + t)dx dt RN

RN

= (I) + (II). R On the one hand one has (I) = Ω [uε ∗ (vε − ψ ε0 )](x)f ε (x)dx and kuε kLp (Ω) kvε − ψ ε0 kLq (RN ) kf ε kLm′ (Ω)

|(I)| ≤

c kvε − ψ ε0 kLq (RN )

≤

where c is a positive constant independent of ε. It follows from (6.5) that |(I)| ≤ cη for 0 < ε ≤ α.

(6.6)

On the other hand, in view of Theorem 9, we have, as ε → 0, Z ZZ b (x, s)fb(x + t, sr)dxdβ(s) ψ ε0 (x)f ε (x + t)dx → ψ 0 RN

RN ×K(∆(A))

where r = lim δ t/ε (for a suitable subsequence of ε → 0) in K(∆(A))-weak∗. So let Φ : RN × ∆(A) → R be defined by ZZ b (x, s)fb(x + t, sr)dxdβ(s), (t, r) ∈ RN × ∆(A). Φ(t, r) = ψ 0 RN ×K(∆(A))

Then we easily check that Φ ∈ K(RN ; C(∆(A))), so that there is a function Ψ ∈ K(RN ; A) with Φ = G ◦ Ψ. We can therefore define the trace Ψε (t) = Ψ(t, t/ε) (t ∈ RN ) and we have E     D b t, δ t = Φ t, δ t Ψε (t) = δ εt , Ψ(t, ·) = Ψ ε ε ZZ b (x, s)fb(x, sδ t )dxdβ(s). = ψ 0 ε RN ×K(∆(A))

Next, we have Z Z uε (t) (II) = RN

=

RN

ψ ε0 (x)f ε (x

(II1 ) + (II2 ).

 Z + t)dx − Ψ (t) dt + ε

uε (t)Ψε (t)dt

RN

As far as (II1 ) is concerned, set Z ψ ε0 (x)f ε (x + t)dx − Ψε (t) for a.e. t ∈ RN . Vε (t) = RN

Then the following claims hold.

Claim 1. For a.e. t, Vε (t) → 0 as ε → 0 (possibly up to a subsequence)

INTROVERTED ALGEBRAS WITH MEAN VALUE

Claim 2.

R

RN

25

uε (t)Vε (t)dt → 0 as ε → 0.

Indeed, for Claim 1, applying Theorem 9 leads one to Z

RN

ψ ε0 (x)f ε (x

+ t)dx →

ZZ

RN ×K(∆(A))

b (x, s)fb(x + t, sr)dxdβ(s) as ε → 0 ψ 0

where r is such that δ t/ε → r in K(∆(A)) weak∗ for some subsequence of ε. Moreover, since Ψε (t) = Φ(t, δ t/ε ), we have by the continuity of Φ(t, ·) that, for the same subsequence, Ψε (t) →

ZZ

RN ×K(∆(A))

b (x, s)fb(x + t, sr)dxdβ(s). ψ 0

Whence Claim 1 is justified. As for Claim 2, first and foremost we have |Vε (t)| ≤ c for a.e. t ∈ Ω where c is a positive constant independent of t and ε. Since f and ψ 0 belong to K(RN ; A) we have that f ε and ψ ε0 lie in K(RN ) and their support are contained in a fixed compact set of RN . Therefore ψ ε0 ∗ f ε ∈ K(RN ). As a result, Vε ∈ K(RN ) and further its support is contained in a fixed compact set K ⊂ RN independent of ε. This being so, let γ > 0. From Egorov’s theorem there exists D ⊂ RN such that meas(RN \D) < γ and Vε converges uniformly to 0 on D. We have the following series of inequalities Z

RN

uε (t)Vε (t)dt

≤

kuε kLp (D) kVε kLp′ (D) + kuε kLp (RN \D) kVε kLp′ (RN \D)

≤

C kVε kLp′ (D∩K) + Cmeas(RN \D)

≤

C1 meas(K) sup |Vε (t)| + C1 γ t∈D

where C1 is a positive constant independent of both ε and D. It emerges from the uniform continuity of Vε in D that there exists ε0 > 0 with α1 ≤ α such that Z

RN

uε (t)Vε (t)dt ≤ C2 γ provided 0 < ε ≤ α1 ,

where C2 > 0 is independent of ε. This shows Claim 2. Thus (II1 ) → 0 as ε → 0. As for (II2 ), using once again the weak Σ-convergence of (uε )ε , we get Z

Ω

uε (t)Ψε (t)dt →

ZZ

Ω×K∆(A))

b r)dtdβ, u b0 (t, r)Ψ(t,

26

JEAN LOUIS WOUKENG

and

ZZ

Ω×K(∆(A))

=

ZZ

Ω×K(∆(A))

=

ZZ

Ω×K(∆(A))

=

ZZ

Ω×K(∆(A))

=

ZZ

Ω×K(∆(A))

b r)dtdβ u b0 (t, r)Ψ(t, u b0 (t, r)Φ(t, r)dtdβ u b0 (t, r) "Z Z

"Z Z

RN ×K(∆(A))

RN ×K(∆(A))

# b b ψ 0 (x, s)f (x + t, sr)dxdβ(s) dtdβ(r)

b (x − t, sr u b0 (t, r)ψ 0

b )(x, s)fb(x, s)dxdβ(s). ∗∗ψ (b u0 c 0

−1

#

)dtdβ(r) fb(x, s)dxdβ(s)

Thus, there is 0 < α2 ≤ α1 such that Z ZZ b )fbdxdβ ≤ η for 0 < ε ≤ α2 . ∗∗ψ (b u0 c (uε ∗ ψ ε0 )f ε dx − 0 Ω 2 Ω×K(∆(A))

(6.7)

Now, let 0 < ε ≤ α2 be fixed. Finally the decomposition ZZ Z ∗∗b v0 )fbdxdβ (b u0 c (uε ∗ vε )f ε dx − Ω×K(∆(A)) Ω Z ZZ i h b −b ∗∗(ψ v0 ) fbdxdβ = [uε ∗ (vε − ψ ε0 )] f ε dx + u b0 c 0 Ω Ω×K(∆(A)) ZZ Z ε ε b )fbdxdβ, ∗∗ψ (b u0 c + (uε ∗ ψ 0 )f dx − 0 Ω

Ω×K(∆(A))

associated to (6.5)-(6.7) allow one to see that Z ZZ ∗∗b v0 )fbdxdβ ≤ Cη for 0 < ε ≤ α2 . (b u0 c (uε ∗ vε )f ε dx − Ω Ω×K(∆(A))

Here C is a positive constant independent of ε. This concludes the proof.



Remark 6. Theorem 10 generalizes its homologue Theorem 2 in [31] which is concerned with the special case when q = 1 and Ω bounded. Though the above proof is very similar to the one in [31], it is different from the latter in the sense that it takes full advantage of Egorov’s theorem. In practise, we deal in this work with the evolutionary version of the concept of Σ-convergence. This requires some further notion such as the one related to the product of algebras with mean value. Let A1 (resp. A2 ) be an algebra with mean value on Rm1 (resp. Rm2 ). We define their product denoted byPA1 ⊙ A2 as the closure in BUC(Rm1 × Rm2 ) of the tensor product A1 ⊗ A2 = { finite ui1 ⊗ ui2 : uij ∈ Aj , j = 1, 2}. It is a well known fact that A1 ⊙ A2 is an algebra with mean value on Rm1 × Rm2 ; see e.g. [21, 22]. With this in mind, let A = Ay ⊙Aτ where Ay (resp. Aτ ) is an algebra with mean value on RN y (resp. Rτ ). The same letter G will denote the Gelfand transformation on Ay , Aτ and A, as well. Points in ∆(Ay ) (resp. ∆(Aτ )) are denoted by s (resp. s0 ). The compact space ∆(Ay ) (resp. ∆(Aτ )) is equipped with the M -measure β y (resp. β τ ), for Ay (resp. Aτ ). We have ∆(A) = ∆(Ay )×∆(Aτ ) (Cartesian product)

INTROVERTED ALGEBRAS WITH MEAN VALUE

27

and the M -measure for A, with which ∆(A) is equipped, is precisely the product measure β = β y ⊗ β τ (see [21]). Finally, let 0 < T < ∞. We set QT = Ω × (0, T ), an open cylinder in RN +1 . This being so, a sequence (uε )ε>0 ⊂ Lp (QT ) (1 ≤ p < ∞) is said to weakly p Σ-converge in Lp (QT ) to some u0 ∈ Lp (QT ; BA ) if as ε → 0,   ZZ Z x t u b0 (x, t, s, s0 ) fb(x, t, s, s0 ) dxdtdβ dxdt → uε (x, t) f x, t, , ε ε QT ×K(∆(A)) QT ′

for all f ∈ Lp (QT ; A).

Remark 7. The conclusions of Theorems 7 and 8 are still valid mutatis mutandis in the present context (change there Ω into QT , W 1,p (Ω) into Lp (0, T ; W 1,p (Ω)), p 1,p p p 1,p W 1,p (Ω; IA ) × Lp (Ω; BA ) into Lp (0, T ; W 1,p(Ω; IA )) × Lp (QT ; BA (Rτ ; BA ))). τ y Now, assume that Ay and Aτ are introverted. Let (a, τ ) ∈ RN × R, and let (uε )ε>0 ⊂ Lp (QT ) (1 ≤ p < ∞) be a weakly Σ-convergent subsequence in Lp (QT ) p to u0 ∈ Lp (QT ; BA ). Set vε (x, t) = uε (x + a, t + τ ) for (x, t) ∈ QT − (a, τ ) ≡ (Ω − a) × (−τ , T − τ ).

p Then vε → v0 in Lp (QT − (a, τ ))-weak Σ where v0 ∈ Lp (QT − (a, τ ); BA ) is defined by

vb0 (x, t, s, s0 ) = u b0 (x + a, t + τ , sr, s0 r0 ), (x, t, s, s0 ) ∈ [QT − (a, τ )] × ∆(A),

the micro-translations r and r0 being determined as follows: δ a/ε → r in K(∆(Ay )) and δ τ /ε → r0 in K(∆(Aτ )) up to a subsequence of ε. A similar conclusion holds in Theorem 10 mutatis mutandis. 7. Homogenization of a parameterized Wilson-Cowan type equation with finite and infinite delays We consider the parameterized Wilson-Cowan model with delay [36, 37]   R ξ ∂uε ε N K (x − ξ)f (x, t) = −u (x + a, t) + , u (ξ, t) dξ in RN ε ε N T = R × (0, T ) ∂t ε R uε (x, 0) = u0 (x), x ∈ RN (7.1) where a ∈ RN is fixed, uε denotes the electrical activity level field, f the firing rate function and K ε = K ε (x) = K(x, x/ε) the connectivity kernel. We assume that K ∈ K(RN ; A) (A algebra with mean value on RN ) is nonnegative R an introverted ε and is such that RN K (x)dx ≤ 1, f : RN × R → R is a nonnegative Carath´eodory function satisfying the following conditions: (

(H1) For almost all y ∈ RN , the function f (y, ·) : λ 7→ f (y, λ) is continuous; for all λ ∈ R, the function f (·, λ) : y 7→ f (y, λ) is measurable and f (·, 0) lies in L1 (RN ) ∩ L2 (RN ); there exists a positive constant k1 such that |f (y, µ1 ) − f (y, µ2 )| ≤ k1 |µ1 − µ2 | for all y ∈ RN and all µ1 , µ2 ∈ R.

(H2) f (·, µ) ∈ A for all µ ∈ R. 1 N (H3) For any sequence (vε )ε>0 ⊂ L1 (RN T ) such that vε → v0 in L (RT )-weak Σ, ε 1 N we have f (·, vε ) → f (·, v0 ) in L (RT )-weak Σ.

28

JEAN LOUIS WOUKENG

Assumption (H1) is used to derive the existence and uniqueness result while assumptions (H2) and (H3) are the cornerstone in the homogenization process. It is worth noticing that assumption (H3) is meaningful. It concerns the convergence of fluxes to flux. Indeed the convergence result vε → v0 in L1 (RN T )-weak Σ does not in general ensure the convergence result f ε (·, vε ) → f (·, v0 ) in L1 (RN T )-weak Σ. However, under some circumstances, this becomes true; see e.g. [31, Section 4] for some concrete examples. In [31], Eq. (7.1) with a = 0 has been considered. Here we aim at showing how the translation induces a memory effect at the microscopic level. Before we can do that, however, we need to provide an existence result. But by repeating the proof of Theorem 3 in [31], we get the following result. Theorem 11. Assume that u0 ∈ L1 (RN ) ∩ L2 (RN ). For each ε > 0, there exists a unique solution uε ∈ C([0, ∞); L1 (RN ) ∩ L2 (RN )) to (7.1) satisfying h i sup sup kuε (·, t)kL1 (RN ) + kuε (·, t)kL2 (RN ) ≤ C (7.2) ε>0 0≤t≤T

where C is a positive constant depending only on u0 and on T . Moreover the sequence (uε )00 ⊂ Lp (RN T ) is weakly Σ-convergent in L (RT ) if Z Z Z  x uε (x, t) f x, t, dxdt → u b0 (x, t, s) fb(x, t, s) dxdtdβ (s) ε RN RN T T ×K(∆(A))
′ N b for all f ∈ Lp RN T ; A where f (x, t, ·) = G(f (x, t, ·)) a.e. in (x, t) ∈ RT . With this in mind, we can now state and prove the homogenization result.

Theorem 12. Let (uε )ε>0 be the sequence of solutions to (7.1). Then there exists a subsequence of ε still denoted by ε such that, when ε → 0, it holds that δ aε → r in K(∆(A)) weak∗

(7.3)

uε → G1−1 ◦ u0 in L1 (RN T )-weak Σ

(7.4)

and 1

N

where u0 ∈ C([0, T ]; L (R ×∆(A))) is the unique solution to the following equation  ∂u N 0 b b ∂t (x, t, s) = −u0 (x + a, t, sr) + (K ∗ ∗f (·, u0 ))(x, t, s), (x, t, s) ∈ RT × ∆(A) 0 N u0 (x, 0, s) = u (x), (x, s) ∈ R × ∆(A). (7.5) Proof. Let E be an ordinary sequence of positive real numbers ε. We know from Theorem 11 that the sequence (uε )ε∈E is uniformly integrable in L1 (RN T ). As a result of [part (ii) of] Theorem 7, there exist a subsequence E ′ of E and a function 1 ′ v0 ∈ L1 (RN T ; BA ) such that, as E ∋ ε → 0, we have (7.4) with u0 = G1 ◦ v0 . On the other hand, in view of Theorem 5, (δ a/ε )ε∈E ′ is a sequence in the compact space K(∆(A)), so that there exists a subsequence of E ′ not relabeled and r ∈ K(∆(A)) such that (7.3) holds true. The theorem will be proved once we will check that u0 ∈ C([0, T ]; L1(RN × ∆(A))) and satisfies (7.5). In order to do that, first recall that K ∈ K(RN × (0, T ); A) since K(RN ; A) ⊂ K(RN × (0, T ); A), so that we have ′ K ε → K in L1 (RN T )-strong Σ as E ∋ ε → 0.

INTROVERTED ALGEBRAS WITH MEAN VALUE

29

On the other hand, using the assumption (H3) in conjunction with (7.4), we are led to ′ f ε (·, uε ) → f (·, v0 ) in L1 (RN T )-weak Σ as E ∋ ε → 0.

We infer from Theorem 10 that

′ K ε ∗ f ε (·, uε ) → K ∗ ∗f (·, v0 ) in L1 (RN T )-weak Σ when E ∋ ε → 0.

It also holds that ′ uε (· + a, ·) → w0 in L1 (RN T )-when Σ when E ∋ ε → 0

where w b0 (x, t, s) = u0 (x + a, t, sr) ((x, t, s) ∈ RN T × ∆(A)) with r determined by (7.3); this is a consequence of [part (ii) of] Theorem 9. Finally, noting that Eq. (7.1) is equivalent to the following integral equation Z t uε (x, t) = u0 (x) + [(K ε ∗ f ε (·, uε ))(x, τ ) − uε (x + a, τ )] dτ , 0

we are led (after passing to the limit when E ′ ∋ ε → 0 and using Fubini and Lebesgue dominated convergence results in the integral term) to Z t 0 v0 (x, t, y) = u (x) + [(K ∗ ∗f (·, v0 ))(x, τ , y) − w0 (x, τ , y)] dτ . 0

Therefore, composing both members of the above equality by G1 , we end up with Z th i 0 b ∗ ∗fb(·, u0 ))(x, τ , s) − u0 (x + a, τ , sr) dτ (K u0 (x, t, s) = u (x) + 0

which is nothing else but the integral form of (7.5). We also conclude from the preceding equation that u0 lies in C([0, T ]; L1(RN × ∆(A))) as expected. Whence the proof.  8. Homogenization of a nonlocal nonlinear heat equation We consider the following non local boundary value problem ε ε ε ε ρε ∂u ∂t − div a (·, ·, ∇uε ) + K ∗ a0 (·, ·, ∇uε ) = f in QT uε = 0 on ∂Ω × (0, T ) uε (x, 0) = u0 (x) in Ω

(8.1)

where Ω is a bounded smooth open set in RN , ρε (x) = ρ(x/ε), aε (·, ·, ∇uε ) (x, t) = a (x/ε, t/ε, ∇uε (x, t)) (same definition for aε0 (·, ·, ∇uε )), K ε (x, t) = K (x/ε, t/ε),   Z ∞   x τ x t−τ ε ε a0 , , , ∇uε (x, τ ) dτ , (K ∗ a0 (·, ·, ∇uε ))(x, t) = K ε ε ε ε 0 the functions a : RN × R × RN → RN , a0 : RN × R × RN → R, K : RN × R → R and ρ : RN → R being constrained as follows: A1 For each λ ∈ RN , the functions a(·, ·, λ) and a0 (·, ·, λ) are measurable; a(y, τ , 0) = 0 almost everywhere (a.e.) in (y, τ ) ∈ RN × R;

(8.2) (8.3)

30

JEAN LOUIS WOUKENG

There are three constants c0 , c1 , c2 > 0 such that 2

(i) a(y, τ , λ) − a(y, τ , λ′ ) · λ − λ′ ≥ c1 λ − λ′ (ii) |a(y, τ , λ)| + |a0 (y, τ , λ)| ≤ c2 (1 + |λ|) (iii) a(y, τ , λ) − a(y, τ , λ′ ) + a0 (y, τ , λ) − a0 (y, τ , λ′ ) ≤ c0 λ − λ′ for all λ, λ′ ∈ RN and a.e. in (y, τ ) ∈ RN × R, where the dot denotes the usual Euclidean inner product in RN and |·| the associated norm.

(8.4)

A2 K ∈ L1 (RN +1 ), ρ ∈ L∞ (RN ) and there exists Λ > 0 such that Λ−1 ≤ ρ(y) ≤ Λ for a.e. y ∈ RN . It is well known that the functions aε (·, ·, Dv) and aε0 (·, ·, Dv) (for fixed v ∈ L (0, T ; W01,2 (Ω))), K ε and ρε are well defined as elements of L2 (QT )N , L2 (QT )N , L1 (QT ) and L∞ (Ω) respectively and satisfy properties of the same type as in A1A2. Finally, choose f in L2 (QT ) and u0 ∈ L2 (Ω). Our first objective in this section is to provide an existence and uniqueness result for problem (8.1). 2

Theorem 13. For any fixed ε > 0, the problem (8.1) possesses a unique solution uε ∈ L2 (0, T ; W01,2 (Ω)) ∩ C(0, T ; L2(Ω)) and the following a priori estimates holds: Z T 2 2 sup kuε (t)kL2 (Ω) ≤ C and kuε (t)kW 1,2 (Ω) dt ≤ C (8.5) 0≤t≤T

0

0

where C is a positive constant which does not depend on ε. Proof. Let z ∈ L2 (0, T ; W01,2 (Ω))∩C(0, T ; L2 (Ω)), and consider the following boundary value problem ε ε ε ε ρε ∂u ∂t − div a (·, ·, ∇uε ) = f − K ∗ a0 ·, ·, ∇z) in QT uε = 0 on ∂Ω × (0, T ) uε (x, 0) = u0 (x) in Ω.

(8.6)

Using a standard fashion (see e.g. [29]), we derive the existence and uniqueness of a solution uε ∈ L2 (0, T ; W01,2(Ω)) ∩ C(0, T ; L2(Ω)) to (8.6). Thus we have defined a mapping z 7→ uε from X = L2 (0, T ; W01,2 (Ω)) ∩ C(0, T ; L2(Ω)) into itself. We need to show that this mapping is contractive. To this end, let us endow X with the norm  12  2 2 (u ∈ X). kukX = sup ku(t)kL2 (Ω) + kukL2 (0,T ;W 1,2 (Ω)) 0

0≤t≤T

Let z1 , z2 ∈ X, and consider, for j = 1, 2, the solution uj of the corresponding PDE. We have, for any φ ∈ L2 (0, T ; W01,2(Ω)) and any 0 ≤ t ≤ T ,  Z Z t ∂uj aε (·, ·, ∇uj ) · ∇φdxds , φ ds + ρε ∂t Qt 0 Z Z (K ε ∗ aε0 (·, ·, ∇zj ))φdxds, f φdxds + = Qt

Qt

where Qt = Ω × (0, t), hence  Z Z t ∂(u1 − u2 ) (aε (·, ·, ∇u1 ) − aε (·, ·, ∇u2 )) · ∇φdxds , φ ds + ρε ∂t Qt 0 Z ε ε ε (K ∗ (a0 (·, ·, ∇z1 ) − a0 (·, ·, ∇z2 )))φdxds. = Qt

INTROVERTED ALGEBRAS WITH MEAN VALUE

31

Taking φ = u1 − u2 , assumptions A1-A2 entail Z 1 −1 2 2 Λ ku1 (t) − u2 (t)kL2 (Ω) + c1 |∇u1 − ∇u2 | dxds 2 Qt ≤

kK ε ∗ (aε0 (·, ·, ∇u1 ) − aε0 (·, ·, ∇u2 ))kL2 (Qt ) ku1 − u2 kL2 (Qt ) .

But, in view of Poincar´e’s inequality, there is a positive constant α depending only on Ω such that ku1 − u2 kL2 (Qt ) ≤ α k∇u1 − ∇u2 k2L2 (Qt ) for any 0 ≤ t ≤ T , hence ≤

kK ε ∗ (aε0 (·, ·, ∇u1 ) − aε0 (·, ·, ∇u2 ))kL2 (Qt ) ku1 − u2 kL2 (Qt )

α kK ε ∗ (aε0 (·, ·, ∇u1 ) − aε0 (·, ·, ∇u2 ))kL2 (Qt ) k∇u1 − ∇u2 kL2 (Qt )

c1 α2 2 2 kK ε ∗ (aε0 (·, ·, ∇z1 ) − aε0 (·, ·, ∇z2 ))kL2 (Qt ) + k∇u1 − ∇u2 kL2 (Qt ) . 2c1 2 It readily holds that Z 1 −1 c1 2 Λ ku1 (t) − u2 (t)kL2 (Ω) + |∇u1 − ∇u2 |2 dxds 2 2 Qt ≤

α2 kK ε k2L1 (Qt ) kaε0 (·, ·, ∇z1 ) − aε0 (·, ·, ∇z2 )k2L2 (Qt ) 2c1 α2 2 2 ≤ kKkL1 (RN +1 ) k∇z1 − ∇z2 kL2 (QT ) for a.e. 0 ≤ t ≤ T. 2c1 Taking the supremum over [0, T ] we end up with    Z 1 −1 c1 2 2 |∇u1 − ∇u2 | dxds sup ku1 (t) − u2 (t)kL2 (Ω) + min Λ , 2 2 0≤t≤T QT ≤

≤

α2 T kKk2L1 (RN +1 ) k∇z1 − ∇z2 k2L2 (QT ) , 2c1

hence # 21 2 kKkL1 (RN +1 )
ku1 − u2 kX ≤ C T kz1 − z2 kX where C = . min 12 Λ−1 , c21 √ Consequently, by shrinking T > 0 in such a way that C T < 1, it emerges that the above mapping is contractive. The rest of the proof follows by mere routine.  √

"

α2 2c1

Remark 8. It follows from the inequalities (8.5) that the sequence (uε )ε>0 determined above is relatively compact in the space L2 (QT ). Throughout the rest of this section, Ay and Aτ are algebras with mean value on RN y and Rτ respectively. We set A = Ay ⊙ Aτ and further we assume that Aτ is 2,∞ ) introverted. It is worth noting that property (5.1) is still valid for ψ ∈ C(QT ; BA 2,∞ 2 ∞ N +1 where BA = BA ∩ L (Ry,τ ). Bearing this in mind, let (uε )ε>0 be the sequence of solutions to (8.1). Our main objective here amounts to study the asymptotic behaviour as ε → 0, of (uε )ε>0 . This will of course arise from the following important assumption: 2 2 N a0 (·, ·, λ) ∈ BA and a(·, ·, λ) ∈ (BA ) for any λ ∈ RN , 1 K ∈ BA and ρ ∈ Ay with M (ρ) > 0.

(8.7)

32

JEAN LOUIS WOUKENG

The homogenization of problems of type (8.1) has been left opened in [2] in which the authors considered only the linear version of such type of equations. They used the Laplace transform to perform the homogenization process. Our work is therefore the first one in which the homogenization of (8.1) is considered, even in the periodic setting. This being so, let Ψ ∈ C(QT ; (A)N ). Suppose that (8.7) is satisfied. It can be shown (as in [23]) that the function (x, t, y, τ ) 7→ a(y, τ , Ψ(x, t, y, τ )), denoted 2,∞ N below by a(·, ·, Ψ), lies in C(QT ; (BA ) ), and we can therefore define its trace (x, t) 7→ a(x/ε, t/ε, Ψ(x, t, x/ε, t/ε)) (ε > 0) denoted by aε (·, ·, Ψε ). The same is true for a0 (·, ·, Ψ) and aε0 (·, ·, Ψε ). The proof of the next two results can be found in [23] (see Proposition 3.1 and Corollary 3.1 therein). Proposition 4. Suppose (8.7) holds. For Ψ ∈ C(QT ; (A)N )) we have aε (·, ·, Ψε ) → a(·, ·, Ψ) in L2 (QT )N -weak Σ as ε → 0.

2 N The mapping Ψ 7→ a(·, ·, Ψ) of C(QT ; (A)N )) into L2 (QT ; BA ) extends by conti2 2 N 2 N nuity to a unique mapping still denoted by a, of L (QT ; (BA ) ) into L2 (QT ; BA ) such that

(a(·, ·, v) − a(·, ·, w)) · (v − w) ≥ c1 |v − w|2 a.e. in QT × RN y × Rτ ka(·, ·, v) − a(·, ·, w)kL2 (QT ;B 2 )N ≤ c0 kv − wkL2 (QT ;(B 2 )N ) A

A

a(·, ·, 0) = 0 a.e. in RN y × Rτ

2 N for all v, w ∈ L2 (QT ; (BA ) ).

Corollary 5. Let ψ 0 ∈ C0∞ (QT ) and ψ 1 ∈ C0∞ (QT ) ⊗ A∞ . For ε > 0, let Φε = ψ 0 + εψ ε1 ,

(8.8)

i.e., Φε (x, t) = ψ 0 (x, t) + εψ 1 (x, t, x/ε, t/ε) for (x, t) ∈ QT . Let (vε )ε∈E is a sequence in L2 (QT )N such that vε → v0 in L2 (QT )-weak Σ as E ∋ ε → 0 where 2 v0 ∈ L2 (QT ; BA ), then, as E ∋ ε → 0, ZZ Z b )b b a(·, ·, Dψ 0 + ∂ ψ aε (·, ·, DΦε )vε dxdt → 1 v0 dxdtdβ. QT ×∆(Ay )×K(∆(Aτ ))

QT

Remark 9. The conclusion of the above results also hold true for the mapping a0 , mutatis mutandis. Now, let V

=

F10

=

{u ∈ L2 (0, T ; W01,2 (Ω)) : u′ ∈ L2 (0, T ; W −1,2(Ω))} and

1,2 2 )). (Rτ ; BA V × L2 (QT ; BA τ y

Endowed with its natural topology, F10 is a Hilbert space admitting 
 F0∞ = C0∞ (QT ) × C0∞ (QT ) ⊗ DAτ (Rτ ) ⊗ DAy (RN y )

as a dense subspace. Bearing this in mind, let (uε )ε>0 be a sequence of solutions to (8.1), and let E = (εn )n be an ordinary sequence of positive real numbers converging to zero. Since (uε )ε∈E is relatively compact in L2 (QT ), there exists a subsequence E ′ of E and u0 ∈ L2 (QT ) such that, as E ′ ∋ ε → 0, uε → u0 in L2 (QT ).

(8.9)

INTROVERTED ALGEBRAS WITH MEAN VALUE

33

In view of (8.5) and by the diagonal process, we find a subsequence of (uε )ε∈E ′ still denoted by (uε )ε∈E ′ which weakly converges in L2 (0, T ; W01,2 (Ω)) to u0 , hence u0 ∈ L2 (0, T ; W01,2(Ω)). We infer from both Theorem 8 and Remark 7 the existence 1,2 2 of a function u1 ∈ L2 (QT ; BA (Rτ ; BA )) such that, as E ′ ∋ ε → 0, τ y ∂u0 ∂u1 ∂uε → + in L2 (QT )-weak Σ (1 ≤ j ≤ N ). ∂xj ∂xj ∂yj

(8.10)

This being so, for v = (v0 , v1 ) ∈ F10 we set Dy v = ∇v0 + ∇y v1 and Dv = G1 ◦ Dy v ≡ ∇v0 + ∂b v1 . We consider the variational problem  Find u = (u0 , u1 ) ∈ F10 such that     M (ρ) R T hu′ (t), v (t)i dt + RR a(·, ·, Du) · Dvdxdtdβ 0 0 QT ×∆(Ay )×K(∆(Aτ )) b R RR 0 b  ∗∗b a0 (·, ·, Du))v0 dxdtdβ = QT f v0 dxdt + QT ×∆(Ay )×K(∆(Aτ )) (Kc    for all v = (v0 , v1 ) ∈ F10 , (8.11) where the brackets h·, ·i henceforth stand for the duality paring between W01,2 (Ω) and W −1,2 (Ω) and u′0 (t) = u′0 (·, t) (same definition for v0 (t)), and the convolution is with respect to the time argument, that is, Z Z b s0 r−1 )b b a0 (·, ·, Du))(x, t, s, s0 ) = a0 (s, r0 , Du(x, ζ, s, r0 ))dβ τ (r0 )dζ. K(s, (K∗∗b 0 Rζ

K(∆(Aτ ))

It can be shown that the above problem possesses at most one solution. The following global homogenization result holds.

Theorem 14. Assume (8.7) holds true. Then the couple u = (u0 , u1 ) determined by (8.9)-(8.10) solves the variational problem (8.11). Proof. Let (u0 , u1 ) be as in (8.9)-(8.10). Then it belongs to L2 (0, T ; W01,2(Ω)) × 1,2 2 L2 (QT ; BA )). It remains to check that u′0 ∈ L2 (0, T ; W −1,2 (Ω)) and that (Rτ ; BA τ y u solves (8.11). First, let ψ ∈ C0∞ (QT ). The variational formulation of (8.1) with ψ as a test function yields Z Z Z ∂ψ (K ε ∗ aε0 (·, ·, ∇uε ))ψdxdt ρε u ε aε (·, ·, ∇uε ) · ∇ψdxdt + − dxdt + ∂t QT QT QT Z f ψdxdt. = QT

Thanks to the second estimate in (8.5), the sequences aε (·, ·, ∇uε ) and (K ε ∗ aε0 (·, ·, ∇uε )) are bounded in L2 (QT )N and in L2 (QT ) respectively. Hence there exists a subsequence of E ′ (E ′ determined in (8.10)) not relabeled and two functions z ∈ L2 (QT )N and z0 ∈ L2 (QT ) such that, as E ′ ∋ ε → 0, aε (·, ·, ∇uε ) → z in L2 (QT )N -weak and K ε ∗ aε0 (·, ·, ∇uε ) → z0 in L2 (QT )-weak. Letting E ′ ∋ ε → 0 in the above equation and using (8.9), we obtain Z Z Z ∂ψ u0 −M (ρ) z0 ψdxdt z · ∇ψdxdt + dxdt + ∂t QT QT QT Z f ψdxdt. = QT

It follows that

M (ρ)

∂u0 = div z − z0 + f. ∂t

34

JEAN LOUIS WOUKENG

2 −1,2 0 (Ω)), so that Since M (ρ) 6= 0, it follows by mere routine that ∂u ∂t ∈ L (0, T ; W 1 u0 ∈ V . This shows that u = (u0 , u1 ) ∈ F0 . Let us now check that u satisfies ∞ (8.11). To achieve this, let Φ = (ψ 0 , ̺(ψ 1 )) ∈ F0∞ where ψ 1 ∈ C0∞ (QT ) ⊗ A∞ y ⊗ Aτ 2 2 (̺ being denoting the canonical mapping of BA into BA ), and define Φε as in (8.8) (see Corollary 5). Then we have Φε ∈ C0∞ (QT ), and by the monotonicity of a, we have Z (aε (·, ·, ∇uε ) − aε (·, ·, ∇Φε )) · (∇uε − ∇Φε )dxdt ≥ 0, QT

or, owing to (8.1), 2 R R ε R 2 1 ρ |uε (T )| dx ≤ 12 Ω ρε u0 dx + QT f (uε − Φε )dxdt 2 ΩR R ε ε ε − QT ρε uε ∂Φ ∂t dxdt + QT (K ∗ a0 (·, ·, ∇uε ))(uε − Φε )dxdt R ε − QT a (·, ·, ∇Φε ) · ∇(uε − Φε )dxdt.

(8.12)

We pass to the limit in (8.12) by considering each term separately. First, in view of (8.9), it is an easy exercise to see that ZZ Z b ρ(s) |u0 (T )|2 dxdβ y ≤ lim inf ρε |uε (T )|2 dx, ′ E ∋ε→0

Ω×∆(Ay )

that is,

M (ρ)

Z

Ω

|u0 (T )|2 dx ≤ lim inf ′

E ∋ε→0

Next, we have from the definition of the Z Z ε 0 2 ρ u dx → M (ρ) Ω

Ω

Z

Ω

Ω

ρε |uε (T )|2 dx.

mean value that 0 2 u dx when E ′ ∋ ε → 0.

Considering the next term, we obviously have, as E ′ ∋ ε → 0, Z Z f (u0 − ψ 0 )dxdt. f (uε − Φε )dxdt → QT

QT

In view of (8.9) associated to the convergence result 2



∂ψ 1 ∂τ

ε

→M



∂ψ 1 ∂τ



= 0 in

L (QT )-weak, it holds that Z Z Z T ∂Φε ∂ψ 0 ε ρ uε u0 dxdt → M (ρ) dxdt = M (ρ) hu′0 (t), ψ 0 (t)i dt. ∂t ∂t QT QT 0

Now, because of Corollary 5 we obtain, when E ′ ∋ ε → 0, ZZ Z b a(·, ·, DΦ)·D(u−Φ)dxdtdβ. aε (·, ·, ∇Φε )·∇(uε −Φε )dxdt → QT

QT ×∆(Ay )×K(∆(Aτ ))

Finally, for the last term, due to the estimate (8.5), we have that the sequence 2 ) (aε0 (·, ·, ∇uε ))ε∈E ′ is bounded in L2 (QT ) so that there exist a function z0 ∈ L2 (QT ; BA ′ ε 2 such that, up to a subsequence of E , a0 (·, ·, ∇uε ) → z0 in L (QT )-weak Σ. On the 1 other hand, since K ∈ BA we have that K ε → K in L1 (RN +1 )-strong Σ. It there emerges from Theorem 10 that K ε ∗ aε0 (·, ·, ∇uε ) → K ∗ ∗z0 in L2 (QT )-weak Σ as E ′ ∋ ε → 0.

We use once again (8.9) to get, when E ′ ∋ ε → 0, ZZ Z (K ε ∗aε0 (·, ·, ∇uε ))(uε −Φε )dxdt → QT

QT ×∆(Ay )×K(∆(Aτ ))

b ∗∗b (Kc z0 )(u0 −ψ 0 )dxdtdβ.

INTROVERTED ALGEBRAS WITH MEAN VALUE

35

Putting together all the above convergence results and taking the lim inf E ′ ∋ε→0 in (8.12), we end up with RT 0 ≤ RR0 hf (t) − M (ρ)u′0 (t), u0 (t) − ψ 0 (t)i dt a(·, ·, DΦ) · D(u − Φ)dxdtdβ − QT ×∆(Ay )×K(∆(Aτ )) b (8.13) RR b ∗∗b (Kc z0 )(u0 − ψ 0 )dxdtdβ for all Φ ∈ F0∞ . − QT ×∆(Ay )×K(∆(Aτ ))

Since F0∞ is dense in F10 , after considering a continuity argument, we see that (8.13) still holds for any Φ ∈ F10 . Taking in (8.13) the particular functions Φ = u−λv with λ > 0 and v = (v0 , v1 ) ∈ F10 , then dividing both sides of the resulting inequality by λ, and letting λ → 0, and finally changing v into −v, leads to RT RR a(·, ·, Du) · Dvdxdtdβ M (ρ) 0 hu′0 (t), v0 (t)i dt + QT ×∆(Ay )×K(∆(Aτ )) b RT RR b (Kc ∗∗b z0 )v0 dxdtdβ = hf (t), v0 (t)i dt + QT ×∆(Ay )×K(∆(Aτ ))

0

for all v = (v0 , v1 ) ∈ F10 .

The last part of the proof consists in identifying the function z0 . It is sufficient to check that K ∗ ∗z0 = K ∗ ∗a0 (·, ·, Dy u).

To this end, fix η > 0 and choose ψ 0 ∈ C0∞ (QT ) and ψ 1 ∈ C0∞ (QT ) ⊗ A∞ be such that η η and ku1 − ̺(ψ 1 )kL2 (QT ;B2 (Rτ ;B1,2 )) < . ku0 − ψ 0 kL2 (0,T ;W 1,2 (Ω)) < 0 A A y τ 4c0 kKkB 1 4c0 kKkB 1 A

A

F0∞ ,

Then letting Φ = (ψ 0 , ̺(ψ 1 )) ∈ it holds that




K ∗ ∗ a0 (·, ·, Dy u) − z0 2 2 ) L (QT ;BA


≤ K ∗ ∗ a0 (·, ·, Dy u) − a0 (·, ·, ∇u0 + ∇y ψ 1 ) L2 (Q ;B2 )


T A + K ∗ ∗ a0 (·, ·, ∇u0 + ∇y ψ 1 ) − a0 (·, ·, Dy Φ) L2 (QT ;B2 ) A


+ K ∗ ∗ a0 (·, ·, Dy Φ) − z0 2 2 ≤

η 2

L (QT ;BA )

+ lim inf kK ε ∗ (aε0 (·, ·, ∇Φε ) − aε0 (·, ·, ∇uε ))kL2 (QT ) . ′ E ∋ε→0

But kK ε ∗ (aε0 (·, ·, ∇Φε ) − aε0 (·, ·, ∇uε ))kL2 (QT )

≤ kK ε kL1 (QT ) kaε0 (·, ·, ∇Φε ) − aε0 (·, ·, ∇uε )kL2 (QT )

≤ C k∇Φε ) − ∇uε kL2 (QT )

where C is a positive constant not depending on ε. On the other hand, by part (i) of assumption (8.4), we have Z 2 (aε (·, ·, ∇uε ) − aε (·, ·, ∇Φε )) · (∇uε − ∇Φε )dxdt. c0 k∇Φε ) − ∇uε kL2 (QT ) ≤ QT

Therefore, proceeding exactly as in the proof of Theorem 3.10 in [39] (see also the proof of Theorem 4.1 in [40]), we are readily led to K ∗ ∗z0 = K ∗ ∗a0 (·, ·, Dy u). This concludes the proof of the theorem. 

36

JEAN LOUIS WOUKENG

The global homogenized problem (8.11) is equivalent to the following system  RT ′ RR  a(·, ·, Du) · ∇v0 dxdtdβ hu0 (t), v0 (t)i dt + QT ×∆(Ay )×K(∆(Aτ )) b  M (ρ) R RR 0 b ∗ ∗b a0 (·, ·, Du))v0 dxdtdβ = QT f v0 dxdt + QT ×∆(Ay )×K(∆(Aτ )) (K   for all v0 ∈ L2 (0, T ; W01,2 (Ω)) (8.14) and ZZ 1,2 2 (Rτ ; BA b a(·, ·, Du) · ∂b v1 dxdtdβ = 0, all v1 ∈ L2 (QT ; BA )). τ y QT ×∆(Ay )×K(∆(Aτ ))

(8.15) Let us first deal with (8.15). For that, let λ ∈ RN , and consider the following variational ( 1,2 2 Find v(λ) ∈ BA (Rτ ; BA ): τ y R 1,2 2 d · ∂ wdβ b a(·, ·, λ + ∂ v(λ)) b = 0 for all w ∈ BA ). (Rτ ; BA ∆(Ay )×K(∆(Aτ )) τ y (8.16) Owing to the properties of the function a (see Proposition 4), it holds that (8.16) 1,2 1,2 2 2 2 possesses a solution in BA (Rτ ; BA ) which is unique in the space BA (Rτ ; BA /IA ). τ τ y y Now, taking λ = ∇u0 (x, t) with (x, t) arbitrarily fixed in QT , and then choosing in (8.15) the particular test functions v1 (x, t) = ϕ(x, t)w ((x, t) ∈ QT ) with ϕ ∈ 1,2 2 C0∞ (QT ) and w ∈ BA ), and finally comparing the resulting equation with (Rτ ; BA τ y (8.16), it follows (by the uniqueness argument) that u1 = v(∇u0 ), where the righthand side of this equality stands for the function (x, t) 7→ v(∇u0 (x, t)) from QT 1,2 2 2 into BA (Rτ ; BA /IA ). τ y We can now deal with (8.14). To this end, we define the homogenized coefficients as follows: For λ ∈ RN , Z d b a(·, ·, λ + ∂ v(λ))dβ; b(λ) = ∆(Ay )×K(∆(Aτ ))

b0 (λ) =

Z

∆(Ay )×K(∆(Aτ ))

e ρ =

M (ρ).

d b ∗b (K a0 (·, ·, λ + ∂ v(λ)))dβ;

Then substituting u1 = v(∇u0 ) in (8.14) and choosing there the special test function v0 = ϕ ∈ C0∞ (QT ), we quickly obtain by disintegration, the macroscopic homogenized problem, viz.,  ∂u e ρ ∂t0 − div b(∇u0 ) + b0 (∇u0 ) = f in QT (8.17) u0 (0) = u0 in Ω. By the uniqueness of the solution to (8.11), the existence and the uniqueness of the solution to (8.17) is ensured. We are therefore led to the following Theorem 15. Assume that (8.7) holds. For each ε > 0 let uε be the unique solution to (8.1). Then as ε → 0, uε → u0 in L2 (QT )

where u0 is the unique solution to (8.17).

INTROVERTED ALGEBRAS WITH MEAN VALUE

37

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