Recent Results on a Generalization of the Laplacian
†A.B. SIMAS1* and F.J. VALENTIM2
Received on May 14, 2014 / Accepted on May 23, 2015
ABSTRACT.In this paper we discuss recent results regarding a generalization of the Laplacian. To be more precise, fix a functionW(x1, . . . ,xd)= dk=1Wk(xk), where eachWk :R→ Ris a right continuous
with left limits and strictly increasing function. UsingW, we construct the generalized laplacianLW =
d
i=1∂xi∂Wi, where∂Wi is a generalized differential operator induced by the functionWi. We present results on spectral properties ofLW, Sobolev spaces induced byLW (W-Sobolev spaces), generalized partial differential equations, generalized stochastic differential equations and stochastic homogenization.
Keywords:W-Sobolev space, generalized Laplacian, homogenization, partial differential equations.
1 INTRODUCTION
In the ’50s William Feller introduced a more general concept of differential operators, that is, operators of the type(d/d W)(d/d V)where, typically,W andV are strictly increasing functions withV (but not necessarilyW) being continuous. In this paper we are interested in the formal adjoint of(d/d W)(d/d V), which is simply(d/d V)(d/d W), in the caseV(x)=xis the identity function. For more details on Feller’s operators, we refer the reader to [4, 5, 9].
Recently, some attention has been raised to a class of generalized differential operator involving the derivative with respect to a strictly increasing functionW, we cite [2, 7, 6, 8, 9, 11, 12] as some examples of this fact. On one hand, this operator can be naturally obtained from behavior of some interacting particle systems with random conductances with the interesting feature that, in contrast with usual homogenization phenomena, the entire noise survives in the limit and the differential operator itself depends on the specific realization of random conductance [2, 7]. On the other hand, a second surprising aspect is that the differential equation that appears in the limit is a second-order differential operator in which one of the derivatives is a derivative with respect to functionW, which may have jumps. Even more, the set of jumps ofW can be dense inR.
*Corresponding author: Alexandre B. Simas †Research supported by CNPq and FAPES.
1Departamento de Matem´atica, Universidade Federal da Para´ıba, Cidade Universit´aria, Campus I, 58051-970 Jo˜ao Pessoa, PB, Brasil. E-mail: [email protected]
The goal of this paper is present an overview of the main recent results regarding this differen-tial operator. The rest of the paper unfolds as follows: in Section 2 we present the generalized Laplacian, which we denote byLW; in Section 3 we construct theW-Sobolev spaces and present several properties that are analogous to classical results on Sobolev spaces, we also present re-sults on existence and uniqueness forW-generalized elliptic equations, and a uniqueness result forW-generalized parabolic equations; in Section 4 we present stochastic homogenization re-sults of suitable random operators∇NAN∇WN, that are discretizations of the operatorLW. This paper is essencially inspired in works [11, 12, 3, 13].
2 THE OPERATORLW
Fix a functionW :Rd→Rsuch that
W(x1, . . . ,xd)= d
k=1
Wk(xk) (2.1)
whereWk : R→ Rarestrictly increasingright continuous functions with left limits (c`adl`ag), and periodic in the sense that
Wk(u+1)−Wk(u)=Wk(1)−Wk(0)
for allu ∈ Randk = 1, . . . ,d. To keep notation simple, we assume thatWk vanishes at the origin, that is,Wk(0)=0.
Denote byTthe unidimensional torus and·,·the inner product ofL2(T):
f,g =
T
f(u)g(u)du .
For eachk=1,2, . . . ,d letDWk be the set of functions f inL2(T)such that
f(x) = a + bWk(x) +
(0,x]
Wk(d y) y
0
ᒃ(z)d z
for some functionᒃinL2(T)such that
1
0
ᒃ(z)d z = 0,
(0,1]
Wk(d y)
b+
y
0
ᒃ(z)d z = 0
Define the operatorLWk :DWk →L
2(T)asLW
k f =ᒃ. Formally,
LWk f =
d d x
d d Wk
f ,
where the generalized derivatived/d Wkis defined as
d f
d Wk
(x)= lim
ǫ→0
f(x+ǫ)− f(x)
Wk(x+ǫ)−Wk(x)
,
By a convenient restriction of the operatorsLWkto a dense subspaceDk ⊂DWk, it is not difficult
to prove thatLWk : Dk → L2(T)is symmetric and non-positive. Thus, by using Friedrichs extension (see, for instance, [14, chapter 5]), one obtains that the extended operator, also denoted byLWk,LWk :DWk →L2(T), is self-adjoint and, the setAWk of the eigenvectors ofLWkforms a complete orthonormal system inL2(T), the details of this approach can be found in [6].
Let
AW = {f :Td →R; f(x1, . . . ,xd)= d
k=1
fk(xk),fk∈AWk},
and define the operatorLW : DW :=s pan(AW)→ L2(Td)as follows: for f = dk=1 fk ∈ AW,
LW(f)(x1, . . .xd)= d
k=1
d
j=1,j=k
fj(xj)LWkfk(xk), (2.2)
and extend toDW by linearity. In particular,DW is dense inL2(Td); and the setAW forms a complete, orthonormal, countable system of eigenvectors for the operatorLW.
LetL2
xk⊗W k(T
d)be the Hilbert space of measurable functionsH :Td→Rsuch that
Td
d(xk⊗Wk)H(x)2 < ∞,
whered(xk⊗Wk) = d x1· · ·d xk−1d Wk d xk+1· · ·d xd.
Lemma 2.1.Let f,g∈DW, then for i =1, . . . ,d,
Td
∂xi∂Wi f(x)
g(x)d x = −
Td
(∂Wi f)(∂Wig)d(x
i⊗W i).
In particular,
TdLW
f(x)g(x)d x = −
d
i=1
Td
(∂Wi f)(∂Wig)d(x
i⊗W i).
that is,LW is symmetric and non-positive.
The proof consist in an application of Fubini’s theorem and an approximation of the integral by Riemann sum. Informally,
T
∂xi∂Wi f
gd xi = −
T
∂Wi f ∂xig
d xi ≈ − f
Wi
g xi
xi
= − f
Wi
g Wi
Wi ≈ −
T
∂Wi f ∂Wig
d Wi
where the sum is over partitions of the torusTandhstands for the increment of the function
Consider
DW = {v=
∞
k=1
vkhk ∈ L2(Td);
∞
k=1
v2kαk2<+∞},
wherehk ∈AW andαkis the eigenvalue associated to to eigenvectorhk. Define the operatorLW :DW → L2(Td)by
−LWv=
+∞
k=1
αkvkhk.
The operatorLW is clearly an extension of the operatorLW, and formally,
LW f = d
k=1
LWkf
where
LWk f = ∂xk∂Wk f,
and the partial generalized derivative∂Wk is defined by
∂Wkf(x)= lim
ǫ→0
f(x1, . . . ,xk+ǫ, . . . ,xd)− f(x1, . . . ,xk, . . . ,xd)
Wk(xk+ǫ)−Wk(xk)
,
if the above limit exists and is finite. The following theorem gives us some properties of the operatorLW.
Theorem 2.2.The operatorLW :DW →L2(Td)enjoys the following properties:
i) The domainDW is dense in L2(Td). In particular, the set of eigenvectorsAW forms a complete orthonormal system;
ii) The eigenvalues of the operator−LW form a countable set{αk}k≥0. All eigenvalues have
finite multiplicity, and it is possible to obtain a re-enumeration{αk}k≥0such that
0=α0≤α1≤ · · · and lim
n→∞αn= ∞;
iii) The operatorI−LW :DW →L2(Td)is bijective;
iv) LW :DW → L2(Td)is self-adjoint and non-positive:
−LW f, f ≥ 0;
v) LW is dissipative.
We will now provide an outline of the proof. The details can be found in [13]. SinceDW ⊂DW, we have thatDW is dense inL2(Td). The properties of the eingevalues follows from properties ofLWkand the definition ofLW. From ii) we have thatI−LW is injective. Forv∈L
2(Td), we
have that
v=
+∞
k=1
vkhk ∈L2(Td) , where thevk are such that
∞
k=1
v2k <+∞.
Then,
u=
+∞
k=1
vk
γk
hk∈DW,
and(I−LW)u=v. HenceI−LW is bijective.
LetL∗W : DW∗ ⊂ L2(Td) → L2(Td)be the adjoint ofLW. SinceLW is symmetric, we have
DW ⊂ DW∗. So, to show thatLW = L∗
W, it is enough to show thatDW∗ ⊂ DW. Letϕ = +∞
k=1ϕkhk ∈ DW∗ be given, and letLW∗ϕ = ψ ∈ L2(Td). Then, for allv =+∞
k=1vkhk ∈ DW,
v, ψ = v,LW∗ϕ = LWv, ϕ =
+∞
k=1
−αkvkϕk.
Henceψ =+∞
k=1−αkϕkhk. In particular,+∞k=1αk2ϕ
2
k <+∞ and ϕ ∈ DW. Thus,LW is self-adjoint. By ii)LW is non-positive.
Finally, fix a functionginDW, letλ >0, and let also f =(λI−LW)g. So
λg,g + −LWg,g = g,f ≤ g,g1/2f, f1/2.
Using iv), the second term on the left hand side is non-negative. Thus,λg ≤ f = (λI−
LW)g, that is,LW is dissipative.
3 W-SOBOLEV SPACE AND DIFFERENTIAL EQUATIONS
We construct the W-Sobolev spaces, which consist of functions f having weak generalized gradients
∇W f =(∂W1 f, . . . , ∂Wd f).
Several properties, that are analogous to classical results on Sobolev spaces, are obtained. Ex-istence and uniqueness results for W-generalized elliptic equations, and uniqueness results for W-generalized parabolic equations are also established. More details on these results can be found in [11]
3.1 The definition and properties
Recall the definition of Hilbert spaceL2
xk⊗W k(T
Let L2
xi⊗W i,0(T
d)be the closed subspace of L2
xi⊗W i(T
d)consisting of the functions that have
zero mean with respect to the measured(xj⊗Wj):
Td
f d(xi⊗Wi)=0.
We define the Sobolev space ofW-generalized derivatives as the space of functionsg∈L2(Td)
such that for eachi=1, . . . ,dthere exists a fuctionGi ∈ L2xi⊗W i,0(T
d)satisfying the following
integral by parts identity for every function f ∈DW:
Td
∂xi∂Wi f
g d x = −
Td
(∂Wi f)Gid(x
i⊗W
i). (3.1)
We denote this space byH1,W(Td). For each functiong ∈H1,W we denoteGi by∂Wig, and we
call it theithgeneralized weak derivativeof the functiongwith respect toW.
In [11] it is shown that the setH1,W(Td)is a Hilbert space with respect to the inner product
f,g1,W = f,g + d
i=1
Td
(∂Wi f)(∂Wig)d(x
i⊗W
i), (3.2)
and we obtain the following properties:
Proposition 3.1.On the space H1,W(Td)we have
• (Poincar´e’s Inequality) Let f ∈H1,W(Td). Then, there exists a finite constant C such that
f − Td
f d x
2
L2(Td)
≤C
n
i=1
Td
∂Wi f 2
d(xi⊗Wi):=C∇W f2L2 W(Td)
.
• (Rellich-Kondrachov’s embedding) The embedding H1,W(Td)⊂L2(Td)is compact.
• Denote by HW−1(Td)the dual space of H
1,W(Td). Thus f ∈ HW−1(Td)if and only if there
exist functions f0∈L2(Td), and fk ∈L2xk⊗W k,0(T
d), such that
f = f0−
d
i=1
∂xi fi, (3.3)
in the sense that forv ∈H1,W(Td)
(f, v)=
Td
f0vd x+
d
i=1
Td
fi(∂Wiv)d(x
i⊗W i),
with(·,·)being the dual pairing. Furthermore,
fH−1 W =inf
⎧ ⎨ ⎩ Td d
i=0 |fi|2d x
1/2
; f satisfies(3.3) ⎫ ⎬
⎭
3.2 The elliptic equations
In this subsection we investigate the solvability of uniformly elliptic generalized partial differen-tial equations. Energy methods within Sobolev spaces are, essendifferen-tially, the techniques exploited.
LetA=(aii(x))d×d,x∈Td, be a mensurable diagonal matrix function satisfying the ellipticity condition, that is, there exists a constantθ >0 satisfying
θ−1≤aii(x)≤θ , (3.4)
for everyx∈Tdandi=1, . . . ,d. To keep notation simple, we writeai(x)to meanaii(x).
Our interest lies on solving the problem
Tλu= f, (3.5)
onu, whereu :Td → R, and f : Td →Ris given. HereTλdenotes the generalized elliptic
operator
Tλu := λu− ∇A∇Wu := λu− d
i=1
∂xi
ai(x)∂Wiu
. (3.6)
The bilinear formBλ[·,·]associated with the elliptic operatorTλis given by
Bλ[u, v] = u, v1,W =λu, v + d
i=1
ai(x)(∂Wiu)(∂Wiv)d(Wi⊗xi), (3.7)
whereu, v∈ H1,W(Td).
Let f ∈ HW−1(Td). A functionu ∈ H1,W(Td)is said to be a weak solution of the equation
Tλu= f if
Bλ[u, v] = (f, v) for all v∈H1,W(Td).
Denote byH1⊥,W(Td)be the set of functions inH1,W(Td)which are orthogonal to the constant functions:
H1⊥,W(Td) = {f ∈H1,W(Td);
Td
f d x=0}.
Proposition 3.2 (Energy estimates forλ=0).Let B0be the bilinear form on H1,W(Td)defined
in(3.7)withλ=0. There exist constantsα >0andβ >0such that for all u, v∈H1,W(Td),
|B0[u, v]| ≤αu1,W v1,W
and for all u∈ H1⊥,W
B0[u,u] ≥βu21,W.
Corollary 3.3.Let f ∈L2(Td). There exists a weak solution u ∈H1,W(Td)for the equation
∇A∇Wu = f (3.8)
if and only if
Td
f d x = 0.
In this case, we have uniquenesses of the weak solutions if we disregard addition by constant functions. Also, let u be the unique weak solution of (3.8)in H⊥
1,W(Td), then
u1,W ≤CfL2(Td),
for some constant C independent of f .
To prove this result we begin by noting that, from Proposition 3.2, Bsatisfies the hypotheses of the Lax-Milgram’s Theorem, [1, chapter 6]. This implies that there exists a weak solution
u ∈ H1,W(Td)of (3.8). Since the functionv ≡1∈ H1,W(Td), and∂Wi1=0, we have from the
definition of weak solution that
Td
f d x = B0[u, v] = 0.
Furthermore, the Proposition 3.2 also implies that there is aβ >0 such that
βu21,W ≤B0[u,u] = f,u ≤ fL2(Td)uL2(Td)≤ fL2(Td)u1,W. (3.9)
The existence of weak solutions and the boundCin the statement of the Corollary follows from the previous expression.
Proposition 3.4 (Energy estimates forλ > 0). Let f ∈ L2(Td)andλ > 0. There exists a unique weak solution u∈H1,W(Td)for the equation
λu− ∇A∇Wu = f. (3.10)
This solution enjoys the following bounds
u1,W ≤CfL2(Td)
for some constant C >0independent of f , and
u ≤λ−1fL2(Td).
To obtain this result, note that an elementary computation shows that
|Bλ[u, v]| ≤αu1,W v1,W and Bλ[u,u] ≥βu21,W,
Observe that, forλ > 0, the operatorλI−LAW : DW → L2(Td)is bijective. Therefore, the equation
λu− ∇A∇Wu= f
has strong solution inDW if and only if f ∈ (λI−LWA)(DW), whereIis the identity operator and(λI−LWA)(DW)stands for the range ofDW under the operatorλI−LAW. Moreover, this strong solution coincides with the weak solution obtained in Proposition 3.4.
3.3 W-evolution equations
We study a class ofW-generalized PDEs that evolves in time: the parabolic equations. We begin by introducing the class ofW-generalized parabolic equations we are interested. Then, we define what is meant by weak solution of such equations using theW-Sobolev spaces.
FixT >0 and let(B, · B)be a Banach space. We denote byL2([0,T],B)the Banach space of measurable functionsU : [0,T] → Bsuch that
U2L2([0,T],B) :=
T
0
Ut2Bdt<∞.
LetA= A(t,x)be a diagonal matrix satisfying the ellipticity condition (3.4) for allt ∈ [0,T], and let: [l,r] →Rbe a continuously differentiable function such that
B−1< ′(x) <B,
for allx, whereB>0,l,r∈Rare constants. We will consider the equation
∂tu = ∇A∇W(u) in(0,T] ×Td,
u=γ in{0} ×Td. (3.11)
whereu : [0,T] ×Td→Ris the unknown function, andγ :Td →Ris given.
We say that a functionρ=ρ(t,x)is a weak solution of the problem (3.11) if:
• For everyH ∈DW the following integral identity holds:
Td
ρ(t,x)H(x)d x−
Td
γ (x)H(x)d x=
t
0
Td
(ρ(s,x))∇A∇WH(x)d x ds
• (ρ(·,·))andρ(·,·)belong toL2([0,T],H1,W(Td)):
T
0
(ρ(s,x))2
L2(Td)+ ∇W(ρ(s,x))2L2 W(Td)
ds<∞,
and
T
0
ρ(s,x)2L2(Td)+ ∇Wρ(s,x) 2
L2 W(Td)
We define the energy in jth direction of a functionuas
Qj(u)= sup H∈DW
2
T
0
Td
(∂xj∂WjH)(s,x)u(s,x)d x ds
−
T
0
ds
Td
[∂WjH(s,x)]
2d(xj⊗W j)
,
and the total energy of a functionuas
Q(u)=
d
j=1
Qj(u).
There is a connection between the functions of finite energy and functions in the Sobolev space
H1,W(Td). In fact, from [11, Lemma 4.1], a functionu ∈ L2([0,T],L2(Td))has finite energy if and only ifubelongs toL2([0,T],H1,W(Td)). In the case the energy is finite, we have
Q(u)=
T
0
∇Wu2L2 W(Td)
dt.
Moreover, we have uniqueness of weak solutions of the problem (3.11):
Lemma 3.5. Fixλ > 0, two density profilesγ1,γ2 : T → [l,r] and denote byρ1,ρ2weak solutions of(3.11)with initial valueγ1,γ2, respectively. Then,
ρt1−ρt2, ρ1t,λ−ρ2t,λ
≤
γ1−γ2, γ1,λ−γ2,λ
eBλt/2
for all t >0. In particular, there exists at most one weak solution of (3.11).
The proof can be found in [11].
4 HOMOGENIZATION OF RANDOM OPERATORS
In this section we describe stochastic homogenization results for theW-generalized elliptic dif-ferential operator. This approach is closely related to the one considered in [10]. The focus of this approach is to study the asymptotic behavior of effective coefficients for a family of random difference schemes whose coefficients can be obtained by the discretization of random high-contrast lattice structures. The study of homogenization is motivated by several applications in mechanics, physics, chemistry and engineering. The details on this section can be found in [12].
4.1 Discrete approximation
LetTNbe the one-dimensional discrete torus withNpoints:
Let, also,TdN = TN ×. . .×TN thed-dimensional discrete torus with Nd points. Let f :
1
NTdN →Rbe a function and define the difference operators:
∂xNj f(x/N) = N[f((x+ej)/N)− f(x/N)] and
∂WN
j f(x/N) =
f((x+ej)/N)− f(x/N)
Wj((x+ej)/N)−Wj(x/N)
.
Consider∇WNf =(∂WN
1f, . . . , ∂
N
Wd f)and∇
Nf =(∂N
x1 f, . . . , ∂
N xd f).
We introduce now three inner products:
f,gN :=
1
Nd
x∈Td N
f(x)g(x),
f,gWj,N :=
1
Nd−1
x∈Td N
f(x)g(x) W((x+ej)/N)−W(x/N),
f,g1,W,N := f,gN + d
j=1 ∂WN
j f, ∂
N
WjgWj,N,
and its induced norms:
f2
L2(Td N)
= f, fN, f2L2 W j(TdN)
= f, fWj,N andf 2
H1,W(TdN)
= f, f1,W,N.
These norms are natural discretizations of the norms introduced in the previous sections.
LetA=(ai j)n×nbe a diagonal matrix and letTλNdenote the discrete generalized elliptic opera-tor
TλNu := λu− ∇NA∇WNu, (4.1)
with
∇NA∇WNu :=
d
i=1
∂xNi
ai(x/N)∂WNiu
.
The bilinear formBN[·,·]associated with the elliptic operatorTλNis given by
BN[u, v] = λu, vN
+ 1
Nd−1
d
i=1
x∈Td N
ai(x/N)(∂WNiu)(∂
N
Wiv)[Wi((xi+1)/N)−Wi(xi/N)],
(4.2)
whereu, v:N−1TdN →R.
A functionu: N−1TdN→Ris said to be a weak solution of the equation
whereu: N−1TdN →Ris the unknown function, and f : N−1TdN →Ris given, if
BN[u, v] = f, vN for all v:N−1TdN →R.
We say that a function f : N−1TdN →Rbelongs to the discrete space of functions orthogonal to the constant functionsHN⊥(TdN)if
1
Nd
x∈Td N
f(x/N)=0.
Note that in the set of functions inTdN we have a “Dirac measure” concentrated in a pointx as a function: the function that takes value Nd inx and zero elsewhere. Therefore, we may integrate these weak solutions with respect to this function to obtain that every weak solution is, in fact, a strong solution. Moreover, many properties of the Lebesgue’s measure also holds for the normalized counting measure. In particular, many results stated in Section 3 can be formuled and provedmutatis mutandisto this discrete setup.
Lemma 4.1.The equation
∇NA∇WNu = f,
has a weak solution u:N−1Td
N →Rif and only if
1
Nd
x∈Td N
f(x)=0.
In this case we have uniqueness of the solution disregarding addition by constants. Moreover, if u ∈HN⊥(TdN)we have the bound
uH
1,W(TdN)≤CfL2(TdN), and uL2(TdN)≤λ −1f
L2(Td N),
where C>0does not depend on f nor N .
Lemma 4.2.Letλ >0. There exists a unique weak solution u:N−1TdN →Rof the equation
λu− ∇NA∇WNu = f. (4.3)
Moreover,
uH
1,W(TdN)≤CfL2(TdN), and uL2(TdN)≤λ −1f
L2(Td N),
where C>0does not depend neither on f nor N .
Note that if f ∈ L2(Td)in Lemma 4.2, andu is a weak solution of the problem (4.3), then following bound holds:
uH
1,W(TdN)≤CfL2(Td),
sincefL2(Td N)
4.2 Connections between the discrete and continuous Sobolev spaces
Given a function f ∈ H1,W(Td), we can define its restriction, fN, to the latticeN−1TdNas
fN(x)= f(x) if x∈ N−1TdN.
However, given a function f :N−1TdN →Rit is not straightforward how to define an extension belonging to H1,W(Td). To do so, we need the definition of W-interpolation, which we give below.
Let fN : N−1TN →RandW :R→R, astrictly increasingright continuous function with left limits (c`adl`ag), and periodic. TheW-interpolation fN∗ of fN is given by:
fN∗(x+t) := W((x+1)/N)−W((x+t)/N)
W((x+1)/N)−W(x/N) f(x)
+W((x+t)/N)−W(x/N)
W((x+1)/N)−W(x/N)f(x+1),
for 0≤t <1. Note that
∂fN∗
∂W(x+t)=
f(x+1)− f(x)
W((x+1)/N)−W(x/N) = ∂
N W f(x).
Using the standard construction ofd-dimensional linear interpolation, it is possible to define the
W-interpolation of a function fN :TdN →R, withW(x)=di=1Wi(xi)as defined in (2.1).
We now establish the connection between the discrete and continuous Sobolev spaces by showing how a sequence of functions defined inTdN can converge to a function inH1,W(Td).
We say that a family fN ∈L2(TdN)converges strongly (resp. weakly) to the function f ∈L2(Td) as N → ∞, if fN∗ converges strongly (resp. weakly) to the function f. From now on we will omit the symbol “∗” in theW-interpolated function, and denote them simply by fN.
The convergence inHW−1(Td)can be defined in terms of duality. Namely, we say that a functional
fN onTdN converges to f ∈ HW−1(Td)weakly (resp. strongly) if for any sequence of functions
uN :TdN → Randu ∈ H1,W(Td)such thatuN → uweakly (resp. strongly) inH1,W(Td), we have
(fN,uN)N −→(f,u), asN → ∞.
4.3 Random environment
In this subsection we introduce the statistically homogeneous rapidly oscillating coefficients that will be used to define the randomW-generalized difference elliptic operators.
Let(,F, µ) be a standard probability space and {Tx : → ;x ∈ Zd}be a group of F-measurable and ergodic transformations which preserve the measureµ:
• µ(TxA)=µ(A), for anyA∈Fandx∈Zd,
• T0=I, Tx◦Ty =Tx+y,
• For any f ∈ L1() such that f(Txω) = f(ω) µ-a.s for each x ∈ Zd, is equal to a constantµ-a.s.
Note that the last condition implies that the groupTxis ergodic. Let us now introduce the vector-valuedF-measurable functions{aj(ω); j =1, . . . ,d}such that there existsθ >0 with
θ−1≤aj(w)≤θ ,
for allω ∈ and j = 1, . . . ,d. Then, define the diagonal matrices AN whose elements are given by
aNj j(x):=aNj =aj(TN xω) , x∈TNd, j=1, . . . ,d. (4.4)
Letλ >0, and let fN be a functional on the space of functionshN :TdN →R, f ∈ HW−1(Td). Let, also,uN be the unique weak solution of
λuN− ∇NAN∇WNuN = fN,
andu0be the unique weak solution of
λu0− ∇A∇Wu0= f. (4.5)
We say that the diagonal matrix Ais ahomogenizationof the sequence of random matrices AN
if the following conditions hold:
• For each sequence fN → f in HW−1(Td), uN converges weakly in H1,W tou0, when
N → ∞;
• aiN∂WN
iu
N →a
i∂Wiu, weakly inL 2
xi⊗W i(T
d)whenN → ∞.
The following homogenization theorem is proved in [12]:
Theorem 4.3. Let AN be a sequence of ergodic random matrices, such as the one that defines our random environment. Then, almost surely, AN(ω)admits a homogenization, where the ho-mogenized matrix A does not depend on the realizationω.
Note that ifu∈DW is a strong solution (or weak) of
λu− ∇A∇Wu= f
anduNis strong solution of the discrete problem
λuN− ∇NAN∇WNuN = f
RESUMO.Neste artigo discutimos recentes resultados sobre uma generalizac¸˜ao do
Lapla-ciano. Mais precisamente, fixe uma func¸˜ao W(x1, . . . ,xd) = dk=1Wk(xk), onde cada Wk :R →R´e uma func¸˜ao cont´ınua ´a direita com limites a esquerda e estritamente
cres-cente. UsandoW, constru´ımos o laplaciano generalizadoLW = d
i=1∂xi∂Wi, onde∂Wi denota o operador diferencial induzido porWi. Apresentamos resultados sobre propriedades
espectrais deLW, espac¸os de Sobolev induzidos porLW (espac¸osW-Sobolev), equac¸˜oes
diferenciais parciais generalizadas, equac¸˜oes diferenciais estoc´asticas e homogeinizac¸˜ao
es-toc´astica.
Palavras-chave:Espac¸osW-Sobolev, Laplaciano generalizado, Homogeinizac¸˜ao, Equac¸˜oes
diferenciais parciais.
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