Recent Results on a Generalization of the Laplacian^†

Abstracts

In this paper we discuss recent results regarding a generalization of the Laplacian. To be more precise, fix a function W(x ₁, ..., x_d ) = Σ^d_k=1 W_k (x_k ), where each W_k : ℝ → ℝ is a right continuous with left limits and strictly increasing function. Using W, we construct the generalized laplacian ℒW = Σ^d_i=1 ∂xi ∂wi, where ∂wi is a generalized differential operator induced by the function W_i . We present results on spectral properties of ℒW, Sobolev spaces induced by ℒW(W-Sobolev spaces), generalized partial differential equations, generalized stochastic differential equations and stochastic homogenization.

W-Sobolev space; generalized Laplacian; homogenization; partial differential equations

Neste artigo discutimos recentes resultados sobre uma generalização do Laplaciano. Mais precisamente, fixe uma função W(x ₁, ..., x_d ) = Σ^d_k=1 W_k (x_k ), onde cada W_k : ℝ → ℝ é uma função contínua á direita com limites a esquerda e estritamente crescente. Usando W, construímos o laplaciano generalizado ℒW = Σ^d_i=1 ∂xi ∂wi, onde ∂wi denota o operador diferencial induzido por W_i . Apresentamos resultados sobre propriedades espectrais de ℒW, espaços de Sobolev induzidos por ℒW (espaços W-Sobolev), equações diferenciais parciais generalizadas, equações diferenciais estocásticas e homogeinização estocástica.

Espaços W-Sobolev; Laplaciano generalizado; Homogeinização; Equações diferenciais parciais

1 INTRODUCTION

In the '50s William Feller introduced a more general concept of differential operators, that is, operators of the type (d/dW)(d/dV) where, typically, W and V are strictly increasing functions with V (but not necessarily W) being continuous. In this paper we are interested in the formal adjoint of (d/dW)(d/dV), which is simply (d/dV)(d/dW), in the case V(x) = x is 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 function W, 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 function W, which may have jumps. Even more, the set of jumps of W can be dense in ℝ.

The goal of this paper is present an overview of the main recent results regarding this differential operator. The rest of the paper unfolds as follows: in Section 2 we present the generalized Laplacian, which we denote by ℒW; in Section 3 we construct the W-Sobolev spaces and present several properties that are analogous to classical results on Sobolev spaces, we also present results on existence and uniqueness for W-generalized elliptic equations, and a uniqueness result for W-generalized parabolic equations; in Section 4 we present stochastic homogenization results of suitable random operators , that are discretizations of the operator ℒW.

This paper is essencially inspired in works (¹¹, ¹², ³, ¹³).

2 THE OPERATOR ℒW

Fix a function W : ℝd → ℝ such that

where Wk : ℝ → ℝ are strictly increasing right continuous functions with left limits (càdlàg), and periodic in the sense that

Wk(u + 1) - Wk(u) = Wk(1) - Wk(0)

for all u?ℝ and k = 1,..., d. To keep notation simple, we assume that Wk vanishes at the origin, that is, Wk(0) = 0.

Denote by 𝕋 the unidimensional torus and ??,?? the inner product of L2(𝕋):

For each k =1, 2, ..., d let be the set of functions f in L ²(𝕋) such that

for some function f in L ²(𝕋) such that

Define the operator as . Formally,

where the generalized derivative d/dWk is defined as

if the above limit exists and is finite.

By a convenient restriction of the operators to a dense subspace 𝒟k ⊂ , it is not difficult to prove that : 𝒟k → L ²(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 by , : → L ²(𝕋), is self-adjoint and, the set of the eigenvectors of forms a complete orthonormal system in L ²(𝕋), the details of this approach can be found in ⁶.

Let

and define the operator 𝕃W : 𝔻W:= span() → L ²(𝕋d) as follows: for ( ,

and extend to 𝔻W by linearity. In particular, 𝔻W is dense in L2(𝕋d); and the set forms a complete, orthonormal, countable system of eigenvectors for the operator 𝕃W.

Let be the Hilbert space of measurable functions H : 𝕋d → ℝ such that

Where d(xk ⊗ W_k ) = dx ₁ ... dx_k-1 dW_k dx_k+1 ... dx_d .

Lemma 1 Let f, g ( 𝔻_W, then for i=1,..., d,

In particular,

that is, 𝕃_W 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,

where the sum is over partitions of the torus 𝕋 and Δhstands for the increment of the function h with respect to the partition. The details can be found in ¹³.

Consider

where h_k ( and (_k is the eigenvalue associated to to eigenvector h_k .

Define the operator ℒW : 𝒟W → L ²(𝕋d) by

The operator ℒW is clearly an extension of the operator 𝕃W, and formally,

where

and the partial generalized derivative is defined by

if the above limit exists and is finite. The following theorem gives us some properties of the operator ℒW.

Theorem 2.2 The operator ℒW : 𝒟W → L ²(𝕋d) enjoys the following properties:

• i) The domain 𝒟_W is dense in L ²(𝕋d). In particular, the set of eigenvectors forms a complete orthonormal system;

• ii) The eigenvalues of the operator -ℒ_W form a countable set {(_k }_k>0. All eigenvalues have finite multiplicity, and it is possible to obtain a re-enumeration {(_k }_k>0 such that

• iii) The operator 𝕀 -ℒW : 𝒟W → L ²(𝕋d) is bijective;

• iv) ℒW : 𝒟W → L ²(𝕋d) is self-adjoint and non-positive:

⟨-ℒW f, f⟩ > 0

• v) ℒ_W is dissipative.

In view of i), iii) and iv) we may use Hille-Yosida theorem to conclude that ℒW is the generator of a strongly continuous contraction semigroup {P_t : L ²(𝕋d) → L ²(𝕋d)}t>0.

We will now provide an outline of the proof. The details can be found in ¹³. Since 𝔻W ⊂ 𝒟W, we have that 𝒟W is dense in L ²(𝕋d). The properties of the eingevalues follows from properties of and the definition of ℒW. From ii) we have that 𝕀 - ℒWis injective. For υ ( L ²(𝕋d), we have that

Then,

and (𝕀 - ℒW)u = υ. Hence 𝕀 - ℒW is bijective.

Let : 𝒟W ⊂ L ²(𝕋d) → L ²(𝕋d) be the adjoint of ℒW. Since ℒW is symmetric, we have 𝒟W ⊂ 𝒟W*. So, to show that ℒW = , it is enough to show that 𝒟W* ⊂ 𝒟W. Let be given, and let ℒ_W*φ = ψ ( L ²(𝕋d). Then, for all ,

Hence . In particular, and φ ( 𝒟W. Thus, ℒWis self-adjoint. By ii) ℒW is non-positive.

Finally, fix a function g in 𝒟W, let λ > 0, and let also f = (λ𝕀 - ℒW)g. So

λ⟨g, g⟩ + ⟨-ℒ_W g, g⟩ = ⟨g, f⟩ < ⟨g, g⟩^1/2 ⟨f, f⟩^1/2.

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 generalizedgradients

Several properties, that are analogous to classical results on Sobolev spaces, are obtained. Existence 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 space given in Section 2.

Let be the closed subspace of consisting of the functions that have zero mean with respect to the measure d(xj ⊗ W_j ):

We define the Sobolev space of W-generalized derivatives as the space of functions g ( L ²(𝕋d) such that for each I = 1, ..., d there exists a fuction G_i ( satisfying the following integral by parts identity for every function f( 𝔻W:

We denote this space by H_1,W (𝕋d). For each function g ( H_1,W we denote G_i by , and we call it the ith generalized weak derivative of the function g with respect to W.

In (¹¹) it is shown that the set H1,W(𝕋d) is a Hilbert space with respect to the inner product

and we obtain the following properties:

Proposition 3 1 On the space H1,W(𝕋d) we have

• ? (Poincaré's Inequality) Let f ( H1,W(𝕋d). Then, there exists a finite constant C such that

• ? (Rellich-Kondrachov's embedding) The embedding H1,W(𝕋d) ⊄ 𝕃2(𝕋d) is compact.

• ? Denote by the dual space of H_1,W (𝕋d). Thus f ( if and only if there exist functions f ₀ ( L ²(𝕋d), and f_k ( , such that

in the sense that for υ ( H1,W(𝕋d)

with (?,?) being the dual pairing. Furthermore,

3.2 The elliptic equations

In this subsection we investigate the solvability of uniformly elliptic generalized partial differential equations. Energy methods within Sobolev spaces are, essentially, the techniques exploited.

Let A = (aii(x))d×d, x ( 𝕋d, be a mensurable diagonal matrix function satisfying the ellipticity condition, that is, there exists a constant θ > 0 satisfying

for every x ( 𝕋d and I = 1, ..., d. To keep notation simple, we write ai(x) to mean aii(x).

Our interest lies on solving the problem

on u, where u : 𝕋d → ℝ, and f : 𝕋d → ℝ is given. Here Tλ denotes the generalized elliptic operator

The bilinear form Bλ[·, ·] associated with the elliptic operator Tλ is given by

where u, υ ( H1,W(𝕋d).

Let f ( . A function u ( H1,W(𝕋d) is said to be a weak solution of the equation Tλu = f if

Bλ[u, υ] = (f, υ) for all υ ( H1,W(𝕋d).

Denote by be the set of functions in H_1,W (𝕋d) which are orthogonal to the constant functions:

Proposition 3.2 (Energy estimates for λ = 0) Let B ₀ be the bilinear form on H_1,W (𝕋d) defined in (3.7) with λ = 0. There exist constants α > 0 and β > 0 such that for all u, υ ( H_1,W (𝕋d)

|B ₀[u, υ]| < α‖u‖1,W‖υ‖1,W

and for all

The proof of this result follows from Poincaré's inequality and (3.4).

Corollary 3.3 Let f ( L2(𝕋d). There exists a weak solution u ( H1,W(𝕋d) for the equation

if and only if

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 , then

for some constant C independent of f.

To prove this result we begin by noting that, from Proposition 3.2, B satisfies 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 function υ ≡ 1 ( H1,W(𝕋d), and , we have from the definition of weak solution that

Furthermore, the Proposition 3.2 also implies that there is a β > 0 such that

The existence of weak solutions and the bound C in the statement of the Corollary follows from the previous expression.

Proposition 3.4 (Energy estimates for λ > 0) Let f ( L2(𝕋d) and λ > 0. There exists a unique weak solution u ( H1,W(𝕋d) for the equation

This solution enjoys the following bounds

for some constant C > 0 independent of f, and

To obtain this result, note that an elementary computation shows that

where β = min{λ, θ-1} > 0, α = max{λ, θ}< ? and θ is given in (3.4). The proof follows from Lax-Milgram's Theorem and an estimate similar to (3.9).

Observe that, for λ > 0, the operator λ𝕀 - : 𝒟W → L ²(𝕋d) is bijective. Therefore, the equation

λu - ∇A∇_Wu = f

has strong solution in 𝔻W if and only if f ( (λ𝕀 - )(𝔻W), where 𝕀 is the identity operator and (λ𝕀 - )(𝔻W) stands for the range of 𝔻W under the operator λ𝕀 - . Moreover, this strong solution coincides with the weak solution obtained in Proposition 3.4.

3.3 W-evolution equations

We study a class of W-generalized PDEs that evolves in time: the parabolic equations. We begin by introducing the class of W-generalized parabolic equations we are interested. Then, we define what is meant by weak solution of such equations using the W-Sobolev spaces.

Fix T > 0 and let (B, ‖·‖B) be a Banach space. We denote by L2([0, T], B) the Banach space of measurable functions U : [0, T] → B such that

Let A = A(t, x) be a diagonal matrix satisfying the ellipticity condition (3.4) for all t ? [0, T], and let Φ : [l, r] → ℝ be a continuously differentiable function such that

B-1 < Φ' (x) < B,

for all x, where B > 0, l, r ? ℝ are constants. We will consider the equation

where u : [0, T] × Td → ℝ is the unknown function, and γ : 𝕋d → ℝ is given.

We say that a function ρ =ρ(t, x) is a weak solution of the problem (3.11) if:

• ? For every H ( 𝔻W the following integral identity holds:

• ? Φ(ρ (?, ?)) and ρ (?, ?) belong to L2 ([0, T], H1,W(𝕋d)):

and

We define the energy in jth direction of a function u as

and the total energy of a function u as

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 function u ( L2([0, T], L2(𝕋d)) has finite energy if and only if u belongs to L2([0, T], H1,W(𝕋d)). In the case the energy is finite, we have

Moreover, we have uniqueness of weak solutions of the problem (3.11):

Lemma 3.5 Fix λ > 0, two density profiles γ1, γ2 : 𝕋 → [l, r] and denote by ρ1, ρ2 weak solutions of (3.11) with initial value γ1, γ2, respectively. Then,

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 the W-generalized elliptic differential 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

Let 𝕋N be the one-dimensional discrete torus with N points:

𝕋N = ℝ/Nℤ ≃ {0, 1, ..., N - 1}.

Let, also, = 𝕋N × ... × 𝕋N the d-dimensional discrete torus with Ndpoints. Let f : → ℝ be a function and define the difference operators:

Consider and .

We introduce now three inner products:

and its induced norms:

These norms are natural discretizations of the norms introduced in the previous sections.

Let A = (aij)n×n be a diagonal matrix and let denote the discrete generalized elliptic operator

with

The bilinear form B_N [·, ·] associated with the elliptic operator is given by

where u, υ : .

A function u : is said to be a weak solution of the equation

where u : is the unknown function, and f : is given, if

We say that a function f : belongs to the discrete space of functions orthogonal to the constant functions if

Note that in the set of functions in we have a "Dirac measure" concentrated in a point x as a function: the function that takes value Nd in x 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 proved mutatis mutandis to this discrete setup.

Lemma 4.1 The equation

has a weak solution u : if and only if

In this case we have uniqueness of the solution disregarding addition by constants. Moreover, if u ( we have the bound

where C > 0 does not depend on f nor N.

Lemma 4.2 Let λ > 0. There exists a unique weak solution u : of the equation

Moreover,

where C > 0 does not depend neither on f nor N.

Note that if f ( L ²(𝕋d) in Lemma 4.2, and u is a weak solution of the problem (4.3), then following bound holds:

since as N → ?.

4.2 Connections between the discrete and continuous Sobolev spaces

Given a function f ( H_1,W (𝕋d), we can define its restriction, f_N , to the lattice as

However, given a function f : it is not straightforward how to define an extension belonging to H_1,W (𝕋d). To do so, we need the definition of W-interpolation, which we give below.

Let f_N : N ^-1𝕋N → ℝ and W : ℝ → ℝ, a strictly increasing right continuous function with left limits (càdlàg), and periodic. The W-interpolation of f_N is given by:

for 0 ? t < 1. Note that

Using the standard construction of d-dimensional linear interpolation, it is possible to define the W-interpolation of a function , with W(x) = 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 in can converge to a function in H_1,W (𝕋d).

We say that a family f_N ( L ²() converges strongly (resp. weakly) to the function f ( L ²(𝕋d) as N → ?, if converges strongly (resp. weakly) to the function f. From now on we will omit the symbol " " in the W-interpolated function, and denote them simply by f_N .

The convergence in can be defined in terms of duality. Namely, we say that a functional f_N on converges to f ( weakly (resp. strongly) if for any sequence of functions u_N : → ℝ and u ( H_1,W (𝕋d) such that u_N → u weakly (resp. strongly) in H_1,W (𝕋d), we have

(f_N, u_N )N → (f, u), as N → ?.

4.3 Random environment

In this subsection we introduce the statistically homogeneous rapidly oscillating coefficients that will be used to define the random W-generalized difference elliptic operators.

Let (Ω, ℱ, μ) be a standard probability space and {T_x : Ω → Ω; x ( ℤd} be a group of ℱ -measurable and ergodic transformations which preserve the measure μ:

• ? T_x : Ω → Ω is ℱ-measurable for all x ( ℤd,

• ? μ (T_x A) = μ (A), for any A? ℱ and x ( ℤd,

• ? T ₀ = I, T_x ͦ T_y = T_x+y ,

• ? For any f ( L ¹(Ω) such that f(T_xω) = f(ω) μ-a.s for each x ( ℤd, is equal to a constant μ-a.s.

Note that the last condition implies that the group T_x is ergodic. Let us now introduce the vector-valued ℱ-measurable functions {a_j (ω); j = 1, ..., d} such that there exists θ >0 with

θ^-1 < a_j (ω) < θ,

for all ω ? Ω and j = 1, ..., d. Then, define the diagonal matrices AN whose elements are given by

Let λ > 0, and let f_N be a functional on the space of functions h_N : → ℝ, f ( . Let, also, be the unique weak solution of

and u0 be the unique weak solution of

We say that the diagonal matrix A is a homogenization of the sequence of random matrices AN if the following conditions hold:

• ? For each sequence fN → f in , uN converges weakly in H1,W to u0, when N ? ?;

• ? , weakly in when N ? ?.

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, A}_N(ω) admits a homogenization, where the homogenized matrix A does not depend on the realization ω

Note that if u ( 𝔻W is a strong solution (or weak) of

λu - ∇A∇_Wu = f

and u_N is strong solution of the discrete problem

then, the homogenization theorem also holds, that is, u_N also converges weakly in H1,W to u.

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• †
  Research supported by CNPq and FAPES.

Publication Dates

• Publication in this collection
  Aug 2015

History

• Received
  14 May 2014
• Accepted
  23 May 2015