Characterizing the strange term in critical size homogenization: Quasilinear equations with a general microscopic boundary condition

1 Introduction

A well-known effect in homogenization theory is the appearance of some changes in the structural modelling of the homogenized problem for suitable critical size of the elements configuring the “micro-structured” material. It seems that the first result in this direction was presented in the pioneering paper by Marchenko and Hruslov [27]. A more popular presentation of the appearance of some “strange terms” was due to Cioranescu and Murat [4]. Both articles dealt with linear equations with Neumann and Dirichlet boundary conditions, respectively. Since then many papers were devoted to different formulations, e.g., more general elliptic partial differential equations (possibly of quasilinear type), Robin type and other boundary conditions of different nature, etc. It is impossible to mention all of them here (a few of them will be mentioned in the rest of the introduction) but the reader may imagine that the nature of this “strange term” may be completely different according to the peculiarities of the formulation in consideration (something that was already indicated at the end of the introduction of the paper by Cioranescu and Murat [4]).

The main goal of this paper is to characterize the change of structural behavior arising in the homogenization process when applied to chemical reactions taking place on fixed-bed nanoreactors, at the microscopic level, on the boundary of the particles

for a very general type of chemical kinetics (here given by the maximal monotone graph σ of ℝ 2 ). Thanks to this generality on the maximal monotone graph σ, our treatment also includes the case of microscopic Dirichlet boundary conditions. In this way we unify the treatment of apparently different formulations, which before were considered separately.

The diffusion is modeled by the quasilinear operator Δ p ⁢ u ε ≡ div ⁢ ( | ∇ ⁡ u ε | p - 2 ⁢ ∇ ⁡ u ε ) , with p > 1 . Notice that p = 2 corresponds to the linear diffusion operator, and that p ≠ 2 appears in turbulent regime flows or non-Newtonian flows (see [8]). As it is well known, this operator appears in many other contexts and is one of the best examples of quasilinear operators leading to a formulation in terms of nonlinear monotone operators (see, e.g., [26, 1, 7]).

The “normal derivative” must be then understood as ∂ ν p ⁡ u ε = | ∇ ⁡ u ε | p - 2 ⁢ ∇ ⁡ u ε ⋅ 𝝂 , where 𝝂 is the outward unit normal vector on the boundary of the particles S ε ⊂ ∂ ⁡ Ω ε . In fact, we will consider the structural assumption

The cases p ≥ n are completely different, see [31, 32] (see also, for instance, the study made for a general monotone quasilinear equation with Dirichlet boundary conditions in [7]).

As mentioned before, the generality assumed on the maximal monotone graph σ of ℝ 2 allows to treat, in a unified way, different cases as the case of Dirichlet boundary conditions, which corresponds to the choice of σ given by

(see, e.g. [1]), and the case of nonlinear Robin type boundary conditions, which corresponds (see, e.g. [24]) to the case in which D ⁢ ( σ ) = ℝ and σ is a continuous nondecreasing function.

The domain Ω ε ⊂ ℝ n is assumed to have an ε-periodical structure. Since our main goal is to get a very precise description of the so-called “strange term” in the homogenized problem, we shall assume that the particles are balls of radius a ε = C 0 ⁢ ε α , where α > 1 . One of the interesting properties that arise from our precise characterization is that there is uniqueness of solutions of the homogenized problem. This was not always proved in previous results (cf. the general framework considered in [7], and how their characterization, given in their Lemma 5.1, is not enough to get the uniqueness of solution of their homogenized problem). The consideration of particles of a general shape is a difficult task, especially the exact identification of the “strange terms”. A similar formulation to the one considered in this paper for that case can be obtained, at least for continuous σ, and has been the subject of a different paper (see [15]).

The problem has two different parameters: α, the size of the particles, and γ, the normalization factor of the boundary condition on S ε . When they have critical values

then our main result in this paper shows that the homogenized problem involves a different distributed chemical kinetics nonlinearity:

where

and H : ℝ → ℝ is given by

with

and

where ω n is the surface area of the unit sphere in ℝ n . We show that, for any maximal monotone graph σ, H is a nondecreasing contraction, and thus the existence, uniqueness and continuous dependence of solutions of the homogenized problem is consequence of well-known results on monotone operator theory.

The change of behavior from the nonlinearity of type σ in the nonhomogeneous problem to the nonlinearity H in the homogeneous problem is one of the characteristics of the nanotechnological effects (see, e.g., [33]) and does not appear if 1 ≤ α < n n - p (see [5, 34]).

Before presenting the details of the notation used above, let us mention that our main aim is to provide a common roof and extend (under different points of view) some previous results in the literature concerning different structural assumptions (i.e., the functions σ and H) after the homogenization process.

The case of Robin boundary conditions ∂ n ⁡ u + β ⁢ ( ε ) ⁢ σ ⁢ ( u ) = 0 on S ε was first studied by Marchenko and Hruslov in a series of papers dealing mainly with the linear case σ ⁢ ( u ) = λ ⁢ u , see [19, 20, 21, 27]. Some references on different choices of smooth functions σ can be found in [6, 18, 24, 25, 37, 22, 29] and the references therein. For further references, see [13, 23, 17, 16]. Some previous results by the authors [11], formulated there for some not necessarily Lipschitz functions σ and p ∈ [ 2 , n ) , will be here extended to the case a general maximal monotone graph σ (which includes the case of Dirichlet boundary conditions) and p ∈ ( 1 , n ) .

The special case of Dirichlet boundary condition u ε = 0 on S ε , covered by (1.2), gives σ - 1 ⁢ ( s ) = 0 for any s ∈ ℝ , and so H ⁢ ( r ) = r for any r ∈ ℝ . Therefore, the “strange term” arising in the homogenized equation becomes 𝒜 ⁢ | u | p - 2 ⁢ u . This was shown for p = 2 in the pioneering paper by Cioranescu and Murat [4]. However, even in this simple case, the treatment in [7] for the case p ≠ 2 is not as sharp as in our case. In [7], Dal Maso and Skrypnik do not provide an explicit expression for this strange term. In fact, their characterization (see [7, Lemma 5.1]) does not guaranty uniqueness of solutions of the homogenized problem.

The case of the boundary condition

which was studied for smooth σ 0 in [22] by ad hoc techniques, is also covered by the common proof provided in this paper, by taking

See also [12, 28].

The choice of the critical values of α and γ might appear arbitrary. Let us give some reasons why this is a good choice. First, if N ⁢ ( ε ) is the number of particles, then N ⁢ ( ε ) ∼ ε - n . It is easy to see that | S ε | = N ⁢ ( ε ) ⁢ | ∂ ⁡ ( a ε ⁢ G 0 ) | ∼ ε α ⁢ ( n - 1 ) - n , where G 0 is the unit ball centered at 0. Let us analyze the choice of γ. If we consider the reaction term on the weak formulation, with σ ⁢ ( u ε ) a bounded sequence in L ∞ and v a bounded test function, then

is a bounded sequence. Hence, if the sequence u ε is bounded in L ∞ and v is a bounded test function, then

can only be expect to tend to either 0 or + ∞ if γ ≠ α ⁢ ( n - 1 ) - n , and hence we will lose the reaction term on the equation on the homogenized equation or we lose the equation altogether. If the macroscopic behavior is given by a reaction diffusion equation (with nontrivial reaction), then the choice scaling γ as ε → 0 can be no other.

The appearance of the critical value of α has to do with a property of traces. It is known (see [30]) that

As it turns out, the critical scale is the one in which both terms in the right-hand side have the same order of convergence. Notice that, in the critical case α = n n - p , we have γ = α ⁢ ( p - 1 ) .

Notice that for a Newtonian fluid in ℝ 3 ( n = 3 , p = 2 ), the critical size corresponds to α = 3 . Obviously the critical value of α is an increasing function of p. Therefore, for non-Newtonian dilatant fluids or a Newtonian flow in turbulent regime ( p > 2 ), our assumption means α > 3 , the particles are tiny with respect to their repetition, whereas for pseudoplastic fluids ( p < 2 ), the critical particles satisfy α < 3 , and hence are not so tiny with respect to their repetition.

A relevant application of our results is the following. Let us consider the usual formulation in Chemical Engineering (see [35, 9]) with a constant external supply

where g is a nondecreasing real function such that g ⁢ ( 0 ) = 0 . In order to adapt our results, we introduce the change in variable u = 1 - w and σ ⁢ ( u ) = g ⁢ ( 1 ) - g ⁢ ( 1 - u ) , and the problem becomes

Notice that the presence of w ε = 1 on ∂ ⁡ Ω is translated to a source in S ε for u ε . We will see later (Theorem 6.2) that the new equation for H, when α = n n - 2 , is

that is,

so that an extension of w ε converges weakly in H 1 ⁢ ( Ω ) to w crit , the solution of

and h is given by

Notice that in the case of Neumann problems, σ ⁢ ( s ) ≡ 0 for any s ∈ R , and although σ - 1 is a well-known maximal monotone graph, the more direct identification of the “strange term” H ⁢ ( u ) is obtained trough the implicit equation (1.7), since in this case we get that

In the noncritical cases, 1 < α < n n - 2 , we will show that an extension of w ε converges weakly in H 1 ⁢ ( Ω ) to w non ⁢ - ⁢ crit , the solution of

with 𝒜 ^ = C 0 n - 1 ⁢ | ∂ ⁡ G 0 | . Finally, we will show in Theorem 6.3 that

so we have a pointwise “better” reaction in the critical case [10]. We point out that a different criterion to establish the optimality of the reaction in terms of the so-called “chemical effectiveness” was considered by the authors in [14].

The plan of the rest of the paper is the following: Section 2 will be devoted to the statement of the main results, Section 3 contains the proof of the existence results for equation (1.1) and the characterization of H, Section 4 is devoted to the proof of the main result, Theorem 2.4, and Section 5 contains the proof of the auxiliary Theorem 2.9, which studies the limit of the diffusion. We conclude the paper with Section 6, where we study the noncritical case and the pointwise comparison of its homogenized solution with the critical case.

2 Statement of the main results

Let Ω be a bounded domain in ℝ n , n ≥ 3 , with a smooth boundary ∂ ⁡ Ω , and let Y = ( - 1 2 , 1 2 ) n . Denote by G 0 = B 1 ⁢ ( 0 ) the unit ball centered at the origin. This plays a crucial role in the proof. As far as we known, no results are known in the critical cases if G 0 is not a ball. For δ > 0 and ε > 0 , we define sets δ ⁢ B = { x : δ - 1 ⁢ x ∈ B } and Ω ~ ε = { x ∈ Ω : ρ ⁢ ( x , ∂ ⁡ Ω ) > 2 ⁢ ε } . Let

where α > 1 and C 0 is a given positive number. Define

where Υ ε = { j ∈ ℤ n : ( a ε ⁢ G 0 + ε ⁢ j ) ∩ Ω ~ ¯ ε ≠ ∅ } , N ⁢ ( ε ) = | Υ ε | ≅ ε - n , and ℤ n denotes the set of vectors z with integer coordinates. Define Y ε j = ε ⁢ Y + ε ⁢ j , where j ∈ Υ ε and note that G ¯ ε j ⊂ Y ¯ ε j and center of the ball G ε j coincides with the center of the cube Y ε j . Our “microscopic domain” is defined as

We define the space W 0 1 , p ⁢ ( Ω ε , ∂ ⁡ Ω ) as the completion, with respect to the norm of W 1 , p ⁢ ( Ω ε ) , of the set of infinitely differentiable functions in Ω ¯ ε equal to zero in a neighborhood of ∂ ⁡ Ω , that is,

Concerning the solvability of problem (1.1), we start by introducing the notion of weak solution. Since we assume that σ : ℝ → 𝒫 ⁢ ( ℝ ) , where 𝒫 ⁢ ( ℝ ) denotes the set of subsets of ℝ , we recall, by well-known results (see, e.g., [2]), that

and that there exists a convex lower semicontinuous function Ψ : ℝ → ( - ∞ , + ∞ ] , with Ψ ⁢ ( 0 ) = 0 , such that σ = ∂ ⁡ Ψ is its subdifferential. We also know that if we define

where ∅ denotes the empty set, and

then D ⁢ ( σ ) ⊂ D ⁢ ( Ψ ) ⊂ D ⁢ ( Ψ ) ¯ = D ⁢ ( σ ) ¯ .

In the rest of the paper we will always assume that f ∈ L p ′ ⁢ ( Ω ) , where, as usual, p ′ = p p - 1 .

Since u ε is the minimizer of the following energy functional in W 1 , p ⁢ ( Ω ε , ∂ ⁡ Ω ) (see [26, 1]):

we consider the following definition of weak solution.

The existence and uniqueness of a weak solution to problem (2.2) is an easy consequence of well-known results:

The homogenized problem will involve the function H : ℝ → ℝ given by (1.5). Let us present some of the properties satisfied by H.

Concerning the homogenized problem (1.3), we point out that since H is a nondecreasing nonexpansion on ℝ , for the parameters 𝒜 and ℬ 0 given by (1.4) and (1.6), and for f ∈ L p ′ ⁢ ( Ω ) , there exists a unique weak solution u ∈ W 0 1 , p ⁢ ( Ω ) of problem (1.3). Moreover, | H ⁢ ( u ) | p - 2 ⁢ H ⁢ ( u ) ∈ L p ′ ⁢ ( Ω ) . For the proof it is enough to set V = W 0 1 , p ⁢ ( Ω ) and define the operator A : V → V ′ by

Notice that, since H is Lipschitz, H ⁢ ( v ) ∈ L p ⁢ ( Ω ) for any v ∈ L p ⁢ ( Ω ) . Then A is a hemicontinuous strictly monotone coercive operator, and the existence and uniqueness of a weak solution u is standard (see, e.g., [26]).

We will make fundamental use of the following reformulation of a weak solution. Since the limit operator A : V → V ′ , with V = W 0 1 , p ⁢ ( Ω ) , given by (2.5), is hemicontinuous and monotone, we can use the Brezis–Sibony characterization (see [3, Lemma 1.1] or [26, Chapter 2, Theorem 2.2]), that is, u ∈ W 0 1 , p ⁢ ( Ω ) is a weak solution of (1.3) if and only if

The main result of this paper is the following convergence result.

The other key result we will prove in this paper is Theorem 2.9 below, the statement of which requires some preliminary lemmas. The extension u ~ ε of solutions u ε can be obtained by applying the methods of [30].

Moreover, by applying this extension theorem and the methods introduced in [30], we can prove the following useful estimates.

Thanks to the a priori estimate (2.3) and the properties of the extension operator P ε : W 0 1 , p ⁢ ( Ω ε , ∂ ⁡ Ω ) → W 1 , p ⁢ ( Ω ) , we know that and there exists u ∈ W 0 1 , p ⁢ ( Ω ) such that

The difficult task is to show that u ∈ W 0 1 , p ⁢ ( Ω ) is the weak solution of problem (1.3) such as it is ensured in Theorem 2.4.

Motivated by this and (2.6), we will also use the fact that if u ε ∈ W 0 1 , p ⁢ ( Ω ε , ∂ ⁡ Ω ) is the weak solution of problem (1.1), then

for any test function v ∈ W 1 , p ⁢ ( Ω ε , ∂ ⁡ Ω ) .

The problematic term, in order to pass to the limit, is the boundary integrals over S ε . Here we will follow a technique of proof introduced by the last author (Shaposhnikova) in collaboration with different co-authors (see, e.g., [28, 37, 34]), which can be applied in different frameworks.

This lemma (which we remark is independent of α and γ, see the proof in [37]) is the key point of the homogenization technique in the critical case. It is based in the general idea that if P ε j is the center of the ball G ε j = { x ∈ Y ε j : | x - P ε j | < a ε } and if T ε j denotes the ball of radius ε / 4 centered at the point P ε j , then we can get several explicit estimates on the solution w ε j ⁢ ( x ) for j = 1 , … , N ⁢ ( ε ) of the auxiliary cellular boundary value problem

One of the many remarkable properties of this cellular problem is that its (unique) weak solution, w ε j , is radially symmetric (recall that G 0 is a ball) and satisfies that ∂ ν p ⁡ w ε j is constant on ∂ ⁡ T ε j and on ∂ ⁡ G ε j . Due to the divergence theorem,

Furthermore, we can make explicitly several computations. Hence, we have an explicit way to compare the reaction term on S ε with an auxiliary term on balls with radius C ⁢ ε , and Lemma 2.8 becomes very useful.

Another key idea of our proof is to relate a general test function v ∈ W 0 1 , p ⁢ ( Ω ) , used to check the limit characterization (2.6), with some suitable correction v ε , which is a better fitted test function in the microscopic weak formulation (2.7). In fact, by density, it will be enough to do that with a smooth test function v ∈ 𝒞 c ∞ ⁢ ( Ω ) . We will construct such adaptation among test functions in the form v ε = v - h ⁢ W ε , where, for the moment, h ∈ W 1 , ∞ ⁢ ( Ω ) without any other property, and, which is crucial, W ε ∈ W 0 1 , ∞ ⁢ ( Ω ) defined as

with w ε j the solution of the auxiliary cellular boundary value problem (2.8). The following technical result will explain why the function H, arising in the limit problem (1.3), was taken in this concrete form (more precisely, so that (2.4) holds), different from the boundary kinetics σ.

The aforementioned corrector term in the form h ⁢ W ε , where h ∈ W 1 , ∞ ⁢ ( Ω ε , ∂ ⁡ Ω ) will be taken to satisfy the condition h ⁢ ( x ) = H ⁢ ( v ⁢ ( x ) ) for a.e. x ∈ Ω , with H given by (2.4). These conditions rise naturally so that the term I 2 , ε above cancels out with the reaction term.

3 Existence of u ε and characterization of the function H