1 Introduction and statement of results
We present new results concerning the asymptotic behavior, as , of the solution of a family of boundary value problems formulated in a cavity (or plant) represented by a bounded domain in which a linear diffusion equation is satisfied. The boundary is split into two regions. On one of the regions, homogeneous Dirichlet conditions are specified. On the other one, some small subsets are ε-periodically distributed and some unilateral boundary conditions are specified on them. We also assume that a possible “reaction” may take place on a net of small pieces of the boundary given by the periodic repetition of a rescaled particle .
There are several relevant problems in a wide spectrum of applications leading to such type of formulations, ranging from water and wastewater treatment to food and textile engineering as well as pharmaceutical and biotechnology applications (for a recent review see ). One of them concerns the reverse osmosis when we apply it, for instance, to desalination processes (see, e.g.,  and the references therein). Without intending to use here a “realistic model”, we shall present an over-simplified formulation that, nonetheless, preserves most of the mathematical difficulties concerning the passing to the limit as . Some examples of more complex formulations covering different aspects of the problems considered here can be found, for instance, in  and the many references therein.
We start by recalling that, roughly speaking, semipermeable membranes allow the passing of a certain type of molecules (the so-called “solvents”) but block another type of molecules (the “solutes”). The solvents flow from the region of smaller concentration of solutes to the region of higher concentration (the difference of the concentrations produces the phenomenon known as osmotic pressure). Nevertheless, by creating a very high pressure it is possible to produce an inverse flow, such as the one being used in desalination plants: it is the so-called “reverse osmosis”. Since in many cases the semipermeable membrane contains some chemical products (e.g., polyamides; see ), our formulation will contain also a nonlinear kinetic reaction term in the flux given by a continuous nondecreasing function . Let us call the solvent concentration corresponding to the membrane periodicity scale ε. Let us modulate the intensity of the reaction in terms of a factor , with k to be analyzed later. So, for a critical value of the solvent concentration ψ (associated to the osmotic pressure) the flux (including the reaction kinetic term) is an incoming flux with respect to the solvents plant Ω if the concentration of the solvent molecules on the semipermeable membrane is smaller than or equal to this critical value, but it remains isolated (with no boundary flow, excluding the reaction term, on the membrane, i.e. when the concentration is ). So, if ν is the exterior unit normal vector to the membrane surface, we have
for some parameter called the “finite permeability coefficient of the membrane” (usually, in practice, μ takes big values). We assume a simplified linear diffusion equation on the solvent concentration
and some boundary conditions on the rest of the boundary . For instance, we can distinguish some subregions where Dirichlet or Neumann types of boundary conditions hold, and so, if we introduce the partition and assume that, in fact, , then we can imagine that
Figure 1 presents a simplified case of the above-mentioned framework.
We are especially interested in the study of new behaviors arising in the reverse osmosis membranes having a periodicity ε of the order of nanometers (see, e.g.,  and its references). Mathematically, we shall give a sense to those extremely small scales by demanding that the diameter of these subsets included in is of order , where .
Sometimes it is interesting to consider semipermeable membranes with an “infinite permeability coefficient” (formally , but only for the case ), and thus ψ becomes an obstacle which is periodically repeated in . By following the approach presented in , this can be formulated as
Now, to carry out our mathematical treatment, it is quite convenient to work with the new unknown
and thus if we assume (again for simplicity) that and , we simplify the formulation to arrive at the following formulation which will be the object of study in this paper:
Notice that in the reaction kinetics we made emerge a re-scaling factor , where . The relation between the exponent k and the diameter of the chemical particles (which we shall assume to be given by , where and ) will be discussed later. This relation will depend on the dimension of the space . The case is rather special and will require a different treatment: we shall assume that and .
Homogenization results for boundary value problems with alternating types of boundary conditions, including Robin-type conditions, were widely considered in the literature. We refer, for instance, to the papers [34, 8, 5, 1] which already contain an extensive bibliography on the subject. Huge attention was drawn to similar homogenization problems but in domains perforated by the tiny sets on which some nonlinear Robin-type condition is specified on their boundaries. Some pioneering works in this direction are the papers by Kaizu [25, 26] (see also ). There he investigated all the possible relations between parameters except one: the case of the “critical” relation between parameters α and , i.e. . Later on, this critical case was considered in  for and for the sets given by balls. It seems that it was in the paper  where the effect of “nonlinearity change due to the homogenization process” was discovered for the first time. After that, by using some different methods of proof, the critical case was solved for in  (see also ). The consideration of the case and for arbitrary shaped domains was carried out in . More recently, many results concerning the asymptotic behavior of solutions of problems similar to (1.1) were published in the literature [35, 23, 19, 20, 9, 11, 10, 12]. Nevertheless, in all the above-mentioned works the particles (or perforations, according to the physical model used as motivation of the mathematical formulation) of subsets where assumed to be balls (having a critical radius). We also mention here the paper  that describes the asymptotic behavior of some related problem for the case of arbitrary shaped sets and for . One of the main goals of this paper is to extend some of the techniques of  to problem (1.1), where the periodically distributed reactions arise merely on some part of the global boundary for the critical scaled case. This is the case for which some phenomenological properties which arise at the nano-scale can be simulated and justified by means of homogenization processes.
In order to present the main results of this paper, and their application to the reverse osmosis framework, we need to introduce some auxiliary notations. We start by considering the case . We assume that Ω is a bounded domain in , , with a piecewise-smooth boundary that consists of two parts and , with the property that
We consider a model such that with being diffeomorphic to a ball. We define , . Let
Notice that , . Later it will be useful to observe that if we denote by the ball in of radius r centered at a point , and if we define the boundary points
and the set , then we have . In this geometrical setting (see Figure 2), the so-called “strong formulation” of the problem for which we want to study the asymptotic behavior of its solutions is the following:
where is a locally Hölder continuous nondecreasing function and, at most, super-linear at infinity, i.e. such that
for all , where , . In problem (1.3), is the unit outward normal vector Ω at and is the normal derivative of u at this part of the boundary.
Notice that examples of such functions cover
Furthermore, the behavior at infinity may be superlinear as, for example, in
We say that is a weak solution of (1.1) if
for all .
By we denote the closure in of the set of infinitely differentiable functions in , vanishing on the boundary .
where, here and below, the constant K is independent of ε. Hence, there exists a subsequence (denoted as the original sequence by ) such that, as , we have
By using the monotonicity of the function one can show (see some references in ) that satisfies the following “very weak formulation”:
where φ is an arbitrary function from .
The main goal of this paper is to consider this critical relation between parameters. This scale is characterized by the fact that the resulting homogenized problem will contain a so-called “strange term” expressing the fact that the character of some nonlinearity arising in the homogenized problem differs from the original nonlinearity appearing in the boundary condition of (1.3). Still focusing first on the case , one can show that the critical scale of the size of the holes is given by (see, e.g., the arguments used in [28, 10])
The appropriate scaling of the reaction term so that both the diffusive and nonlinear characters are preserved at the limit is, as usual, driven by . Therefore,
In the present paper, we construct a homogenized problem with a nonlinear Robin-type boundary condition that contains a new nonlinear term, and prove the corresponding theorem stating that the solution of the original problem converges, as , to the solution of the homogenized problem.
We point out that the main difficulties to get a homogenized problem associated to (1.3) come from the following different aspects:
The low differentiability assumed on the function σ (since it is non-Lipschitz continuous at and it has quadratic growth at infinity).
The unilateral formulation of the boundary conditions on .
The general shape assumed on the sets .
The critical scale of the sets .
Some of those difficulties where already in the previous short presentation paper by the authors , but only for , without (ii) and by assuming that σ is Lipschitz continuous. Our main goal is to extend our techniques to the above-mentioned more general framework.
To build the homogenized problem we still need to introduce some “capacity-type” auxiliary problems. Given , for we introduce the new auxiliary function , depending also on and σ, as the (unique) solution of the problem
Remember that was given in the structural assumption (1.2). The existence and uniqueness of is given in Lemma 2.2 below. Let us introduce also the auxiliary function , as the unique solution of the problem
We then define, for , the possibly nonlinear function
and the scalar
where in both definitions . Notice that can be extended by symmetry to as a harmonic function. Moreover, by the maximum principle, reaches its maximum in , and so by the strong maximum principle . Then we know that
Some properties of the function will be presented later. In particular, in Lemma 2.8 below we will show that is always a Lipschitz continuous function (even if σ is merely Hölder continuous) such that
The following theorem gives a description of the limiting function obtained in (1.7).
As mentioned before (see also ), the case requires to introduce some slight changes. The domain is given now in the following way: We consider Ω to be a bounded domain in , the boundary of which consists of two parts and , , . We set
For a small parameter and , we introduce the sets
where is a set of vectors and is a whole number. Set
It is easy to see that . Set . Note that for for all we have and ; see Figure 3.
The formulation of the problem starts by searching
being a solution to the following variational inequality:
where φ is an arbitrary function from . This time, for simplicity, we assume merely that the function is continuously differentiable, and there exist positive constants , such that the following condition is satisfied:
(more general terms where considered in , but without the Signorini-type constraints). Therefore, we have
Our homogenized result in this case is the following theorem.
Let , , , and let be a solution to problem (1.16). Then there exists a subsequence such that , strongly in and weakly in as , and the function is a weak solution to the following boundary value problem:
where verifies the functional equation
The special case of dimension is illustrative in order to get a complete identification of the function . Curiously enough, the characterization condition for and (see, e.g., the functional equations (1.10) and (1.17), respectively) is quite related (but not exactly the same) to the one obtained in , although the problem under consideration in that paper was not the same as problem (1.3). It was shown in  that if is the solution of a problem
then H is given by
for . When the Signorini condition is included, σ can be generalized to the maximal monotone graph
It was proven in  that the corresponding zero-order term is given by
This behavior is interesting because matches (1.18) formally. The maximal monotone graph
has, formally, the inverse
In this way, it is clear that
For the case we will make further comments in this direction in Section 6.1.
Coming back to the framework of the semipermeable membranes problems, what we can conclude is that the homogenization of a set of periodic semipermeable membranes with an “infinite permeability coefficient”, in the critical case, leads to a homogenized formulation which is equivalent to having a global semipermeable membrane, at , with a “finite permeability coefficient of this virtual membrane” , which is the best we can get, even if the original problem involves a finite permeability μ. Indeed, this comes from the properties of the function (which can also be computed for the case of a microscopic membrane with finite permeability), for instance, on the subpart of the boundary (with , i.e. where ). Moreover, by using that and the decomposition , we know now that , and thus, for the case of a microscopic finite permeability membrane, it is not difficult to show that we get a homogenized permeability membrane coefficient given by , which is larger than μ (see more details in Section 6.1 below).
The plan of the rest of the paper is the following: Part I (containing Sections 2–6) is devoted to the study of the case and contains also some comments and possible extensions (for instance, we give some link between the present homogenization results and the homogenization of some problems involving non-local fractional operators). Part II (containing Section 7) is devoted to the proof of the convergence result for .
2 Estimates on the auxiliary functions
2.1 On the auxiliary function
2.2 On the auxiliary function
Arguing as above, in terms similar to the paper , we get the following lemma.
Let σ be a maximal monotone graph. There exists a unique solution of problem (1.9). Furthermore, the following assertions hold:
If , then .
If , then .
Concerning the dependence with respect to the parameter u, we start by considering the case in which σ is a maximal monotone graph associated to a continuous function.
Let σ be a nondecreasing continuous function such that . Then
for all .
Let , be two solutions of problem (1.9) with parameters . Consider the function . This function is a solution of the following exterior problem:
First consider the case . If we choose as a test function in the integral identity for problem (2.1), we arrive at
The second integral in the obtained expression can be nonzero only if , i.e. . By combining this inequality with the condition , we get . This inequality and the monotonicity of the function σ imply that the second integral is non-negative. Hence, two integrals must be equal to zero, so -a.e. in and in . But we have as , and hence , i.e. .
One can construct a function such that if , and if . We take as a test function in the integral identity for problem (2.1) and obtain
Since σ is monotone .
For the first integral we have
where . We have that
For the second integral we derive the estimation
where . Thereby, as , we have , , , and so
Taking into account that , we derive from the last corollary that , i.e. in . Moreover, we have that -a.e. in .
The case is analogous to the one above, so we have in and -a.e. in . This concludes the proof. ∎
The use of the comparison principle leads to an additional conclusion.
Let . Then
The functions and can be extended (by symmetry) as harmonic functions to such that and on . The comparison principle proves the result. ∎
A more regular dependence with respect to u can also be proved under additional regularity of the function σ.
Lemma 2.5 (Differentiable dependence of solutions).
Suppose that and . Then the map is differentiable for every smooth bounded set K such that . Furthermore, if we define
for , and . In particular,
Considering the difference of two solutions, we obtain
for some in between and . From this we obtain
Taking , and using the fact that can be bounded and is continuous, we obtain
for h small. Thus admits a weak limit as in ; let it be denoted by . Thus, up to a subsequence, it admits a pointwise limit and strong limit in . It is clear that
As , we deduce the result . ∎
Notice that is the unique solution of
Assume . Then , and does not depend on μ. Therefore, . Furthermore,
2.3 On the regularity of the function H
i.e. in the notation of weak derivatives,
Notice that (and thus ) does not depend on σ, but only on .
First, let σ be smooth and . Again, let . Taking derivatives in (1.10), we have that
Since , we have that . Using as a test function in (2.3), we obtain that
using the facts that and .
If σ is a general maximal monotone graph, estimate (2.4) is maintained by approximation by a smooth sequence of the function . ∎
3 Convergence of the boundary integrals where
3.1 On the auxiliary function
We introduce a function as a solution of the boundary value problem
where is a parameter. We will compare this auxiliary function with the functions
The function is a weak solution of problem (3.1) if it satisfies the integral identity
for an arbitrary function .
From the uniqueness of problem (3.1) and the method of sub- and supersolutions, we have the following lemma.
The function satisfies the following estimations:
If , then .
If , then .
Since σ is monotone, from the previous result we obtain
We define the function
Notice that for all . The following lemma proposes an estimate of the introduced function and its gradient.
The following estimations for the function , which was defined in (3.3), are valid:
From the weak formulation of problem (3.1) for we know
We take as a test function in this expression and obtain
Then we can transform the obtained relation to the following expression:
By using the monotonicity of , we derive the following inequality:
Due to the monotonicity of σ and (3.2), we have that
Hence, the following estimate is valid:
Adding over all cells, we get
Friedrich’s inequality implies
Summing over all cells and using the obtained estimations, we derive
which concludes the proof. ∎
Hence, as we have
3.2 The comparison between and
As an immediate consequence of Lemma 3.1 we have the following lemma.
For all and a.e. , we have
The following lemma gives an estimate of the proximity of the functions and .
For the introduced functions and following estimations hold:
The function is a solution to the following boundary value problem:
Applying the comparison principle, we have a.e. in . We take as a test function in the corresponding weak solution integral expression for the above problem:
We transform the right-hand side expression of the inequality in the following way:
Let us estimate the obtained terms. We can extend by the symmetry for , which is harmonic in . By using some estimates on the derivatives of harmonic functions and the maximum principle, for , we get
since, on , from the maximum principle we have
This last estimate implies that for . Therefore, we get an estimate of the second term:
Then we estimate the first term:
Combining the obtained estimates and using the properties of the function σ, we get
Friedrich’s inequality implies that
This concludes the proof. ∎
We have that1
3.3 Convergence to the “strange term”
The following result plays a crucial role in the proof of Theorem 1.3.
Let H be the function defined by (1.10), and let φ be an arbitrary function in . Then for any test function we have
as , where and is the unit outward normal to .
Consider the cylinder
We define the auxiliary function as the unique solution to the following boundary value problem:
The constant is defined from the solvability condition for problem (3.7):
We take as a test function in the integral identity associated to problem (3.7) and obtain
Using the embedding theorems, we obtain the estimate
Taking into account that
and using some estimates proved in , we derive
From estimate (3.10) it follows that
Adding all the above integral identities for problem (3.7), we derive that for the following inequality holds:
Then we have
Let us estimate the terms on the right-hand side of the obtained inequality. For the first term we have
By using the continuity in -norm on the hyperplanes of the functions from , we estimate the second term:
Hence we have
4 Convergence of the boundary integrals where
4.1 The auxiliary function
We introduce the function as the unique solution of the following problem:
and then we define
It is easy to see that and
4.2 Estimate of the difference between and
Let and be as above. Then
The function satisfies the following problem:
We take as a test function in an integral identity for the above problem:
We transform the right-hand side expression of the identity in the following way:
For an arbitrary point we have
The last estimate implies that for . Therefore, we can estimate the second term in the following way:
Then we estimate the first term:
By combining acquired estimations, we derive
Friedrich’s inequality implies
This concludes the proof. ∎
4.3 Convergence to the “strange term”
Let be given by (1.11). Then for all functions we have
as , where ν is the unit outward normal to .
By analogy with the proof of Lemma 3.7, we define the function as a solution to the following boundary value problem:
The constant μ is defined from the solvability condition for problem (4.2):
By using the same technique as in the proof of the Lemma 3.6, we have
Summing up all integral identities for problem (4.2), we derive that for the arbitrary function from the following inequality is true:
Then we have
By using the continuity in -norm on the hyperplanes of the functions from , we estimate the second term:
Hence we have
5 Proof of Theorem 1.3
For different reasons it is convenient to introduce some new notation: instead of using the decomposition mentioned in Section 1 (see the statement of Theorem 1.3), we shall use the alternative decomposition (i.e. , but ).
Let be an arbitrary function from . We choose a point such that
where and . Define the function
From estimates (3.4) we conclude that
in as . We set
Hence we get
Considering the first integral of the right-hand side of the inequality above, we have
Then we proceed by transforming in the following way:
Lemma 3.5 implies that
By using Green’s formula, we have the following decomposition of the second integral:
From Lemma 3.7 we have
Now we consider the third term of identity (5.2). By using the fact that
we deduce that
By using the fact that and , a.e. in , we have that
Hence we conclude
Lemma 4.2 implies
We first notice that , and so
Thus we only need to study the term
On the other hand,
Since σ is Hölder continuous and , we have that, for a.e. ,
By using the same reasoning, estimate (3.4) implies that
Then by Lemma 3.5 we have that
which, by applying the embedding for , can be estimated as
By Lemma 3.6 and using , we obtain
Combining these estimates with (5.10), we derive, since , that
holds for any .
Finally, given , we consider the test function , in (5.13) and we pass to the limit as . By doing so, we get that satisfies the integral condition
for any . This concludes the proof. ∎
6 Possible extensions and comments
6.1 Extension to the case of σ as a maximal monotone graph
In , the authors showed that a similar problem, although restricted to the case of spherical particles distributed through the whole domain, could be treated in the general framework of maximal monotone graphs σ, which allow for a common roof between the Dirichlet, Neumann and Signorini boundary conditions and many more. We have restricted ourselves here to the case of Hölder continuous σ (see (1.4)), but this condition is only used at the very end, in estimates (5.11) and (5.12) to compute the last term of (5.10). The superlinearity condition is only used to obtain Lemma 3.6. These seem to be only technical difficulties, and can probably be avoided. Let us introduce what results can be expected if these problems could be circumvented.
Maximal monotone graph of .
A monotone graph of is a map (or operator) such that
The set is called domain of the multivalued operator σ. Some authors define maximal monotone graphs as maps and define .
A monotone graph σ is extended by another monotone graph if and for all . A monotone graph is called maximal if it admits no proper extension; for further references, see .
Definition of solution.
The auxiliary functions.
The equation of is well-defined when σ is a maximal monotone graph. As we have proved in this paper, the estimate is independent of σ, and so H is Lipschitz continuous for any maximal monotone graph σ.
Signorini boundary conditions.
This is the case under study in this paper. Nonetheless, let us study in the general setting. For this kind of boundary condition, we need to consider the following maximal monotone graph:
and . Let us compute in this setting:
For , we can see what happens explicitly. We have that . Thus . Since , we must have that on . But then . Hence when .
When , we have that . Thus only the values of σ affect .
We conclude that
Dirichlet boundary conditions.
In this case, we would have and
By the same reasoning, we have that
for all . In this case of Dirichlet boundary conditions, the critical case generates a linear term in the homogenized equation. This type of phenomena was already noticed by Cioranescu and Murat .
Cases of finite and infinite permeable coefficient.
The Signorini boundary condition imposed as a maximal monotone graph (6.1) is the extreme case of infinite permeability, aiming to represent the behavior of very large finite permeability given by a reaction term of the form
where μ is very large. As in Remark 2.7, it is easy to show that the corresponding kinetic will be of the form
Moreover, since we have proven that the Signorini boundary condition is an extremal case (i.e. ), we have that .
6.2 On the super-linearity condition
is only used in the proof of Lemma 3.6. However, it is our belief that this condition can be removed and that the result can still be obtained. We provide here a proof for and a ball .
We define the auxiliary function as the unique solution of
where is given by
Using prolate ellipsoidal coordinates, we can give an explicit expression of . These coordinates are given by
where , and . By defining , it can be proven through symmetry that . Furthermore, is the unique solution of the one-dimensional problem
Integrating this simple one-dimensional boundary value problem, we obtain
since we can recover from the change in variable
Due to mirror symmetry it is clear that . Thus we have
Using the explicit expression of , we can compute that
Summing over , we deduce that
It is easy to prove that
for any . We now apply that
6.3 Connections to fractional operators
Let us consider a domain , where is a smooth bounded domain. Then and . The related problem
is very relevant because it can be linked to the study of the fractional Laplacian . In fact, the boundary conditions on can be written compactly as
where χ is the indicator function. This boundary condition can be written as an equation of not involving the interior part of the domain, , by understanding the normal derivative of problem (6.5) as the fractional Laplace operator in (see [3, 15] and the references therein). Then (6.6) can be written as
is expected, where H and h will depend on σ and . This could provide some new results of critical size homogenization for the fractional Laplacian (in the spirit of the important work , where some random aspects on the net, and for a general fractional power of the Laplacian, are also considered).
7 Proof of Theorem 1.4
Here and below, the constants K and are independent of ε.
Hence there exists a subsequence (denote as the original sequence ) such that, as , we have
We introduce auxiliary functions and as weak solutions to the following problems:
Note that and are also a solutions of the following boundary value problems in the domains and (see Figure 4), respectively, where ,
where and . Define
where , and
We have and
Note that for an arbitrary function such that on , we have
We consider as a test function in the obtained equality and get
In addition, we have
Hence, (7.6) implies that
Given that if and using the embedding theorem, we get
From here we derive the estimate
From this estimation it follows that
This concludes the proof. ∎
We introduce the function as the weak solution to the following boundary value problem:
The function verifies the problem
Let and let be a bounded sequence. Then the following estimate holds:
Proof of Theorem 1.4.
First of all, equation (1.17) has a unique solution that is a Lipschitz continuous function in and satisfies
for all and a certain constant .
We rewrite inequality (7.10) in the following way:
From the fact that as weakly in , we have
Lemma 7.1 implies that
Consider the remaining integrals in (7.11). Set
where as .
It is easy to see that
where , .
using the results of , we derive
Let us find the limit of the expression
where as .
To conclude the proof we will estimate the limit of . We have
Lemma 7.2 implies that
Then vanishes due to equation (1.17). Using that , on and the fact that , we have
Hence we have that and
for any , where satisfies the functional equation (1.17). This concludes the proof. ∎
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About the article
Published Online: 2018-10-21
Published in Print: 2019-03-01
Funding Source: Ministerio de Ciencia e Innovación
Award identifier / Grant number: MTM 2014-57113-P
Award identifier / Grant number: MTM2017-85449-P
The research of the first and second authors was partially supported by the projects ref. MTM 2014-57113-P and MTM2017-85449-P of the DGISPI (Spain).
Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 193–227, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2018-0158.
© 2020 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0