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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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Homogenization of a net of periodic critically scaled boundary obstacles related to reverse osmosis “nano-composite” membranes

Jesús Ildefonso DíazORCID iD: https://orcid.org/0000-0003-1730-9509 / David Gómez-CastroORCID iD: https://orcid.org/0000-0001-8360-3250
  • Instituto de Matemática Interdisciplinar, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain; and Departamento de Matemática Aplicada, E.T.S. de Ingeniería – ICAI, Universidad Pontificia de Comillas
  • orcid.org/0000-0001-8360-3250
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/ Alexander V. Podolskiy / Tatiana A. Shaposhnikova
Published Online: 2018-10-21 | DOI: https://doi.org/10.1515/anona-2018-0158

Abstract

One of the main goals of this paper is to extend some of the mathematical techniques of some previous papers by the authors showing that some very useful phenomenological properties which can be observed at the nano-scale can be simulated and justified mathematically by means of some homogenization processes when a certain critical scale is used in the corresponding framework. Here the motivating problem in consideration is formulated in the context of the reverse osmosis. We consider, on a part of the boundary of a domain Ωn, a set of very small periodically distributed semipermeable membranes having an ideal infinite permeability coefficient (which leads to Signorini-type boundary conditions) on a part Γ1 of the boundary. We also assume that a possible chemical reaction may take place on the membranes. We obtain the rigorous convergence of the problems to a homogenized problem in which there is a change in the constitutive nonlinearities. Changes of this type are the reason for the big success of the nanocomposite materials. Our proof is carried out for membranes not necessarily of radially symmetric shape. The definition of the associated critical scale depends on the dimension of the space (and it is quite peculiar for the special case of n=2). Roughly speaking, our result proves that the consideration of the critical case of the scale leads to a homogenized formulation which is equivalent to having a global semipermeable membrane, at the whole part of the boundary Γ1, with a “finite permeability coefficient of this virtual membrane”, which is the best we can get, even if the original problem involves a set of membranes of any arbitrary finite permeability coefficients.

Keywords: Homogenization; critical scale; reverse osmosis; Signorini boundary conditions; elliptic partial differential equations; strange term

MSC 2010: 35B27; 76M50; 35M86; 35J87

1 Introduction and statement of results

We present new results concerning the asymptotic behavior, as ε0, of the solution uε of a family of boundary value problems formulated in a cavity (or plant) represented by a bounded domain Ωn in which a linear diffusion equation is satisfied. The boundary Ω is split into two regions. On one of the regions, homogeneous Dirichlet conditions are specified. On the other one, some small subsets Gε 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 Gε of small pieces of the boundary given by the periodic repetition of a rescaled particle G0.

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 [29]). One of them concerns the reverse osmosis when we apply it, for instance, to desalination processes (see, e.g., [24] 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 ε0. Some examples of more complex formulations covering different aspects of the problems considered here can be found, for instance, in [17] 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 [18]), our formulation will contain also a nonlinear kinetic reaction term in the flux given by a continuous nondecreasing function σ(s). Let us call wε the solvent concentration corresponding to the membrane periodicity scale ε. Let us modulate the intensity of the reaction in terms of a factor ε-k, 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 w(x) on the semipermeable membrane GεΩ 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 w(x)<ψ). So, if ν is the exterior unit normal vector to the membrane surface, we have

νwε+ε-kσ(ψ-wε)=0on {xGεΩ:wε(x)>ψ},νwε+ε-kσ(ψ-wε)=-ε-kμ(ψ-wε)on {xGε:wε(x)ψ},

for some parameter μ>0 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

-Δw=F in Ω,

and some boundary conditions on the rest of the boundary Ω. For instance, we can distinguish some subregions where Dirichlet or Neumann types of boundary conditions hold, and so, if we introduce the partition Ω=Γ1Γ2 and assume that, in fact, GεΓ1, then we can imagine that

νwε(x)=h(x)on xΓ1Gε¯

and

wε(x)=g(x)on xΓ2.

Figure 1 presents a simplified case of the above-mentioned framework.

A simple illustration of a plant with a reverse osmosis membrane.
Figure 1

A simple illustration of a plant with a reverse osmosis membrane.

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., [4] and its references). Mathematically, we shall give a sense to those extremely small scales by demanding that the diameter of these subsets included in Gε is of order aε, where aεε.

Sometimes it is interesting to consider semipermeable membranes with an “infinite permeability coefficient” (formally μ=+, but only for the case wε(x)=ψ), and thus ψ becomes an obstacle which is periodically repeated in Gε. By following the approach presented in [16], this can be formulated as

νwε+ε-kσ(ψ-wε)=0on {xGε:wε(x)>ψ},νwε+ε-kσ(ψ-wε)0on {xGε:wε(x)=ψ},(wε-ψ)(νwε+ε-kσ(ψ-wε))=0on Gε.

Now, to carry out our mathematical treatment, it is quite convenient to work with the new unknown

uε(x):=ψ-wε(x),

and thus if we assume (again for simplicity) that h=g=0 and f:=-F, we simplify the formulation to arrive at the following formulation which will be the object of study in this paper:

{-Δuε=f(x),xΩ,uε0,νuε+ε-kσ(uε)0,uε(νuε+ε-kσ(uε))=0,xGε,νuε=0,xΓ1G¯ε,uε=0,xΓ2.(1.1)

Notice that in the reaction kinetics we made emerge a re-scaling factor β(ε):=ε-k, where k. The relation between the exponent k and the diameter of the chemical particles (which we shall assume to be given by aε=C0εα, where C0>0 and α>1) will be discussed later. This relation will depend on the dimension of the space n3. The case n=2 is rather special and will require a different treatment: we shall assume that aε=C0εe-α2/ε and β(ε)=eα2/ε.

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 [7]). There he investigated all the possible relations between parameters except one: the case of the “critical” relation between parameters α and β(ε), i.e. α=k=nn-2. Later on, this critical case was considered in [22] for n=3 and for the sets Gε given by balls. It seems that it was in the paper [22] 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 n3 in [35] (see also [21]). The consideration of the case n=2 and for arbitrary shaped domains Gε was carried out in [33]. 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 Gε where assumed to be balls (having a critical radius). We also mention here the paper [14] that describes the asymptotic behavior of some related problem for the case of arbitrary shaped sets Gε and for n3. One of the main goals of this paper is to extend some of the techniques of [14] to problem (1.1), where the periodically distributed reactions arise merely on some part of the global boundary Ω 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.

Case n3.

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 n3. We assume that Ω is a bounded domain in n{x1>0}, n3, with a piecewise-smooth boundary Ω that consists of two parts Γ1 and Γ2, with the property that

Γ1=Ω{xn:x1=0}.

We consider a model G0 such that G0¯{xn:x1=0,|x|<14} with G0¯ being diffeomorphic to a ball. We define δB={x:δ-1xB}, δ>0. Let

Gε~=j(aεG0+εj)=jGεj,

where

={0}×n-1

and

aε=C0εk,k=n-1n-2 and C0>0.(1.2)

A justification of the above choice of exponent k can be found, for instance, in [28] (see also [32]). We define the net of sets Gε as the union of sets GεjGε~ such that Gεj¯Γ1 and ρ(Γ1,Gεj¯)2ε, i.e.

Gε=jΥεGεj,

where

Υε={j:ρ(Γ1,Gεj¯)2ε}.

Domain Ω when n≥3{n\geq 3}.
Figure 2

Domain Ω when n3.

Notice that |Υε|dε1-n, d=const>0. Later it will be useful to observe that if we denote by Tr(x0) the ball in n of radius r centered at a point x0, and if we define the boundary points

Pεj=εj=(0,Pε,2j,,Pε,nj)for j,

and the set Tε/4j=Tε/4(Pεj) , then we have Gεj¯Tε/4j. 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:

{-Δuε=f(x),xΩ,uε0,νuε+ε-kσ(uε)0,uε(νuε+ε-kσ(uε))=0,xGε,νuε=0,xΓ1G¯ε,uε=0,xΓ2,(1.3)

where σ: is a locally Hölder continuous nondecreasing function and, at most, super-linear at infinity, i.e. such that

k1|s-t||σ(t)-σ(s)|K1|t-s|ρ1+K2|s-t|ρ2for some ρ1,ρ2(0,2](1.4)

for all t,s0, where k1,K1,K2>0, σ(0)=0. In problem (1.3), ν=(-1,0,,0) is the unit outward normal vector Ω at {x1=0} and νu=-ux1 is the normal derivative of u at this part of the boundary.

Example 1.1.

Notice that examples of such functions cover

σ(s)=s.

Furthermore, the behavior at infinity may be superlinear as, for example, in

σ(s)={s,0ss0,s0+(s-s0)2,s>s0.

The weak formulation of (1.3) is the following definition (see, e.g., [16]).

Definition 1.2.

We say that uε is a weak solution of (1.1) if

uεKε={gH1(Ω,Γ2):g0 a.e. on Gε}

and

Ωuε(φ-uε)dx+ε-kGεσ(uε)(φ-uε)𝑑xΩf(φ-uε)𝑑x(1.5)

for all φKε.

By H1(Ω,Γ2) we denote the closure in H1(Ω) of the set of infinitely differentiable functions in Ω¯, vanishing on the boundary Γ2.

It is well known (see, e.g., some references in [10]) that problem (1.5) has a unique weak solution uεKε. From (1.5) we immediately deduce that

uεL2(Ω)K,(1.6)

where, here and below, the constant K is independent of ε. Hence, there exists a subsequence (denoted as the original sequence by u~ε) such that, as ε0, we have

uεu0weakly in H1(Ω,Γ2),uεu0strongly in L2(Ω).(1.7)

By using the monotonicity of the function σ(u) one can show (see some references in [10]) that uε satisfies the following “very weak formulation”:

Ωφ(φ-uε)dx+ε-kGεσ(φ)(φ-uε)𝑑xΩf(φ-uε)𝑑x,(1.8)

where φ is an arbitrary function from Kε.

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 n3, 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])

α=n-1n-2.

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 εk|Gε|. Therefore,

k=n-1n-2.

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 ε0, 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:

  • (i)

    The low differentiability assumed on the function σ (since it is non-Lipschitz continuous at u=0 and it has quadratic growth at infinity).

  • (ii)

    The unilateral formulation of the boundary conditions on Gε.

  • (iii)

    The general shape assumed on the sets Gε.

  • (iv)

    The critical scale of the sets Gεj.

Some of those difficulties where already in the previous short presentation paper by the authors [13], but only for n=2, 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 u, for y(n)+=n{y1>0} we introduce the new auxiliary function w^(y;u), depending also on G0 and σ, as the (unique) solution of the problem

{-Δw^=0,y(n)+,νw^-C0σ(u-w^)=0,yG0,νw^=0,yG0,y1=0,w^0as |y|.(1.9)

Remember that C0>0 was given in the structural assumption (1.2). The existence and uniqueness of w^(y;u) is given in Lemma 2.2 below. Let us introduce also the auxiliary function κ^(y), y(n)+, as the unique solution of the problem

{Δκ^=0,y(n)+,κ^=1,yG0,νκ^=0,yG0,y1=0,κ^0,as |y|.

We then define, for u, the possibly nonlinear function

HG0(u):=G0νw^(u,y)𝑑y=C0G0σ(u-w^(u,y))𝑑y(1.10)

and the scalar

λG0:=G0νκ^(y)𝑑y,(1.11)

where in both definitions y=(0,y2,,yn). Notice that κ^(y) can be extended by symmetry to nG¯0 as a harmonic function. Moreover, by the maximum principle, κ^(y) reaches its maximum in G0, and so by the strong maximum principle νκ^=-x1κ^>0. Then we know that

λG0>0.

Some properties of the function HG0 will be presented later. In particular, in Lemma 2.8 below we will show that HG0 is always a Lipschitz continuous function (even if σ is merely Hölder continuous) such that

0HG0λG0.(1.12)

The following theorem gives a description of the limiting function u0 obtained in (1.7).

Theorem 1.3.

Let n3, let α=k=n-1n-2 and let uε be a weak solution of problem (1.3). Then the function u0 defined in (1.7) is a weak solution of the following problem:

{-Δu0=f,xΩ,νu0+C0n-2HG0(u0,+)-λG0C0n-2u0,-=0,xΓ1,u0=0,xΓ2,

where HG0 is defined by (1.10) and λG0 by (1.11). Here, as usual, u0,+:=max{u0,0} and u0,-=max{-u0,0}, so that u0=u0,+-u0,-.

Case n=2.

As mentioned before (see also [13]), the case n=2 requires to introduce some slight changes. The domain is given now in the following way: We consider Ω to be a bounded domain in 2{x2>0}, the boundary of which consists of two parts Ω=Γ1Γ2 and Γ1=Ω{x2=0}=[-l,l], l>0, Γ2=Ω{x2>0}. We set

Y1={(y1,0):-12<y1<12},l^0={(y1,0):-l0<y1<l0}Y1,l0(0,12).

For a small parameter ε>0 and 0<aε<ε, we introduce the sets

G~ε=j(aεl^0+εj)=jlεj,

where is a set of vectors j=(j1,0) and j1 is a whole number. Set

Υε={j:lεj¯{x=(x1,0):x1[-l+2ε,l-2ε]}}.

Consider Yεj=εY0+εj and

lε=jΥεlεj.

It is easy to see that lεj¯Yεj. Set γε=Γ1lε¯. Note that for for all j we have |lεj|=2aεl0 and |lε|daεε-1; see Figure 3.

Domain Ω in two dimensions.
Figure 3

Domain Ω in two dimensions.

The formulation of the problem starts by searching

uεKε={gH1(Ω,Γ2):g0 a.e. on lε},(1.13)

being a solution to the following variational inequality:

Ωuε(φ-uε)dx+eα2εGεσ(uε)(φ-uε)𝑑x1Ωf(φ-uε)𝑑x,(1.14)

where φ is an arbitrary function from Kε. This time, for simplicity, we assume merely that the function σ: is continuously differentiable, σ(0)=0 and there exist positive constants k1, k2 such that the following condition is satisfied:

k1uσ(u)k2for all u.(1.15)

(more general terms σ(u) where considered in [13], but without the Signorini-type constraints). Therefore, we have

k1u2uσ(u)k2u2for all u.

Note that (1.13), (1.14) is a weak formulation of the following strong formulation of the problem:

{-Δuε=f,xΩ,uε0,x2uε-β(ε)σ(uε)0,uε(x2uε-β(ε)σ(uε))=0,xlε,x2uε=0,xγε,uε=0,xΓ1.(1.16)

Our homogenized result in this case is the following theorem.

Theorem 1.4.

Let aε=C0εe-α2/ε, β(ε)=eα2/ε, α0, C0>0 and let uε be a solution to problem (1.16). Then there exists a subsequence such that uεu0, strongly in L2(Ω) and weakly in H1(Ω,Γ1) as ε0, and the function u0H1(Ω,Γ1) is a weak solution to the following boundary value problem:

{-Δu0=f,xΩ,x2u0=πα2(Hl0(u0,+)-u0,-),xΓ1,u0=0,xΓ2,

where Hl0(u) verifies the functional equation

πHl0(u)=2l0α2C0σ(u-Hl0(u)).(1.17)

The special case of dimension n=2 is illustrative in order to get a complete identification of the function HG0. Curiously enough, the characterization condition for HG0 and Hl0 (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 [10], although the problem under consideration in that paper was not the same as problem (1.3). It was shown in [10] that if H: is the solution of a problem

H(u)=Cσ(u-H(u)),

then H is given by

H(u)=(I+(Cσ)-1)-1(u)(1.18)

for u. When the Signorini condition is included, σ can be generalized to the maximal monotone graph

σ~(u)={σ(u),u>0,(-,0],u=0,,u<0.

It was proven in [10] that the corresponding zero-order term is given by

H~(u)={H(u),u>0,u,u0.

This behavior is interesting because H~ matches (1.18) formally. The maximal monotone graph

γ(u)={0,u>0,(-,0],u=0,,u<0,

has, formally, the inverse

γ-1(u)={,u>0,[0,+),u=0,0,u<0.

In particular,

Hγ(u):=(I+(Cγ)-1)-1(u)={,u>0,[0,+),u=0,u,u<0.

In this way, it is clear that

H~(u)={H(u),u>0,Hγ(u),u<0.

For the case n=3 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 Γ1, with a “finite permeability coefficient of this virtual membrane” μ, 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 HG0 (which can also be computed for the case of a microscopic membrane with finite permeability), for instance, on the subpart of the boundary {xΓ1:w0(x)ψ} (with w0(x):=ψ-u0(x), i.e. where u0(x)0). Moreover, by using that HG0(0)=0 and the decomposition u0=u0,+-u0,-, we know now that νw0=λG0C0n-2w0, 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 λG0C0n-2, 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 26) is devoted to the study of the case n3 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 n=2.

2 Estimates on the auxiliary functions

2.1 On the auxiliary function κ^

By using the method of sub- and supersolutions as in [14], the following result holds.

Lemma 2.1.

There exists a unique solution κ^X of problem (1.9), where

𝕏={vLloc2((n)+¯):vL2((n)+)n,|v|K|y|n-2}

such that

0κ^(y)1for a.e. y(n)+¯

and

κ^(y)K|y|n-2for a.e. y(n)+¯.

We define also the family of auxiliary functions

κ^εj=κ(x-Pεjaε).

2.2 On the auxiliary function w^

Arguing as above, in terms similar to the paper [14], we get the following lemma.

Lemma 2.2.

Let σ be a maximal monotone graph. There exists a unique solution w^(u,)X of problem (1.9). Furthermore, the following assertions hold:

  • If u0 , then 0w^(u,y)uκ^(y)u.

  • If u0 , then 0w^(u,y)uκ^(y)u.

Hence,

|w^(u,y)||u|κ^(y)for a.e. y(n)+¯.

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.

Lemma 2.3.

Let σ be a nondecreasing continuous function such that σ(0)=0. Then

|w^(u1,y)-w^(u2,y)||u1-u2|

for all u1,u2R.

Proof.

Let w^(u1,y), w^(u2,y) be two solutions of problem (1.9) with parameters u1,u2. Consider the function v=w(u1,y)-w(u2,y). This function is a solution of the following exterior problem:

{Δv=0,y(n)+,νyv=C0(σ(u1-w^(u1,y))-σ(u2-w^(u2,y))),yG0,v0as |y|.(2.1)

First consider the case u1>u2. If we choose v- as a test function in the integral identity for problem (2.1), we arrive at

(n)+|v-|2𝑑x+C0G0(σ(u1-w^(u1,y))-σ(u2-w^(u2,y)))v-𝑑s=0.

The second integral in the obtained expression can be nonzero only if v<0, i.e. w^(u1,y)-w^(u2,y)<0. By combining this inequality with the condition u1>u2, we get u1-w^(u1,y)>u2-w^(u2,y). 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 v-=0 n-1-a.e. in G0 and v-=c in (n)+. But we have v0 as |y|, and hence c=0, i.e. w^(u1,y)-w^(u2,y)0.

One can construct a function φ(r)C0() such that φ=0 if |r|>1, and φ=1 if |r|<0.5. We take (u1-u2-v)-φ(ρ(x,G0)/R) as a test function in the integral identity for problem (2.1) and obtain

-(n)+v(u1-u2-v)-φ(ρ(x,G0)/R)dx-(n)+φ(ρ(x,G0)/R)R(u1-u2-v)-vρdx+C0G0(σ(u1-w^(u1,y))-σ(u2-w^(u2,y)))(u1-u2-v)-𝑑s=I1,R+I2,R+I3=0.

Since σ is monotone I30.

For the first integral we have

I1,R-𝒟1,R|(u1-u2-v)-|2dx0,

where 𝒟1,R=((n)+){xn:ρ(x,G0)<R}. We have that

I1=-(n)+|(u1-u2-v)-|2dx=limRI1,R0.

For the second integral we derive the estimation

|I2,R|K1𝒟2,R|v|R|v|𝑑xK1RvL2(𝒟2,R)vL2((n)+)K2Rn-22vL2((n)+)0as R,

where 𝒟2,R=((n)+){xn:R2<ρ(x,G0)<R}. Thereby, as R, we have I1+I3=0, I10, I30, and so

I1=0,I3=0.

Taking into account that v0 as |y|, we derive from the last corollary that (u1-u2-v)-0, i.e. 0v<u1-u2 in (n)+. Moreover, we have that v<u1-u2 n-1-a.e. in G0.

The case u1<u2 is analogous to the one above, so we have u1-u2v0 in (n)+ and n-1-a.e. in G0. This concludes the proof. ∎

The use of the comparison principle leads to an additional conclusion.

Lemma 2.4.

Let u1>u2. Then

0w^(u1,y)-w^(u2,y)(u1-u2)κ^(y).(2.2)

Proof.

The functions φ1(y)=w^(u1,y)-w^(u2,y) and φ2(y)=(u1-u2)κ^(y) can be extended (by symmetry) as harmonic functions to nG0¯ such that lim|y|(φ2-φ1)=0 and φ2-φ10 on G0. 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 σC1 and σk1>0. Then the map uRw^(u,)Lloc2(K) is differentiable for every smooth bounded set K such that G0K(Rn)+¯. Furthermore, if we define

W^(u,y)=w^(u,y)u,

then

(n)+W^(u,y)φdy=C0G0σ(u-w^(u,y))(1-W^(u,y))φ(y)𝑑Sy(2.3)

for φCc((Rn)+¯), W^L2(Ω)n and 0W^(u,y)κ^(y). In particular,

0w^uκ^(y).

Proof.

Considering the difference of two solutions, we obtain

(n)+w^(u+h,y)-w^(u,y)hφdy=C0G0σ(u+h-w^(u+h,y))-σ(u-w^(u,y))hφ𝑑Sy=C0G0σ(ξh(y))(1-w^(u+h,y)-w^(u,y)h)φ(y)𝑑Sy

for some ξh in between u+h-w^(u+h,y) and u-w^(u,y). From this we obtain

(n)+w^(u+h,y)-w^(u,y)hφdy+C0G0σ(ξh(y))w^(u+h,y)-w^(u,y)hφ(y)𝑑Sy=C0G0σ(ξh(y))φ(y)𝑑Sy.

Taking φ=w^(u+h,y)-w^(u,y)h, and using the fact that w^(u,y) can be bounded and σ is continuous, we obtain

(w^(u+h,y)-w^(u,y)h)L2(Ω)n2+k1C0w^(u+h,y)-w^(u,y)hL2(G0)2C

for h small. Thus w^(u+h,y)-w^(u,y)h admits a weak limit as h0 in H1(K); let it be denoted by W^(u,y). Thus, up to a subsequence, it admits a pointwise limit and strong limit in L2(K). It is clear that

ξh(y)u-w^(u,y)pointwise as h0.

By passing to the limit for φ𝒞c((n)+¯) fixed, we characterize (2.3). From (2.2) we deduce that, for h>0,

0w^(u+h,y)-w^(u,y)hκ(y).

As h0, we deduce the result 0W^(u,y)κ(y). ∎

Remark 2.6.

Notice that W^ is the unique solution of

{-ΔW^=0,y(n)+,νW^+C0σ(u-w^)W^=C0σ(u-w^),yG0,νW^=0,yG0,y1=0,W^0,|y|+.

Remark 2.7.

Assume σ(u)=μu. Then σ(u-w^)μ, and W^ does not depend on μ. Therefore, w^(x,u)=uW^(x). Furthermore,

H(u)=λμu.

2.3 On the regularity of the function H

Lemma 2.8.

Let σ be a maximal monotone graph. The function HG0(u) defined by (1.10) is nondecreasing and Lipschitz continuous of constant λG0 given by (1.11), i.e. if u1>u2, then

0H(u1)-H(u2)λG0(u1-u2),(2.4)

i.e. in the notation of weak derivatives,

0H(u)λG0for a.e. u.

Remark 2.9.

Notice that κ^ (and thus λG0) does not depend on σ, but only on G0.

Proof.

First, let σ be smooth and σk1>0. Again, let W^=w^u. Taking derivatives in (1.10), we have that

H(u)=C0G0σ(u-w^)(1-W^)𝑑y.

Since W^1, we have that H0. Using κ^ as a test function in (2.3), we obtain that

H(u)=C0G0σ(u-w^)(1-W^)𝑑y=G0(σ(u-w^)(1-W^))κ^𝑑y=(n)+W^κ^dy=G0W^(νκ^)𝑑yG0(νκ^)𝑑y=λ,

using the facts that νκ^0 and 0W^1.

If σ is a general maximal monotone graph, estimate (2.4) is maintained by approximation by a smooth sequence of the function σk. ∎

3 Convergence of the boundary integrals where uε0

3.1 On the auxiliary function wεj

We introduce a function wεj(u,x) as a solution of the boundary value problem

{Δwεj=0,x(Tε/4j)+,νwεj=ε-kσ(u-wεj),xGεj,νwεj=0,x(Tε/4j)0Gεj¯,wεj=0,x(Tε/4j)+,(3.1)

where u is a parameter. We will compare this auxiliary function with the functions

w^εj(u,x)=w^(u,x-Pεjaε).

The function wεjH01(Tε/4j) is a weak solution of problem (3.1) if it satisfies the integral identity

(Tε/4j)+wεjφdx-ε-kGεjσ(u-wεj)φ𝑑x=0

for an arbitrary function φH01(Tε/4j).

From the uniqueness of problem (3.1) and the method of sub- and supersolutions, we have the following lemma.

Lemma 3.1.

The function wεj satisfies the following estimations:

  • (i)

    If u0 , then 0wεj(u,x)w^εj(u,x)u.

  • (ii)

    If u0 , then uw^εj(u,x)wεj(u,x)0.

Remark 3.2.

Since σ is monotone, from the previous result we obtain

|σ(u-wεj)||σ(u)|.(3.2)

We define the function

Wε={wεj(u,x),x(Tε/4j)+,jΥε0,x(n)+jΥε(Tε/4j)+¯.(3.3)

Notice that WεH1(Ω,Γ2) for all u. The following lemma proposes an estimate of the introduced function and its gradient.

Lemma 3.3.

The following estimations for the function Wε, which was defined in (3.3), are valid:

{WεL2(Ω)2K|u||σ(u)|,WεL2(Ω)2Kε2|u||σ(u)|.(3.4)

Proof.

From the weak formulation of problem (3.1) for jΥε we know

(Tε/4j)+wεjφdx-ε-kGεjσ(u-wεj)φ𝑑x=0.

We take φ=wεj as a test function in this expression and obtain

(Tε/4j)+|wεj|2𝑑x-ε-kGεjσ(u-wεj)wεj𝑑x=0.

Then we can transform the obtained relation to the following expression:

(Tε/4j)+|wεj|2𝑑x+ε-kGεjσ(u-wεj)(u-wεj)𝑑x=ε-kGεjσ(u-wεj)u𝑑x.

By using the monotonicity of σ(u), we derive the following inequality:

(Tε/4j)+|wεj|2𝑑x+ε-kGεjσ(u-wεj)(u-wεj)𝑑xε-kGεj|σ(u-wεj)||u|𝑑x.

Due to the monotonicity of σ and (3.2), we have that

wεjL2((Tε/4j)+)2ε-kGεj|σ(u-wεj)||u|𝑑xk2|u||σ(u)|ε-k|Gεj|.

Hence, the following estimate is valid:

wεjL2((Tε/4j)+)2K|u||σ(u)|εn-1.

Adding over all cells, we get

WεL2(Ω)2K|u||σ(u)|.

Friedrich’s inequality implies

wεjL2((Tε/4j)+)2Kε2wεjL2((Tε/4j)+)2.

Summing over all cells and using the obtained estimations, we derive

WεL2(Ω)2Kε2WεL2(Ω)2Kε2|u||σ(u)|,

which concludes the proof. ∎

Hence, as ε0 we have

Wε0weakly in H1(Ω),Wε0strongly in L2(Ω).

3.2 The comparison between wεj and w^εj

As an immediate consequence of Lemma 3.1 we have the following lemma.

Lemma 3.4.

For all uR and a.e. x(Tε/4j)+, we have

|wεj(u,x)||w^εj(u,x)|.

The following lemma gives an estimate of the proximity of the functions wεj and w^.

Lemma 3.5.

For the introduced functions wεj(u,x) and w^(y,u) following estimations hold:

(vεj(u,x))L2((Tε/4j)+)2K|u|2εn,vεj(u,x)L2((Tε/4j)+)2K|u|2εn+2,

where vεj(u,x)=wεj(u,x)-w^εj(u,x).

Proof.

The function vεj is a solution to the following boundary value problem:

{Δvεj=0,x(Tε/4j)+Gεj¯,νvεj=ε-k(σ(u-w^εj)-σ(u-wεj)),xGεj,νvεj=0,x(Tε/4j)0Gεj¯,vεj=-w^εj,x(Tε/4j)+.

Applying the comparison principle, we have |vεj(u,x)||w^εj(u,x)| a.e. in (Tε/4j)+. We take vεj as a test function in the corresponding weak solution integral expression for the above problem:

vεjL2((Tε/4j)+)2+ε-kGεj(σ(u-wεj)-σ(u-w^εj))vεj𝑑x=-(Tε/4j)+νvεjw^εjds.(3.5)

We transform the right-hand side expression of the inequality in the following way:

-(Tε/4j)+νvεjw^εjds=-(Tε/4j)+(Tε/8j)+¯vεjw^εjdx+(Tε/8j)+νvεjw^εjds.

Let us estimate the obtained terms. We can extend vεj by the symmetry vεj(x,u)=vjε(-x,u) for x(Tε/4j)-, which is harmonic in Tε/4jG0¯. By using some estimates on the derivatives of harmonic functions and the maximum principle, for x~Tε/8j, we get

|xivεj(x~)|1|Tε/16(x~)||Tε/16(x~)vεjxi𝑑x|=Kεn|Tε/16(x~)vεjνi𝑑x|K|u|

since, on Tε/16(x~), from the maximum principle we have

|vεj||w^εj|K|u||(x-Pεj)/aε|n-2=K|u|ε2-naεn-2=K|u|ε2-nεn-1=K|u|ε.

This last estimate implies that |vεj(x~)|K for x~Tε/8j. Therefore, we get an estimate of the second term:

|(Tε/8j)+νvεjw^εjds|K|u|maxTε/8j|w^εj||Tε/8j|K|u|2εn.

Then we estimate the first term:

|(Tε/4j)+(Tε/8j)+¯vεjw^dx|K|u|2εaεn-2=K|u|2εn.

Combining the obtained estimates and using the properties of the function σ, we get

vεjL2((Tε/4j)+)2K|u|2εn.

Friedrich’s inequality implies that

vεjL2((Tε/4j)+)2K|u|2εn+2.

This concludes the proof. ∎

Lemma 3.6.

We have that1

1|Gεj|Gεj|vεj(u,x)|2𝑑xε.

Proof.

From (3.5), using (1.4), we deduce that

k1aε-1Gεj|vεj|2𝑑xK|u|2εn.

Thus,

1|Gεj|Gεj|vεj(u,x)|2𝑑xKaε1-nGεj|vεj(u,x)|2𝑑xKaε2-naε-1Gεj|vεj(u,x)|2𝑑xK|u|2aε2-nεn=K|u|2ε.

This completes the proof. ∎

3.3 Convergence to the “strange term”

The following result plays a crucial role in the proof of Theorem 1.3.

Lemma 3.7.

Let H be the function defined by (1.10), and let φ be an arbitrary function in C(Ω). Then for any test function hH1(Ω,Γ2) we have

|jΥε(Tε/4j)+νxw^εj(φ(P~εj),s)h(s)𝑑s+C0n-2Γ1H(φ(x))h(x)𝑑x|0(3.6)

as ε0, where P~εjGεj and ν=(-1,0,,0) is the unit outward normal to (Tε/4j)+.

Proof.

Consider the cylinder

Qεj={xn:0<x1<ε,-ε2<xi-(P~εj)i<ε2,i=2,,n}.

We define the auxiliary function θεj as the unique solution to the following boundary value problem:

{Δθεj=0,Yεj=Qεj(Tε/4j)+¯,νθεj=-νw^εj(φ(P~εj),x),x(Tε/4j)+,θεjx1=μεj,xγεj=Qεj{x:x1=ε},θεjx1=0,on the rest of the boundary Qεj,θεjYεj=0.(3.7)

The constant μεj is defined from the solvability condition for problem (3.7):

μεj=-C0n-2H(φ(P~εj)).

We take θεj as a test function in the integral identity associated to problem (3.7) and obtain

Yεj|θεj|2𝑑x=-μεjγεjθεj𝑑x+Sε/4j,+νw^θεjds.

Using the embedding theorems, we obtain the estimate

γεj|θεj|𝑑xKε(n-1)/2θεjL2(γεj)Kεn/2θεjL2(Yεj).(3.8)

Taking into account that

max(Tε/4j)+|νw^εj(φ(P~εj),x)|Kaε-1|x-Pεjaε|n-1=Kaεn-2ε1-nK

and using some estimates proved in [31], we derive

(Tε/4j)+|(νw^εj(φ(P~εj),s))θεj|𝑑sK(Tε/4j)+|θεj|𝑑sKε(n-1)/2θεjL2((Tε/4j)+)Kε(n-1)/2{ε-1/2θεjL2(Yεj)+εθεjL2(Yεj)}Kεn/2θεjL2(Yεj).(3.9)

From the above estimates (3.8) and (3.9) we get

θεjL2(Yεj)2Kεn.(3.10)

From estimate (3.10) it follows that

jΥεθεjL2(Yεj)2Kε.

Adding all the above integral identities for problem (3.7), we derive that for hH1(Ω) the following inequality holds:

|jΥε(Tε/4j)+νw^εj(φ(P~εj),s)h𝑑s+C0n-2Γ1H(φ(x))h𝑑x||jΥεY^εjθεjhdx|+|jΥεμεjγεjh𝑑x+C0n-2Γ1H(φ(x))h𝑑x|.(3.11)

Let us estimate the terms on the right-hand side of inequality (3.11). By using estimate (3.10), we get the following inequality for the first term:

|jΥεYεjθεjhdx|KεhH1(Ω).(3.12)

Set

γ^εj=(Qεj)0,Γ1ε=jΥεγ^εj.

Then we have

|jΥεμεjγεjh𝑑x+C0n-2Γ1H(φ(x))h𝑑x||C0n-2jΥε(γεjH(φ(P~εj))h𝑑x-γ^εjH(φ(x))h𝑑x)|C0n-2|jΥεγ^εj(H(φ(P~εj))-H(φ(x)))h𝑑x|+C0n-2|jΥε(γεjH(φ(P~εj))h𝑑x-γ^εjH(φ(P~εj))h𝑑x)|.

Let us estimate the terms on the right-hand side of the obtained inequality. For the first term we have

|jΥεγ^εj(H(φ(P~εj))-H(φ(x)))h𝑑x|KhL2(Γ1ε)maxjΥεxγ^εj|H(φ(P~εj))-H(φ(x))|KhL2(Γ1ε)maxjΥεxγ^εj|φ(P~εj)-φ(x)|KεhH1(Ω).

By using the continuity in L2-norm on the hyperplanes of the functions from H1(Ω), we estimate the second term:

|jΥε(γεjH(φ(P~εj))h𝑑x-γ^εjH(φ(P~εj))h𝑑x)|KεhH1(Ω).

Hence we have

|jΥεμεjγεjh𝑑x+C0n-2Γ1H(φ(x))h𝑑x|KεhH1(Ω).(3.13)

Combining estimates (3.12) and (3.13), we conclude the proof. ∎

4 Convergence of the boundary integrals where uε0

4.1 The auxiliary function κεj

We introduce the function κεj as the unique solution of the following problem:

{Δκεj=0,x(Tε/4j)+Gεj¯,κεj=1,xGεj,νκεj=0,x(Tε/4j)0Gεj¯,κεj=0,xTε/4j,

and then we define

κε={κεj(x),x(Tε/4j)+,jΥε,0,xnjΥε(Tε/4j)+¯.

It is easy to see that κεH1(Ω) and

κε0weakly in H1(Ω) as ε0.(4.1)

4.2 Estimate of the difference between κεj and κ^εj

Lemma 4.1.

Let κεj and κ^ be as above. Then

jΥεκεj-κ^εjH1((Tε/4j)+)2Kε.

Proof.

The function vεj=κεj-κ^ satisfies the following problem:

{Δvεj=0,x(Tε/4j)+,vεj=0,xGεj,νvεj=0,x(Tεj)0Gεj¯,vεj=-κ^εj,x(Tε/4j)+.

We take vεj as a test function in an integral identity for the above problem:

vεjL2((Tε/4j)+)=-(Tε/4j)+νvεjκ^ds.

We transform the right-hand side expression of the identity in the following way:

-(Tε/4j)+νvεjκ^ds=-(Tε/4j)+Tε/8j¯vεjκ^dx+Tε/8jνvεjκ^ds.

For an arbitrary point x0Tε/8j we have

|xivεj(x0)|1|Tε/16(x0)||Tε/16(x0)vεjxi𝑑x|=Kεn|Tε/16(x0)vεjνi𝑑x|K.

The last estimate implies that |vεj(x~)|K for x~Tε/8j. Therefore, we can estimate the second term in the following way:

|Tε/8jνvεjκ^ds|KmaxTε/8j|κ^||Tε/8j|Kεn.

Then we estimate the first term:

|(Tε/4j)+Tε/8j¯vεjκ^dx|Kεaεn-2=Kεn.

By combining acquired estimations, we derive

vεjL2((Tε/4j)+)2Kεn

Friedrich’s inequality implies

vεjL2((Tε/4j)+)2Kεn+2.

This concludes the proof. ∎

4.3 Convergence to the “strange term”

Lemma 4.2.

Let λG0 be given by (1.11). Then for all functions hH1(Ω,Γ2) we have

|jΥε(Tε/4j)+Ωνxκ^(s)εjh(s)ds+C0n-2λG0Γ1hdx|0

as ε0, where ν is the unit outward normal to Tε/4jΩ.

Proof.

By analogy with the proof of Lemma 3.7, we define the function θεj as a solution to the following boundary value problem:

{Δθεj=0,Yεj,νθεj=-νκ^εj,x(Tε/4j)+,θεjx1=μ,xγεj,θεjx1=0,Qεj((Tε/4j)+γεj),θεjYεj=0.(4.2)

The constant μ is defined from the solvability condition for problem (4.2):

μ=-C0n-2λ.

By using the same technique as in the proof of the Lemma 3.6, we have

θεjL2(Yεj)2Kεn,jΥεθεjL2(Yεj)2Kε.

Summing up all integral identities for problem (4.2), we derive that for the arbitrary function from H1(Ω) the following inequality is true:

|jΥε(Tε/4j)+(νκ^εj)h𝑑s+C0n-2λΓ1h𝑑x||jΥεY^εjθεjhdx|+|jΥεμγεjh𝑑x+C0n-2λΓ1h𝑑x|.(4.3)

Let us estimate the terms on the right-hand side of inequality (4.3). By using estimate (3.10), we get following estimation of the first term:

|jΥεYεjθεjhdx|KεhH1(Ω).(4.4)

Then we have

|jΥεμγεjh𝑑x+C0n-2λΓ1h𝑑x||C0n-2λjΥε(γεjh𝑑x-γ^εjh𝑑x)|.

By using the continuity in L2-norm on the hyperplanes of the functions from H1(Ω), we estimate the second term:

|jΥε(γεjh𝑑x-γ^εjh𝑑x)|KεhH1(Ω).

Hence we have

|jΥεμεγεjh𝑑x+C0n-2Γ1h𝑑x|KεhH1(Ω).(4.5)

Combining estimations (4.4) and (4.5), we conclude the proof. ∎

5 Proof of Theorem 1.3

For different reasons it is convenient to introduce some new notation: instead of using the decomposition u0=u0,+-u0,- mentioned in Section 1 (see the statement of Theorem 1.3), we shall use the alternative decomposition u0=u0++u0- (i.e. u0+=u0,+, but u0-=-u0,-).

Proof.

Let φ(x) be an arbitrary function from C0(Ω). We choose a point P^εjGεj¯ such that

minxGεj¯φ+(x)=φ+(P^εj),

where φ+=max{0,φ(x)} and φ-(x)=φ(x)-φ+(x). Define the function

𝒲ε(φ+,x)={wεj(φ+(P^εj),x),x(Tε/4j)+,jΥε0,xnjΥε(Tε/4j)+¯.

From estimates (3.4) we conclude that

𝒲ε(φ+,x)0(5.1)

in H1(Ω,Γ2) as ε0. We set

v=φ+-𝒲ε(φ+,x)+(1-κε)φ-

as a test function in the integral inequality (1.8), where φ is an arbitrary function from C0(Ω). Notice that vKε. Indeed, according to Lemma 3.1 and using that κε1 in Gε, we have for all xGεj that

v=φ+-𝒲ε(φ+,x)+(1-κε)φ-φ+(P^εj)-wεj(φ+(P^εj),x)0.

Hence we get

Ω{(φ+-𝒲ε(φ+,x)+(1-κε)φ-)(φ+-𝒲ε(φ+,x)+(1-κε)φ--uε)}𝑑x   +ε-kjΥεGεjσ(φ+-wεj(φ+(P^εj),x))(φ+-wεj(φ+(P^εj),x)-uε)𝑑xΩf(φ+-𝒲ε(φ+,x)+(1-κε)φ--uε)𝑑x.

Considering the first integral of the right-hand side of the inequality above, we have

Ω(φ-𝒲ε(φ+,x)-κεφ-)(φ-𝒲ε(φ+,x)-κεφ--uε)dx=Ωφ(φ-𝒲ε(φ+(P^εj),x)-κεφ--uε)dx-Ω𝒲ε(φ+,x)(φ-𝒲ε(φ+,x)-κεφ--uε)dx   -Ω(κεφ-)(φ-𝒲ε(φ+,x)-κεφ--uε)dx=i=13Jεi.(5.2)

By using (4.1) and (5.1), we have

limε0Jε1=Ωφ(φ-u0)dx.(5.3)

Then we proceed by transforming Jε2 in the following way:

Jε2=-jΥε(Tε/4j)+wεj(φ+(P^εj),x)(φ-wεj(φ+(P^εj),x)-κεφ--uε)dx=-jΥε(Tε/4j)+{(wεj(φ+(P^εj),x)-w^εj(φ+(P^εj),x))(φ-wεj(φ+(P^εj),x)-κεφ--uε)}𝑑x-jΥε(Tε/4j)+w^εj(φ+(P^εj),x)(φ-wεj(φ+(P^εj),x)-κεφ--uε)dx=Iε1+Iε2.(5.4)

Lemma 3.5 implies that

Iε10as ε0.(5.5)

By using Green’s formula, we have the following decomposition of the second integral:

Iε2=-jΥε(Tε/4j)+{νw^εj(φ+(P^εj),x)(φ+-wεj(φ+(P^εj),x)-uε)}𝑑s-ε-kjΥεGεj{σ(φ+(P^εj)-w^εj(φ+(P^εj),x))(φ+-wεj(φ+(P^εj),x)-uε)}𝑑x=ε1+ε2.(5.6)

From Lemma 3.7 we have

limε0ε1=C0n-2ΩH(φ+(x))(φ+-u0)𝑑x.(5.7)

Combining (5.4)–(5.7) implies that

limε0Jε2=C0n-2ΩH(φ+(x))(φ+-u0)𝑑x+limε0ε2.(5.8)

Now we consider the third term of identity (5.2). By using the fact that

(κεjφ-)ρε=(κεjρε)φ-+κεjφ-ρε=κεj(φ-ρε)-(κεjφ-)ρε+κεjφ-ρε=κ^εj(φ-ρε)-(κ^εj-κεj)(φ-ρε)-(κεjφ-)ρε+κεjφ-ρε.

we deduce that

Jε3=-jΥε(Tε/4j)+κ^εj(φ-(φ-𝒲ε(φ+,x)-κεφ--uε))dx-jΥε(Tε/4j)+(κεj-κ^εj)(φ-(φ-𝒲ε(φ+,x)-κεφ--uε))dx+Ω(κεφ-)(φ-(φ-𝒲ε(φ+,x)-κεφ--uε))𝑑x-Ωκεφ-(φ-(φ-𝒲ε(φ+,x)-κεφ--uε))dx=𝒬ε1+𝒬ε2+𝒬ε3+𝒬ε4.

Lemma 4.1 implies that 𝒬ε20 as ε0. Then we use (1.6), (1.7), (4.1) and (5.1) to derive that 𝒬ε30 and 𝒬ε40 as ε0. Then we transform 𝒬ε1 using the Green’s formula:

𝒬ε1=-jΥε(Tε/4j)+(νκ^εj)φ-(φ-𝒲ε(φ+,x)-κεφ--uε)𝑑s-jΥεGεj(νκ^εj)φ-(φ-𝒲ε(φ+,x)-κεφ--uε)𝑑x.

By using the fact that νκ^0 and φ-𝒲ε(φ+,x)0, φ-uε0 a.e. in Gε, we have that

-jΥεGεj(νκ^εj)φ-(φ-𝒲ε(φ+,x)-κεφ--uε)𝑑x0.

Hence we conclude

𝒬ε1-jΥεTε/4jΩ(νκ^εj.)φ-(φ-uε)ds.

Lemma 4.2 implies

limε0Jε3=limε0𝒬ε1λC0n-2Ωφ-(φ-uε)𝑑x.(5.9)

From (5.2), (5.3), (5.8) and (5.9) we derive that

Ωφ(φ-u0)dx+C0n-2Γ1H(φ+)(φ-u0)𝑑x+λC0n-2Γ1φ-(φ-u0)𝑑x   +limε0{ε-kjΥεGεjσ(φ+-𝒲ε(φ+,x))(φ+-𝒲ε(φ+,x)-uε)dx   -ε-kjΥεGεjσ(φ+(P^εj)-w^εj(φ+(P^εj),x))(φ+-𝒲ε(φ+,x)-uε)dx}Ωf(φ-u0)𝑑x.(5.10)

We first notice that φ+-wεj(φ+(P^εj),x)φ+(P^εj)-w^εj(φ+(P^εj,x)), and so

{σ(φ+-wεj(φ+(P^εj),x))-σ(φ+(P^εj)-w^εj(φ+(P^εj),x))}uε0.

Thus we only need to study the term

ε-kGεj{σ(φ+-wεj(φ+(P^εj),x))-σ(φ+(P^εj)-w^εj(φ+(P^εj),x))}(φ+-wεj(φ+(P^εj),x))𝑑x.

On the other hand,

|φ+-𝒲ε(φ+,x)||φ+|K,wεj(φ+(P^εj),x)w^εj(φ+(P^εj),x)|φ+|K.

Since σ is Hölder continuous and φ𝒞1(Ω¯), we have that, for a.e. xGεj,

|σ(φ+-𝒲ε(φ+,x))-σ(φ+(P^εj)-𝒲ε(φ+,x))|Ki=12|φ+(x)-φ+(P^εj)|ρiKi=12|x-P^εj|ρi=Ki=12aερi.

By using the same reasoning, estimate (3.4) implies that

|ε-kjΥεGεj(σ(φ+-𝒲ε(φ+,x))-σ(φ+(P^εj)-𝒲ε(φ+,x)))(φ+-wεj(φ+(P^εj),x))𝑑x|K2ε-k|Gε|i=12aερiKi=12aερi.(5.11)

Then by Lemma 3.5 we have that

ε-k|jΥεGεj(σ(φ+(P^εj)-𝒲ε(φ+,x))-σ(φ+(P^εj)-w^εj))(φ+-𝒲ε(φ+,x))𝑑x|Kε-kjΥεi=12Gεj|v(φ+(P^εj),x)|ρi𝑑xKε-kjΥε|Gεj|i=121|Gεj|Gεj|v(φ+(P^εj),x)|ρi𝑑x,

which, by applying the L2(G0)Lρi(G0) embedding for 0<ρi2, can be estimated as

Kε-kjΥε|Gεj|i=12(1|Gεj|Gεj|vεj(φ+(P^εj),x)|2𝑑x)ρi2.

By Lemma 3.6 and using 0<ρ2, we obtain

Kε-kjΥε|Gεj|i=12ερi2Kε-k|Υε||Gεj|i=12ερi2Kε-k+1-n+k(n-1)i=12ερi2=Ki=12ερi20.(5.12)

Combining these estimates with (5.10), we derive, since ρ>0, that

Ωφ(φ-u0)dx+C0n-2Γ1H(φ+)(φ-u0)𝑑x+λC0n-2Γ1φ-(φ-u0)Ωf(φ-u0)𝑑x(5.13)

holds for any φH1(Ω,Γ2).

Finally, given ψH1(Ω,Γ2), we consider the test function φ=u0±δψ, δ>0 in (5.13) and we pass to the limit as δ0. By doing so, we get that u0 satisfies the integral condition

Ωu0ψdx+C0n-2Γ1H(u0+)ψ𝑑x+λC0n-2Γ1u0-ψ𝑑x=Ωfψ𝑑x

for any ψH1(Ω,Γ2). This concludes the proof. ∎

6 Possible extensions and comments

6.1 Extension to the case of σ as a maximal monotone graph

In [10], 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 2.

A monotone graph of 2 is a map (or operator) σ:D(σ)𝒫(){} such that

(ξ1-ξ2)(x1-x2)0for all xiD(σ) and all ξiσ(xi).

The set D(σ) is called domain of the multivalued operator σ. Some authors define maximal monotone graphs as maps σ:𝒫() and define D(σ)={xΩ:σ(x)}.

A monotone graph σ is extended by another monotone graph σ~ if D(σ)D(σ~) and σ(x)σ~(x) for all xD(σ). A monotone graph is called maximal if it admits no proper extension; for further references, see [2].

Definition of solution.

The solution uε is also well-defined, although the set Kε must now be written as

Kε={vH1(Ω,Γ2):for all xGε one has v(x)D(σ)}.

We will have that the integral condition (1.8) turns into

Ωφ(φ-uε)dx+ε-kGεξ(x)(φ-uε)𝑑xΩf(φ-uε)𝑑x

for all φKε and ξL2(Gε) such that ξ(x)σ(φ(x)) for a.e. xGε. The existence and uniqueness of this solution follows as in [10] and the references therein.

The auxiliary functions.

The equation of w^ is well-defined when σ is a maximal monotone graph. As we have proved in this paper, the estimate 0HλG0 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:

σ~(s)={σ(s),s>0,(-,0],s=0,,s<0,(6.1)

and D(σ)=[0,+). Let us compute H~G0 in this setting:

  • For u<0, we can see what happens explicitly. We have that uw^(u,)0. Thus u-w^0. Since D(σ)=[0,+), we must have that w^(u,y)=u on G0. But then w^(u,y)=uκ^(y). Hence H~G0(u)=λG0u when u<0.

  • When u>0, we have that 0u-w^(u,). Thus only the values of σ affect HG0(u).

We conclude that

H~G0(u)={HG0(u),u>0,λG0u,u0.

The computations with maximal monotone graphs yield precisely Theorem 1.3. Notice that the bound on H~ given by (1.12) is sharp.

Dirichlet boundary conditions.

In this case, we would have D(σ)={0} and

σ~(0)=(-,+).

By the same reasoning, we have that

H~G0(u)=λG0u

for all u. 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 [6].

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

σ~μ(u)={σ(u),u>0,μu,u0,

where μ is very large. As in Remark 2.7, it is easy to show that the corresponding kinetic will be of the form

H~μ(u)={H(u),u>0,λμu,u0.

Moreover, since we have proven that the Signorini boundary condition is an extremal case (i.e. H~(u)λG0), we have that λμλG0.

6.2 On the super-linearity condition

The condition

|σ(s)-σ(t)|k1|s-t|

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 n=3 and a ball G0.

We define the auxiliary function wε as the unique solution of

{-Δwεj=0,Tε0,wε=0,Gε0,wε=1,Tε0,

where Tε0 is given by

Tε0={x3:x121-(aεε-1)2+x22+x32<ε2}.

Using prolate ellipsoidal coordinates, we can give an explicit expression of wε. These coordinates are given by

x1=aεsinhψcosθ1,x2=aεcoshψsinθ1cosθ2,x3=aεcoshψsinθ1sinθ2,

where 0ψ<, 0θ1π and 0θ22π. By defining σ=sinhψ, it can be proven through symmetry that wε(x)=Vε(σ). Furthermore, Vε is the unique solution of the one-dimensional problem

{ddσ((1+σ2)dVdσ)=0,σ(0,(aε-1ε)2-1),V(0)=0,V((aε-1ε)2-1)=1.

Integrating this simple one-dimensional boundary value problem, we obtain

V(σ)=arctanσarctan(sinh(aε-1ε)2-1)

since we can recover from the change in variable

σ=sinhψ=|x|2-aε2+(aε2-|x|2)2+4x12aε22aε2.

Due to mirror symmetry it is clear that x1wε|{x1=0}=0. Thus we have

(Tε0)+wε(u2)dx=(Tε0)+x1wεu2dS-Gε0x1wεu2dS.

Using the explicit expression of wε, we can compute that

νwε|(Tε0)+aεε-2,νwε|Gε0-1aε2-|x|2.

Now let

Tεj=Pεj+Tε0,Tε=jΥεTεj,Wε(x)=wε(x-Pεj) for xTεj.

Summing over Υε, we deduce that

(Tε)+Wε(u2)ds=jΥε(Tεj)+νwεju2ds-Gεx1Wεu2ds.(6.2)

It is easy to prove that

(Tεj)+νwεjh2dsKjΥεhH1((Tεj)+)2(6.3)

for any hH1(Ω). We now apply that

uL2(Gε)2K(ε-1uL2(Tε+)+εuL2(Tε+)2).(6.4)

With (6.2)–(6.4) we can prove Lemma 3.6 for k1=0.

6.3 Connections to fractional operators

Let us consider a domain Ω=Ω×(0,+), where Ωn-1 is a smooth bounded domain. Then Γ1=Ω and Γ2=Ω×(0,+). The related problem

{-Δuε=0,(x,y)Ω×(0,+),νuε+ε-kσ(uε)=ε-kgε,xGε,νuε=0,xΩG¯ε,uε=0,(x,y)Ω×(0,+),uε0,|y|+,(6.5)

is very relevant because it can be linked to the study of the fractional Laplacian (-Δ)1/2. In fact, the boundary conditions on Ω can be written compactly as

νuε+ε-kχGεσ(uε)=ε-kχGεgε,xΩ,(6.6)

where χ is the indicator function. This boundary condition can be written as an equation of Ω not involving the interior part of the domain, Ω×(0,+), by understanding the normal derivative of problem (6.5) as the fractional Laplace operator (-Δ)1/2 in Ω (see [3, 15] and the references therein). Then (6.6) can be written as

(-Δ)12uε+ε-kχGεσ(uε)=ε-kχGεgε,xΩ.(6.7)

Thus the study of the limit of (6.5) will provide a homogenization result for (6.7). By applying similar techniques to this paper and previous results in the literature [12], the homogenized problem

(-Δ)12u0+CH(x,u0)=Ch,xΩ,

is expected, where H and h will depend on σ and gε. This could provide some new results of critical size homogenization for the fractional Laplacian (in the spirit of the important work [3], 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

It is well known that problem (1.13), (1.14) has a unique weak solution uεH1(Ω,Γ2). By using (1.14) and condition (1.15), which was set on the function σ, we get the following estimates:

uεL2(Ω)K,eα2εuεL2(Gε)2K1.

Here and below, the constants K and K1 are independent of ε.

Hence there exists a subsequence (denote as the original sequence uε) such that, as ε0, we have

uεu0weakly in H01(Ω),uεu0strongly in L2(Ω).

We introduce auxiliary functions wεj and qεj as weak solutions to the following problems:

{Δwεj=0,xTε/4jTaεj¯,wεj=1,xTaεj,wεj=0,xTε/4j,

and

{Δqεj=0,xTεjlεj¯,qεj=1,xlεj,qεj=0,xTε/4j.(7.1)

Domain Tε+,j∖Taεj¯{T^{{+,j}}_{\varepsilon}\setminus\overline{T^{{j}}_{a_{\varepsilon}}}} and lεj{l^{{j}}_{\varepsilon}}.
Figure 4

Domain Tε+,jTaεj¯ and lεj.

Note that wεj and qεj are also a solutions of the following boundary value problems in the domains (Tε/4j)+Taεj¯ and (Tε/4j)+ (see Figure 4), respectively, where (Trj)+=Trj{x2>0},

{Δwεj=0,x(Tε/4j)+Taεj¯,wεj=0,xTε/4j{x2>0},wεj=1,xTaεj{x2>0},x2wεj=0,x{x2=0}(Tε/4jTaεj¯),{Δqεj=0,x(Tε/4j)+,qεj=0,x(Tε/4j)+,qεj=1,xlεj,x2qεj=0,x(Tε/4j{x2=0})lεj¯,

where jΥε and lεj=aεl^0+εj. Define

Wε(x)={wεj(x),x(Tε/4j)+Taεj¯,jΥε,1,x(Taεj)+,0,x+2jΥεTε/4j¯+,(7.2)

where +2={x2>0}, and

Qε(x)={qεj(x),x(Tε/4j)+,jΥε,0,x+2jΥεTε/4j¯+.(7.3)

We have Wε,QεH01(Ω) and

Wε0weakly in H01(Ω) as ε0.

Lemma 7.1.

Let Wε be a function defined by formula (7.2), and let Qε be a function defined by formula (7.3). Then

Wε-QεH1(Ω)Kε.

Proof.

Note that for an arbitrary function ψH1(Tε/4j) such that ψ=0 on lεj, we have

(Tε/4j)+qεjψdx1dx2=0.

We consider ψ=wεj-qεj as a test function in the obtained equality and get

(Tε/4j)+qεj(wεj-qεj)dx1dx2=0.(7.4)

In addition, we have

(Tε/4j)+wεj(wεj-qεj)dx1dx2=Taεj{x2>0}νwεj(wεj-qεj)𝑑s.(7.5)

By subtracting (7.4) from (7.5), we derive

(Tε/4j)+|(wεj-qεj)|2𝑑x1𝑑x2=Taεj{x2>0}νwεj(wεj-qεj)𝑑s.(7.6)

Note that

wεj(x)=ln(4r/ε)ln(4aε/ε)andνwεj=-1aεln(4aε/ε).

Hence, (7.6) implies that

(wεj-qεj)L2((Tε/4j)+)21aε|ln(4aε/ε)|Taεj{x2>0}|wεj-qεj|𝑑s=1|ln(4aε/ε)|T1j{y2>0}|wεj-qεj|𝑑syJε.

Given that wεj-qεj=0 if yl^0 and using the embedding theorem, we get

JεK|ln(4aε/ε)|((T1j)+|y(wεj-qεj)|2𝑑y)1/2Kε(wεj-qεj)L2((Tε/4j)+).

From here we derive the estimate

(wεj-qεj)L2((Tε/4j)+)Kε.

From this estimation it follows that

Wε-QεH1(Ω)Kε.

This concludes the proof. ∎

We introduce the function m(y)H1((T10)+) as the weak solution to the following boundary value problem:

{Δym=0,yT10{y2>0}=(T10)+,y2m=1,yl^0,νm=2l0π,yT10{y2>0}=(T10)+,y2m=0,y(T10)+l^0(T10)+.

Consider

mεj(x)=εm(x-Pεjaε),x(Taεj)+.

The function mεj(x) verifies the problem

{Δxmεj=0,x(Taεj)+,νmεj=εaε-12l0π,x(Taεj)+,x2mεj=εaε-1,x{x2=0:|x-Pεj|aεl0}=lεj,x2mεj=0on the rest of the boundary.

Lemma 7.2.

Let n=2 and let hεH1(Ω,Γ2) be a bounded sequence. Then the following estimate holds:

|2l0επaεjΥε(Taεj)+hε𝑑s-εaεlεhε𝑑x1|Kε.

Proof.

We have that

|2l0εaε-1π(Taεj)+hε𝑑s-εaε-1lεjhε𝑑x1|=|(Taεj)+xmεjhεdx|xmεjL2((Taεj)+)hεL2((Taεj)+).(7.7)

Due to the fact that

xmεjL2((Taεj)+)2=ε2ym(y)L2((T10)+)2Kε2,

we have

jΥεxmεjL2((Taεj)+)2Kε.(7.8)

From (7.7), (7.8) we derive

|eα2/επ2l0lεhε𝑑x1-eα2/εjΥεTaεj{x2>0}hε𝑑s|δ-1jΥεxmεjL2((Taεj)+)2+δhεL2(Ω)2Kε

if δ=ε. ∎

Proof of Theorem 1.4.

First of all, equation (1.17) has a unique solution H(u) that is a Lipschitz continuous function in and satisfies

(H(u)-H(v))(u-v)k~1|u-v|,|H(u)||u|,(7.9)

for all u,v and a certain constant k~1>0.

We take v=ψ-Qε(H(ψ+)+ψ-) as a test function in (1.14), where ψC(Ω¯), ψ(x)=0 in the neighborhood of Γ2 and H(u) satisfies the functional equation (1.17). Note that from (7.1), (7.3) and (7.9) we have v0 on lε, so vKε. Hence we get

Ω(ψ-Qε(H(ψ+)+ψ-))(ψ-Qε(H(ψ+)+ψ-)-uε)dx+eα2εlεσ(ψ+-H(ψ+))(ψ+-H(ψ+)-uε)𝑑x1Ωf(ψ-Qε(H(ψ+)+ψ-)-uε)𝑑x.(7.10)

We rewrite inequality (7.10) in the following way:

Ω(ψ-Wε(H(ψ+)+ψ-))(ψ-Qε(H(ψ+)+ψ-)-uε)dx   -Ω((Qε-Wε)(H(ψ+)+ψ-))(ψ-Qε(H(ψ+)+ψ-)-uε)dx   +eα2εlεσ(ψ+-H(ψ+))(ψ+-H(ψ+)-uε)𝑑x1Ωf(ψ-Qε(H(ψ+)+ψ-)-uε)𝑑x.(7.11)

From the fact that Qε0 as ε0 weakly in H1(Ω,Γ2), we have

limε0Ωf(ψ-Qε(H(ψ+)+ψ-)-uε)𝑑x=Ωf(ψ-u0)𝑑x,(7.12)limε0ψ(ψ-Qε(H(ψ+)+ψ-)-uε)dx=ε0ψ(ψ-u0)dx.(7.13)

Lemma 7.1 implies that

limε0Ω((Qε-Wε)(H(ψ+)+ψ-))(ψ-Qε(H(ψ+)+ψ-)-uε)dx=0.(7.14)

Consider the remaining integrals in (7.11). Set

Iε-Ω(Wε(H(ψ+)+ψ-))(ψ-Qε(H(ψ+)+ψ-)-uε)dx=-ΩWε{(H(ψ+)+ψ-)(ψ-Qε(H(ψ+)+ψ-)-uε)}dx+αε,

where αε0 as ε0.

It is easy to see that

Iε=-ΩWε{(H(ψ+)+ψ-)(ψ-Qε(H(ψ+)+ψ-)-uε)}dx=-jΥε(Tε/4j)+Taεj¯wεj{(H(ψ+)+ψ-)(ψ-qεj(H(ψ+)+ψ-)-uε)}dx+αε~=-jΥεTε/4j{x2>0}νwεj(H(ψ+)+ψ-)(ψ-uε)𝑑s-jΥεTaεj{x2>0}νwεj(H(ψ+)+ψ-)(ψ+-H(ψ+)-uε)𝑑s+αε~,(7.15)

where αε~0, ε0.

Since

νwεj|Tε/4j=4εln(4aε/ε)=4-α2+εln(4C0),

using the results of [27], we derive

-limε0jΥεTε/4j{x2>0}νwεj(H(ψ+)+ψ-)(ψ-uε)𝑑s=limε04α2-εln(4C0)jΥεTε/4j{x2>0}(H(ψ+)+ψ-)(ψ-uε)𝑑s=πα2Γ2(H(ψ+)+ψ-)(ψ-u0)𝑑s.(7.16)

Let us find the limit of the expression

-jΥεTaεj{x2>0}νwεj(H(ψ+)+ψ-)(ψ+-H(ψ+)-uε)𝑑s+eα2εlεσ(ψ+-H(ψ+))(ψ+-H(ψ+)-uε)𝑑x1=jΥε(α2C0)-1eα2/ε1-εα-2ln(4C0)Taεj{x2>0}(H(ψ+)+ψ-)(ψ+-H(ψ+)-uε)𝑑s   +eα2/εlεσ(ψ+-H(ψ+))(ψ+-H(ψ+)-uε)𝑑x1

=eα2/εlεσ(ψ+-H(ψ+))(ψ+-H(ψ+)-uε)𝑑x1-eα2/εα2C0jΥεTaεj{x2>0}(H(ψ+)+ψ-)(ψ+-H(ψ+)-uε)𝑑s+α^εDε+α^ε,(7.17)

where α^ε0 as ε0.

To conclude the proof we will estimate the limit of Dε. We have

Dε={π2l0α2C0eα2/εlε(H(ψ+)+ψ-)(ψ+-H(ψ)-uε)dx1-eα2/εα2C0jΥεTaεj{x2>0}(H(ψ+)+ψ-)(ψ+-H(ψ+)-uε)ds}+eα2/εlε{σ(ψ+-H(ψ+))-π2l0α2C0H(ψ+)}(ψ+-H(ψ+)-uε)𝑑x1-π2l0α2C0eα2/εlεψ-(ψ+-H(ψ)-uε)𝑑x1=𝒥ε1+𝒥ε2+𝒥ε3.(7.18)

Lemma 7.2 implies that

|𝒥ε1|Kε.(7.19)

Then 𝒥ε2 vanishes due to equation (1.17). Using that uε0, ψ-0 on lε and the fact that ψ-(ψ+-H(ψ+))0, we have

𝒥ε30.(7.20)

Hence we have that limε0Dε0 and

limε0(eα2εlεσ(ψ+-H(ψ+))(ψ+-H(ψ+)-uε)𝑑x1+Iε)πα2Γ2(H(ψ+)+ψ-)(ψ-u0)𝑑s.(7.21)

Therefore, from (7.10)–(7.21) we conclude that u0H1(Ω,Γ2) satisfies the inequality

Ωψ(ψ-u0)dx+πα2Γ1(H(ψ+)+ψ-)(ψ-u0)𝑑x1Ωf(ψ-u0)𝑑x

for any ψH1(Ω,Γ2), where H(u) satisfies the functional equation (1.17). This concludes the proof. ∎

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Footnotes

  • 1

    The function vεj in this section depends on u. 

About the article

Received: 2018-07-17

Accepted: 2018-07-18

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.

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