Homogenization of p-Laplacian in thin domains: The unfolding approach

# Homogenization of p-Laplacian in thin domains: The unfolding approach

## Abstract

In this work we apply the unfolding operator method to analyze the asymptotic behavior of the solutions of the -Laplacian equation with Neumann boundary condition set in bounded thin domains of the type . We take a -periodic function in . The thin domain situation is established passing to the limit in the positive parameter with . Furthermore, we assign to model the oscillatory behavior of the upper boundary of . We consider three cases concerning to the order of oscillations: weak, resonant and high, respectively set by , and .

[intoc]

Keywords: -Laplacian, Neumann boundary condition, Thin domains, Oscillatory boundary, Homogenization.
2010 Mathematics Subject Classification. 35B25, 35B40, 35J92.

## 1 Introduction

Let be the following family of thin domains

 Rε={(x,y)∈R2:x∈(0,1) and 0

where is a strictly positive function, periodic of period , lower semicontinuous which satisfies

 0

with and .

In this work, we are interested in analyzing the asymptotic behavior of the family of solutions set by the following nonlinear elliptic problem

 {−Δpuε+|uε|p−2uε=fε in Rε|∇uε|p−2∇uεηε=0% on ∂Rε (3)

where is the unit outward normal vector to the boundary , with , and

 Δp⋅=div (|∇⋅|p−2∇⋅)

denotes the -Laplacian differential operator. We also assume .

It is known that the variational formulation of (3) is given by

 (4)

for all . Further, the existence and uniqueness of the solutions is guaranteed by Minty-Browder’s Theorem for each fixed . Hence, we are interested here in analyzing the behavior of the solutions as , that is, as the domain gets thinner and thinner although with a high oscillating boundary.

Notice that parameter introduced in (1) models the thin domain situation since . Moreover, we see that has tickness order , and then, it is expected that the sequence of solutions will converge to a function depending just on the first variable as .

On the other hand, the constant sets out the order of the boundary oscillation of . As we will see, the asymptotic behavior of solutions will depend tightly on this positive parameter. We will deal with three distinct cases. With that one called weak oscillation case, as , the resonant one, as , and finally, with the very high oscillation case, . See Figure 1 below where these three cases are illustrated.

We combine techniques as unfolding operator methods for thin domains developed in [6, 7], as well as, that ones presented in [10, 11] in order to analyze monotone operators in perforated domains. We also obtain corrector results for each case studied here.

For , we show that the limit equation is the well-posed problem

 {−∂xB(∂xu)+|u|p−2u=¯f in (0,1),B(∂xu(0))=B(∂xu(1))=0. (5)

Here, function is given by

 B(ξ) = ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩1|Y∗|∫Y∗|∇v|p−2∂y1vdy1dy2, if α=1,1|Y∗|∫L0|∂y1v|p−2∂y1vg(y1)dy1, if α<1,

where is the representative cell of the oscillating set

 Y∗={(y1,y2)∈R2:0

with Lebesgue measure , and function is given by an auxiliar problem which depends on .

If , for each , is taken as the solution of the auxiliar problem

 ∫Y∗|∇v|p−2∇v∇φdy1dy2=0,∀φ∈W1,p#(Y∗),(v−ξy1)∈W1,p#(Y∗),⟨(v−ξy1)⟩Y∗=0, (6)

where and is the average of on .

On the other side, if , is the unique solution of the problem

 ∫L0|∂y1v|p−2∂y1v∂y1ψg(y1)dy1=0,∀ψ∈W1,p#(0,L), (7)

with for a given . Notice that here, the positive function arises as a weight in .

It is worth noting that problems (6) and (7) are well posed due to Minty-Browder’s Theorem, and then, we have that function is also well defined.

For , we obtain that the limit equation is the one-dimensional -Laplacian problem with constant coefficients

 ⎧⎪⎨⎪⎩−g0(|u′|p−2u′)′+|Y∗|L|u|p−2u=^f,x∈(0,1),u′(0)=u′(1)=0. (8)

Recall that is the minimum of the -periodic function . Also, notice that the forcing terms of the limit problems, and , are given as limit of the unfolding operator acting on (see for instance (54) and (20)). The whole statement concerned with limit problems and convergences of the solutions can be seen in Theorem 3.1 for , Theorem 4.1 as , and Theorem 5.3 for .

There are several works in the literature dealing with the problem of studying the effect of thickness and rough boundaries on the feature of the solutions of partial differential equations. We just mention some one of them and references therein [1, 8, 9, 12, 13, 15, 17, 18]. Indeed, we note that thin structures with oscillating boundaries appear in many fields of science: fluid dynamics (lubrication), solid mechanics (thin rods, plates or shells) or even physiology (blood circulation). Therefore, analyzing the asymptotic behavior of models set on thin structures understanding how the geometry and the roughness affects the problem is a very relevant issues in applied science.

We also would like to point out that the particular case, as we take in (3), and then, we have the Laplace differential operator, have been originally discussed in previous works using different techniques and methods. The case where the domain presents weak roughness was treated in [2] using changes of variables and rescaling the thin domain as in the classical work [14]. The resonant case, , has been studied in [3, 4, 15] where techniques from homogenization theory have been used.

The case with fast oscillatory boundary () was obtained in [5] by decomposing the domain in two parts separating the oscillatory boundary. The authors also consider more general and complicated geometries for thin domains which are not given as the graph of certain smooth functions.

In [6, 7], the authors introduce the unfolding method to thin domains tackling these cases to the Laplacian with Neumann boundary condition in a unified way and, with the same milder assumptions on regularity of the domain assumed here.

Concerning with the -Laplacian, we have recently applied techniques from [3, 10, 11] to obtain the limit equation and corrector results in smooth thin domains to the resonant case in [16] improving previous works as [19].

The main goal of this work is to improve the previous mentioned works here concerning with the p-Laplacian. We combine techniques presented in [10, 11] in order to analyze monotone operators and the unfolding method developed in [6, 7] to deal with problem (3) on non-smooth oscillating thin domains for any and .

Notice that this is not a easy task since we study a nonlinear problem posed in non-smooth domains. As a matter of fact, we will be able to deal with a larger class of boundary problems and oscillating thin domains. In fact, since function is just lower semicontinuous, we can set comb-like thin domains or domains where the classical extension operators do not apply (see figure 2). Besides, it is worth observing that the unfolding method also allows us to obtain some new strong convergence results to the solutions.

The paper is organized as follows: In Section 2, we state some notations and basic results. In Section 3, we consider the resonant case . In Section 4, the weak oscillation case is studied. We point out that, in the framework of unfolding method, we easily get a new result of strong convergence, see Corollary 4.1.2. Section 5 concerns with the case of thin domains with very highly oscillatory boundaries. In such situations the period of the oscillations is so small that we have to proceed in a different way than the previous two cases to derive the homogenized limit problem.

## 2 Notations and Basic Facts

To study the convergence of the solutions of (4), we clarify some notation as well some results concerning to monotone operators and the unfolding method that we will be used here throughout the paper.

We will consider two-dimensional thin open sets with oscillatory behavior in its top boundary which are defined as follows

 Rε={(x,y)∈R2:0

The parameters and are positive and the function satisfies the following hypothesis

() is periodic of period and belongs to , that is, there exist positive constants and such that

 0

with and . Moreover, assume is lower semicontinuous, that is, , .

 Y∗={(y1,y2)∈R2:0

is the basic cell of the thin domain .

 ⟨φ⟩O:=1|O|∫Oφ(x)dx

is the average of on an open bounded set .

We also set functional spaces which are defined by periodic functions in the variable . Namely

 Lp#(Y∗)={φ∈Lp(Y∗):φ(y1,y2) is L-periodic in y1 },Lp#((0,1)×Y∗)={φ∈Lp((0,1)×Y∗):φ(x,y1,y2) is L-periodic in y1 },W1,p#(Y∗)={φ∈W1,p(Y∗):φ|∂leftY∗=φ|∂rightY∗}.

For each and any , there exists an integer denoted by such that

 x=εα[xεα]LL+εα{xεα}L, {xεα}L∈[0,L).
 Iε= Int (Nε⋃k=0[kLεα,(k+1)Lεα]),

where is largest integer such that . We still set

 Λε=(0,1)∖Iε=[εαL(Nε+1),1),Rε0={(x,y)∈R2:x∈Iε,0

The following inequalities concerned to monotone operators will be often used throughout this whole paper.

###### Proposition 2.1.

Let .

• If , then

 ⟨|x|p−2x−|y|p−2y,x−y⟩≥cp|x−y|p.
• If , then

 ⟨|x|p−2x−|y|p−2y,x−y⟩ ≥ cp|x−y|2(|x|+|y|)p−2 ≥ cp|x−y|2(1+|x|+|y|)p−2.
###### Corollary 2.1.1.

Let such that and with . Then, is the inverse of . Moreover,

• If (i.e, ), then

 ∣∣|u|p′−2u−|v|p′−2v∣∣≤c|u−v|p′−1.
• If (i.e, ), then

 ∣∣|u|p′−2u−|v|p′−2v∣∣ ≤ c|u−v|(|u|+|v|)p′−2 ≤ c|u−v|(1+|u|+|v|)p′−2.

Now, let us recall the definition to the unfolding operator and some one of its properties. For proofs and details, see [6, 7].

###### Definition 2.2.

Let be a Lebesgue-measurable function in . The unfolding operator acting on is defined as the following function in

 Tεφ(x,y1,y2)=⎧⎨⎩φ(εα[xεα]LL+εαy1,εy2), for (x,y1,y2)∈Iε×Y∗,0, for (x,y1,y2)∈Λε×Y∗. (10)
###### Proposition 2.3.

The unfolding operator satifies the following properties:

1. is linear;

2. , for all , Lebesgue mesurable in ;

3. , ,

 Tε(φ)(x,{xεα}L,yε)=φ(x,y),

for .

4. Let a Lebesgue mesurable function in extended periodically in the first variable. Then, is mesurable in and

 Tε(φε)(x,y1,y2)=φ(y1,y2),∀(x,y1,y2)∈Iε×Y∗.

Moreover, if , then ;

5. Let . Then,

 1L∫(0,1)×Y∗Tε(φ)(x,y1,y2)dxdy1dy2=1ε∫Rε0φ(x,y)dxdy =1ε∫Rεφ(x,y)dxdy−1ε∫Rε1φ(x,y)dxdy;
6. , , . Moreover

 ||Tε(φ)||Lp((0,1)×Y∗)=(Lε)1p||φ||Lp(Rε0)≤(Lε)1p||φ||Lp(Rε).

If ,

 ||Tε(φ)||L∞((0,1)×Y∗)=||φ||L∞(Rε0)≤||φ||L∞(Rε);
7. , ,

 ∂y1Tε(φ)=εαTε(∂xφ) e ∂y2Tε(φ)=εTε(∂y)φ a.e. in (0,1)×Y∗;
8. If , then , . Besides, for , we have

If ,

The above result sets several basic and somehow immediate properties of the unfolding operator.

Notice that, due to the order of the height of the thin set the factor appears in properties 5 and 6. Then, it makes sense to consider the following rescaled Lebesgue measure in the thin domains

 ρε(O)=1ε|O|, ∀O⊂Rε,

which is widely considered in works involving thin domains.

As a matter of fact, from now on, we use the following rescaled norms in the thin open sets

 |||φ|||Lp(Rε)=ε−1/p||φ||Lp(Rε),∀φ∈Lp(Rε),1≤p<∞, |||φ|||W1,p(Rε)=ε−1/p||φ||W1,p(Rε)∀φ∈W1,p(Rε),1≤p<∞.

For completeness we may denote .

From property 6, we have

 ||Tε(φ)||Lp((0,1)×Y∗)≤L1/p|||φ|||Lp(Rε),1≤p<∞.

Property 5 of Proposition 2.3 will be essential to pass to the limit when dealing with solutions of differential equations because it will allow us to transform any integral over the thin sets depending on the parameter into an integral over the fixed set . Notice that, in view of this property, we may say that the unfolding operator “almost preserves” the integral of the functions since the “integration defect” arises only from the unique cell which is not completely included in and it is controlled by the integral on .

Therefore, an important concept for the unfolding method is the following property called unfolding criterion for integrals (u.c.i.).

###### Definition 2.4.

A sequence satisfies the unfolding criterion for integrals (u.c.i) if

 1ε∫Rε1|φε|dxdy→0.
###### Proposition 2.5.

Let be a sequence in , with the norm uniformly bounded. Then, satisfies the (u.c.i).

Furthermore, let be a sequence in , also with uniformly bounded, , with . Then, the product sequence satisfies (u.c.i).

If we still take , then, the sequence satifies (u.c.i).

###### Proposition 2.6.

Let be a sequence in , with uniformly bounded and let be a sequence in set as follows

 ψε(x,y)=ψ(xεα,yε),

where . Then, satisfies (u.c.i).

Now, let us recall some convergence properties of the unfolding operator as goes to zero.

###### Theorem 2.7.

For a measurable function on , -periodic in its first variable and extended by periodicity to , define the sequence by

 fε(x,y)=f(xεα,yε)

a.e. for

 (x,y)∈{(x,y)∈R2:x∈R,0

Then

 Tεfε|(0,1)(x,y1,y2)={f(y1,y2), for (x,y1,y2)∈Iε×Y∗,0, for (x,y1,y2)∈Λε×Y∗. (11)

Moreover, if , then

 Tεfε→f (12)

strongly in .

###### Proposition 2.8.

Let and extend it periodically in -direction defining

 fε(x,y):=f(x,xεα,yε)∈Lp(Rε). (13)

Then,

 Tεfε→f strongly in Lp((0,1)×Y∗). (14)
###### Proof.

It follows from Theorem 2.7 and the density of the tensor product in . ∎

###### Remark 2.1.

Using Proposition 2.8, we also have that, if strongly in , then

 Tε(|fε|p−2fε)→|f|p−2f strongly in Lp′((0,1)×Y∗). (15)

In particular, we can take as in (13).

###### Proposition 2.9.

Let , . Then, considering as a function defined in , we have

 Tεφ→φ strongly in Lp((0,1)×Y∗).
###### Proposition 2.10.

Let be a sequence in , , such that

 φε→φ strongly in Lp(0,1).

Then,

 Tεφε→φ strongly in Lp((0,1)×Y∗).

Next, we recall a convergence result which does not depend on the value of the parameter . To do that, we first introduce a suitable decomposition to functions where the geometry of the thin domains plays a crucial role. We write where is defined as follows

 V(x):=1εg0∫εg00φ(x,s)ds a.e. x∈(0,1). (16)

We set .

###### Proposition 2.11.

Let , , with uniformly bounded and defined as in (16). Then, there exists a function such that, up to subsequences

 Vε⇀φ weakly in W1,p(0,1) and strongly in Lp(0,1), TεVε→φ strongly % in Lp((0,1)×Y∗), |||φε−Vε|||Lp(Rε)→0, |||φε−φ|||Lp(Rε)→0, Tεφε⇀φ% weakly in Lp((0,1);W1,p(Y∗)), Tεφε→φ % strongly in Lp((0,1)×Y∗).

Furthermore, there exists with such that, up to subsequences

 1εTε(φεr)⇀¯¯¯¯φ weakly in Lp((0,1)×Y∗), and Tε(∂yφε)⇀∂y2¯¯¯¯φ weakly in Lp((0,1)×Y∗)

where .

Now, let us recall a compactness result which allows us to identify the limit of the image of the gradient of a uniformly bounded sequence by the unfolding operator method as in (9).

###### Theorem 2.12.

Let , , with uniformly bounded.

Then, there exists and such that (up to a subsequence)

• if , we have

 Tεφε⇀φ% weakly in Lp((0,1);W1,p(Y∗)), Tε∂xφε⇀∂xφ+∂y1φ1 weakly in Lp((0,1)×Y∗), Tε∂yφε⇀∂y2φ1 weakly in Lp((0,1)×Y∗).
• If , we obtain and

 Tεφε⇀φ% , weakly in Lp((0,1);W1,p(Y∗)), Tε∂xφε⇀∂xφ+∂y1φ1, weakly in Lp((0,1)×Y∗).
###### Proof.

See [7, Theorem 3.1 and 4.1] respectively. ∎

Finally, we obtain uniform boundedness to the solutions of the -Laplacian problem (3) for any value of .

###### Proposition 2.13.

Consider the variational formulation of our problem:

 ∫Rε{|∇uε|p−2∇uε∇φ+|uε|p−2uεφ}dxdy=∫Rεfεφdxdy,φ∈W1,p(Rε), (17)

where satisfies

 |||fε|||Lp′(Rε)≤c

for some positive constant independent of . Then,

 |||uε|||Lp(Rε)≤c,|||uε|||W1,p(Rε)≤c,∣∣∣∣∣∣|∇uε|p−2∇uε∣∣∣∣∣∣Lp′(Rε)≤c.
###### Proof.

Take in (17). Then,

 ||uε||pW1,p(Rε)=∫Rε{|∇uε|p+|uε|p}dxdy≤||fε||Lp′(Rε)||uε||Lp(Rε). (18)

Hence,

 |||uε|||W1,p(Rε)≤c. (19)

Therefore, the sequence and , are respectively bounded in and under the norm . ∎

## 3 The resonant case: α=1.

In this section, we use the unfolding operator method introduced in Section 2 in order to pass to the limit in problem (3) assuming . Notice that this case is called resonant since the amplitude, period and oscillation orders in the thin domain are the same one at .

Thus, we consider here in this section, the following two-dimensional thin domain family

 Rε={(x,y)∈R2:0

with satisfying hypothesis (). We have the following result.

###### Theorem 3.1.

Let be the solution of problem (3) with satisfying

 |||fε|||Lp′(Rε)≤c

for independent of . Suppose also that

 Tεfε⇀^f weakly in Lp′((0,1)×Y∗). (20)

Then, there exists such that

 ⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩Tεuε⇀u weakly in Lp((0,1);W1,p(Y∗)),Tε(∂xuε)⇀u′+∂y1u1(x,y1,y2) weakly in Lp((0,1);W1,p(Y∗)),Tε(∂yuε)⇀∂y2u1(x,y1,y2) weakly in Lp((0,1);W1,p(Y∗)),Tε(|∇uε|p−2∇uε)⇀B(∂xu) weakly in Lp((0,1)×Y∗)2

and is the solution of the problem

 {−∂xB(∂