The p-Laplacian in thin channels with locally periodic roughness and different scales

In this work we analyse the asymptotic behaviour of the solutions of the p-Laplacian equation with homogeneous Neumann boundary conditions posed in bounded thin domains as Rε=(x,y)∈R2:x∈(0,1) and 0 0. We take a smooth function G:(0,1)×R↦R , L-periodic in the second variable, which allows us to consider locally periodic oscillations at the upper boundary. The thin domain situation is established passing to the limit in the solutions as the positive parameter ɛ goes to zero and we determine the limit regime for three case: α < 1, α = 1 and α > 1.

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Introduction
Partial differential equations on thin domains (domains in which the size in some directions is much larger than the size in others) appear naturally in biological systems and industrial applications [13,14,24]. In most of the applications, the boundary of those domains is not perfectly flat and one can see irregularities. Then, the influence of such boundary distortions might not be neglected because its effect on the effective equation of the considered system, even far from the rough boundary, can be meaningful [1,9,11]. This motivates researchers to employ different homogenization techniques and try to determine the effective flow behaviour on a lower-dimensional domain which captures the influence of the geometry, roughness and thickness of the perturbed domain on the solutions of such singular boundary value problems. The obtained equations are then suitable for numerical simulations and provide rigorous justification of various natural phenomenon seen in such complex systems.
A simple manner to consider such irregularities is to study domains of type Q ε = (x, y) ∈ R 2 : x ∈ (0, 1) and 0 < y < εg x ε α for ε > 0, where g is a positive, bounded and periodic function satisfying some regularity hypothesis and ε > 0 is a small parameter which goes to zero. Thereby, in the limit ε → 0, the open set Q ε degenerates to the unit interval presenting oscillatory behaviour on the upper boundary (see for instance [1,3,5,[18][19][20][21][22] where similar approach are performed). The periodic rough boundary considered above is certainly a first step, but usually not enough, since most of the irregularities present in real applications are not periodic. In this work we are interested in the following family of rough thin domains R ε = (x, y) ∈ R 2 : x ∈ (0, 1) and 0 < y < εG ε (x) for ε > 0, (1.1) where for some parameter α > 0 with function G satisfying the conditions given by hypothesis (H) set in section 2. This kind of domain perturbation is called in the literature locally periodic thin domain and it is illustrated in figure 1 below.
As an example, one can consider G ε (x) = a(x) + b(x)g(x/ε α ) where a, b : (0, 1) → R are C 1 -piecewise positive functions and g : R → R is a L-periodic function of class C 1 also positive. This includes the case where a, b are positive constants recovering the perturbed regions discussed for instance in [3,5]. Notice that in the case in which α = 0, we also recover the open sets considered in [13] where evolution equations on thin domains without roughness were studied. We observe that the hypothesis (H) considered here is as general as possible for our framework.
In a unified way, we treat the three cases of roughness that can be modelled by the parameter α > 0. We analyse our boundary value problem for 0 < α < 1, α = 1 and α > 1, which represents weak, resonant and strong harshness on the upper boundary respectively. In each case, we have a different effective equation featuring the roughness induced effects on the perturbed model for small values of the parameter ε.
Several references treat issues related to the effect of thickness and rough boundaries on the feature of the solutions of partial differential equations. Indeed, 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, analysing the asymptotic behaviour of models set on thin structures understanding how the geometry and the roughness affect the problem is of considerable current interest in applied science. In these directions, let us mention [7,10,15,21] and references therein.
In this paper, we are interested in analysing the asymptotic behaviour of the solutions of a p-Laplacian equation given by where η ε is the unit outward normal vector to the boundary ∂R ε , 1 < p < ∞ with p −1 + p −1 = 1, and Δ p · = div |∇ · | p−2 ∇· , denotes the p-Laplacian differential operator. We also assume f ε ∈ L p (R ε ) is uniformly bounded. Such quasilinear equations play an important role in applications, given the fact that many models cannot be described by linear equations. In this sense, considering the p-Laplacian equation becomes natural. Moreover, the p-Laplacian is strongly related to non-Newtonian fluids, which arise in many applications related to polymer processing, hydrology, food processing, turbulent filtration, glaciology (see e.g. [6,16,17,25]). Here, differently from many works [11,12], we deal also with the case 1 < p < 2, which is the most relevant range of p in applications (e.g. [6]) and, of course, the case p 2.
We improve the results from [3] (where the Laplacian operator in locally periodic thin domains were considered) dealing with the p-Laplacian equation for any p ∈ (1, ∞). Moreover, we are improving our previous results from [2] where the purely periodic case in bidimensional thin regions were studied. It is worth noticing that the techniques developed in [2,3] cannot be directly applied in this case. On the one hand, the results concerning the unfolding operator obtained in [4] do not guarantee strong convergence in L p for the unfolding operator applied on solutions of quasilinear operators. On the other hand, the analysis performed in [3] just works on L 2 -spaces. Our goal here is to overcome this situation. We discretize the oscillating region passing to the limit using uniform estimates on two parameters: one associated to the roughness, and other given by the variable profile of the thin domain. In this way, a continuous dependence property for the solutions with respect to the function G in L p -norms is crucial and it is obtained in theorem 4.1. We point out that these techniques also work for the dimension reduction from three-dimensional thin sets to two-dimensional ones. The main change is in the limit problem. In 3D, we somehow lose the explicit p-Laplacian form, as in the unidimensional limit, but, clearly, the monotonicity of this limit operator is preserved (it will be done in a forthcoming work).
Notice that our work also goes a step further from [23] where the p-Laplacian operator is studied in standard thin domains. Let us emphasize that the standard thin domains were previously introduced and rigorously studied in the paper [13] of Hale and Raugel where the continuity of the family of attractors set by a semilinear parabolic equation in thin domains was considered.
According to [1] and references therein, it is expected that the sequence u ε will converge to a function of just one variable x ∈ (0, 1) satisfying a one-dimensional equation of the same type. Combining boundary perturbation techniques [3][4][5] and monotone operator analysis [17], we identify the effective limit model of (2.1) at ε = 0.
The paper is organized as follows. In section 2 we state the main result of the paper. In section 3, we introduce some notations and state some basic results which will be needed in the sequel. In section 4, we prove the continuous dependence of the solutions in L p -spaces with respect to the function G uniformly in the parameter ε > 0 improving [3, theorem 4.1] from L 2 to L p -spaces. In section 5, we perform the asymptotic analysis of (1.2) in piecewise periodic thin domains (that is, in thin domains set by functions G which are piecewise constants in the first variable x, and L-periodic in the second one). See figure 2 below which illustrates piecewise periodic open sets.
Next, we provide in section 6 the proof of the main result of the paper (namely theorem 2.1) as a consequence of the analysis performed in the previous sections. Finally, we discuss in section 7, the convergence of the resolvent and semigroup associated to the equation (1.2) under the additional assumption p 2. As we will see, it is obtained combining the classical result [8, theorem 4.2] and our main result theorem 2.1. Furthermore, we include an appendix where the dependence of the auxiliary solution v on admissible functions G is analysed.

Hypothesis on function G and the main result
First, recall that the variational formulation of (1.2) is given by Moreover, existence and uniqueness of the solutions are guaranteed by Minty-Browder's theorem setting a family of solutions u ε .
Next, we state the main hypothesis on function G setting the main conditions on our rough thin domain R ε introduced in (1.1).
Let G : (0, 1) × R → R be a function satisfying that there exist a finite number of points is C 1 and such that G, ∂ x G and ∂ y G are uniformly bounded in (ξ i−1 , ξ i ) × R getting limits when we approach ξ i−1 and ξ i . Further, we assume there exist two constants G 0 and G 1 such that and a real number L > 0 such that G(x, y + L) = G(x, y) for all (x, y) ∈ (0, 1) × R. 3 As we will see, the homogenized limit equation is a one-dimensional p-Laplacian equation with variable coefficients q(x) and r(x). It assumes the following form where the homogenized coefficients are given by and We emphasize here the dependence of the function q(x) with respect to the parameter α > 0 and variable x ∈ (0, 1) which generalizes our previous work [2]. The functionf is the weak limit off ε in L p (0, 1) withf ε defined by the family of known forcing terms f ε ∈ L p (R ε ) in the following waŷ |Y * (x)| denotes the Lebesgue measure of the representative cell which also depends on variable x ∈ (0, 1). The function v used to set the homogenized coefficient q(x) in (2.2) is the unique solution of the problem is the space of periodic functions on the horizontal variable y 1 , and ϕ O denotes the average of any function ϕ ∈ L 1 loc (R M ) on measurable sets O ⊂ R M . It is worth noticing that problem (2.3) is well posed for each x ∈ (0, 1), due to Minty-Browder's theorem, and then, the coefficient q(x) is also well defined. Further, q(x) is a positive function setting a well posed homogenized equation. Indeed, since v is the solution of The main result of the paper is the following: satisfiesf ε f weakly in L p (0, 1). Let u ∈ W 1,p (0, 1) be the unique solution of the homogenized equation where the homogenized coefficients q(x) and r(x) depend on the parameter α > 0 and are given by the expression (2.2). Then, As mentioned before, we are improving the results from [3] where the Laplacian operator in locally periodic thin domains were considered. We recover them taking p = 2 in theorem 2.1. Moreover, we also have improved our previous results from [2] where the purely periodic case in bidimensional thin regions were studied to the p-Laplacian operator where constant homogenized coefficients are obtained. Here, since we are considering locally periodic thin domains, variable homogenized coefficients can be produced. The main step in the proof is to pass to the limit in the solutions with the representative cell depending on variable x ∈ (0, 1) assuming different orders of roughness (different values for the parameter α > 0). To do that, we discretize the oscillating thin region passing to the limit using uniform estimates on two parameters: one associated to the roughness, and other given by the variable profile of the thin domain. In this way, a continuous dependence property for the solutions with respect to the function G in L p -norms is crucial and it is shown in theorem 4.1 below.

Basic facts and the unfolding operator
In this section, we introduce some basic facts, definitions and results concerning to the unfolding method making some straightforward adaptations to our propose. First, let us just recall some basic properties to the p-Laplacian which can be found for instance in [17].

Unfolding operator
Here, we present the unfolding operators for thin domains in the purely and locally periodic settings. We rewrite it to our context in order to simplify our proofs. They were first introduced in [4,5] where details and proofs can be found.
3.1.1. The purely periodic unfolding. Let G i : R → R be a L-periodic function, lower semicontinuous satisfying 0 < g 0,i G i (x) g 1,i with g 0,i = min x∈R G i (x) and g 1,i = sup x∈R G i (x) for any i = 1, . . . , N. Now consider the thin region The basic cell associated to R ε i is By we denote the average of ϕ ∈ L 1 loc (R 2 ) for any open measurable set O ⊂ R 2 . We also set functional spaces which are defined by periodic functions in the variable y 1 ∈ (0, L). Namely For each ε > 0 and any x ∈ (ξ i−1 , ξ i ), there exists an integer denoted by x ε L such that We still set Now we can introduce the unfolding operator. In the sequel, we point out its main properties.
Proposition 3.5. The unfolding operator satisfies the following properties: Then, The above result sets several basic and somehow immediate properties of the unfolding operator. Property four will be essential to pass to the limit when dealing with solutions of differential equations since it allow us to transform any integral over the thin sets depending on the parameter ε and function G i into an integral over the fixed set (

Locally periodic unfolding.
Next we set the locally periodic unfolding operator discussing some properties that will be needed in the sequel.
Definition 3.6. We define the locally periodic unfolding operator T lp ε acting on a measurable function ϕ, as the function T lp where · denotes the extension by zero to the whole space.
As in classical periodic homogenization, we have the unfolding operator reflecting two scales. The macroscopic one, denoted by x which gives the position in the interval (0, 1), and the microscopic scale given by (y 1 , y 2 ) which sets the position in the cell (0, L) × (0, G 1 ). However, due to the locally periodic oscillations of the domain R ε , the definition given here differs from the usual ones. In this case, we do not have a fixed cell that describes the domain R ε which makes the extension by zero needed.
Remark 3.1. We point out that the convergence above cannot be improved because of the definition of locally periodic unfolding operator.

A domain dependence result
In this section we analyse how the solutions of (1.2) depends on the function G ε . Let us take satisfying hypothesis (H) and considering the associated thin domains R ε andR ε by Now, let u ε andû ε be the solutions of (1.2) for the domains R ε andR ε respectively with f ε ∈ L p (R 2 ). We have the following result.
Remark 4.1. The important part of this result is that the function ρ(δ) does not depend on ε. As we will see, it only depends on the positive constants G 0 and G 1 .
In order to prove theorem 4.1, we use the fact that u ε andû ε are minimizers of the functionals that is, We will need to evaluate the minimizers plugging them into different functionals. For this, we set the following operators introduced in [3]: and U ⊂ R 2 is an arbitrary open set. We also consider the following norm in W 1,p (U) We can easily see that , (4.6) and as η 0.
Also, we need the following result about the behaviour of the solutions near the oscillating boundary.
Now, let us first assume p 2. We use the notations of corollary 3.1.1 to simplify proofs. By proposition 3.2, (4.2) and (2.1) for ϕ = P 1+η u ε − u ε , we get Putting together (4.7) and (4.8), we obtain Consequently Now, let us analyse the integral: To do this, notice that putting the power p, multiplying by 1/ε, integrating between 0 and εG ε (x) and using that (y/(1 + η), y) ⊂ (εG ε (x)), we get Thus, we have (4.10) Hence, due proposition 3.3, (4.9) and (4.10), one gets On the other hand, we have Hence, due to (4.8), we get and then, Thus, due proposition 3.3 and (4.10), we get for p > 2 that Notice that to the case p > 2, we have mainly estimated the term |x − y| p . Now, for the case 1 < p < 2, we have to estimate (1 + |x| + |y|) p−2 |x − y| 2 in view of propositions 3.1 and 3.2. Indeed, we can argue as in (4.11) and (4.12), to get, for 1 < p < 2 that and Finally, putting together the last inequality and (4.13), we also obtain cη + cη 1/p + cη + cη 1/p p/2 , for 1 < p < 2 finishing the proof. Now, we are in condition to show theorem 4.1.

The piecewise periodic case
Now, we analyse the limit of {u ε } ε>0 assuming the upper boundary of R ε is piecewise periodic. More precisely, we assume G satisfies (H) being independent on the first variable in each interval (ξ i−1 , ξ i ).
We suppose that G satisfies with G i (y + L) = G i (y) for all y ∈ R. Moreover, we assume the function G i (·) is C 1 for all i = 1, . . . , N and there exist 0 Notice that the domain R ε can now be rewritten as See figure 2 which illustrates this piecewise periodic thin domain. Before proving the main result of this section, let us recall an important result proved, for instance, in [21]. It is concerned to the purely periodic thin domain situation. Proposition 5.1. Assume G satisfies the condition (5.1) and let u ε be the solution of (1.
where ∇ y · = ∂ y 1 ·, ∂ y 2 · and v i is the solution of the auxiliar problem If α > 1, then there exists an unique and the scaling operator Π ε :

Proof. It follows from [21, theorems 3.1, 4.1 and 5.3].
Remark 5.1. We point out that the results in [21] are proved in the unit interval. Here, we just rewrite it to (ξ i−1 , ξ i ). The limit problems are stated in the next result. Now, we are in condition to show the following result.

Theorem 5.2. Suppose G satisfies the assumption (5.1) and let u ε be the solution of problem
and u is the unique solution of the problem where q, r : (0, 1) → R are piecewise constant functions such that with the homogenized constants r i and q i given by where Y * i is the basic cell associated to R ε i Y * i = (y 1 , y 2 ) ∈ R 2 : 0 < y 1 < L and 0 < y 2 < G i (y 1 ) , and v i is the solution of the auxiliary problem Also, u is the unique solution of the problem (5.3) with q(x) = q i and r(x) = r i for x ∈ (ξ i−1 , ξ i ), (5.6) where If α > 1, then there exists a unique u ∈ W 1,p (0, 1) such that Furthermore, u is the unique solution of the problem (5.3) with Proof. By (5.2), we can rewrite (2.1) taking into account the partition Hence, we obtain from (5.7) (with test functions ϕ(x, y) = ϕ(x) ∈ W 1,p (0, 1)) and proposition 3.5 that By proposition 5.1, we can pass to the limit in each subinterval (ξ i−1 , ξ i ). If we assume α 1, we obtain which is equivalent to J C Nakasato and M C Pereira for all ϕ ∈ W 1,p (0, 1). For α < 1, proposition 5.1 guarantees Since (p − 1)(p − 1) = 1, (5.9) can be rewritten as ϕ dx, ∀ϕ ∈ W 1,p (0, 1), where the functions u i are given by proposition 5.1. Notice that q i > 0 for each i. Indeed, by (5.5), we can take (v i − y 1 ) ∈ W 1,p #,0 (Y * i ) as a test function in such way that Consequently, we obtain from the Minty-Browder's theorem that the problem (5.11) has a unique solution in W 1,p (0, 1), and then, we can conclude that u ∈ W 1,p (0, 1) proving the theorem for α 1. Now, let us assume α > 1. Then, from (5.7) and proposition 3.5, we obtain that where Π ε is the scaling operator introduced in proposition 5.1. Hence, by proposition 5.1, we can pass to the limit taking test functions ϕ(x, y) = ϕ(x) ∈ W 1,p (0, 1). We obtain where the functions u i are given by proposition 5.1. Thus, ϕ dx, ∀ϕ ∈ W 1,p (0, 1). (5.12) As G 0 > 0, it follows from Minty-Browder's theorem that (5.12) is well posed. Hence, we get that u ∈ W 1,p (0, 1) is the unique solution concluding the proof of the theorem.

The locally periodic case
In this section, we provide the proof of our main result, theorem 2.1.
Proof of theorem 2.1. Using proposition 3.3 and theorem 3.7, there is u 0 ∈ W 1,p (0, 1) such that, up to subsequences, where χ is the characteristic function of (0, 1) × Y * (x). We show that u 0 satisfies the Neumann problem (5.3). To do this, we use a kind of discretization argument on the oscillating thin domains. We first proceed as in [3, theorem 2.3] fixing a parameter δ > 0 in order to set a piecewise periodic function G δ (x, y) satisfying (5.1) and 0 G δ (x, y) − G(x, y) δ in (0, 1) × R.
Let us construct this function. Recall that G is uniformly C 1 in each of the domains (ξ i−1 , ξ i ) × R. Also, it is periodic in the second variable. In particular, for δ > 0 small enough and for a fixed z ∈ (ξ i−1 , ξ i ) we have that there exists a small interval (z − η, z + η) with η depending only on δ such that |G(x, y) − G(z, y)| + |∂ y G(x, y) − ∂ y G(z, y)| < δ/2 for all x ∈ (z − η, z + η) ∩ (ξ i−1 , ξ i ) and for all y ∈ R. This allows us to select a finite number of points: Notice that this construction can be done for all i = 1, . . . , N. In particular, if we rename all the constructed points ξ k i by 0 = z 0 < z 1 < · · · < z m = 1, for some m = m(δ), we get that G δ (x, y) = G δ i (y) for (x, y) ∈ (z i−1 , z i ) × R and i = 1, . . . , m is a piecewise C 1 -function which is L-periodic in the second variable y.
Since ϕ and η are arbitrary, we conclude that u * = u 0 . Finally, let us see that the convergence holds. Notice that Hence, we can argue as in (6.9) getting (6.10) from (6.2), remark 4.2 and (6.6). And then, we conclude the proof of the theorem.

Convergence of the resolvent and semigroups
In this section, we show the convergence of the resolvent and semigroup associated to the p-Laplacian operator given by the equation (1.2) under the additional condition p 2. For that, let us first consider the operator M ε : L p (R ε ) → L p (0, 1) given by Next, let A ε : W 1,p (R ε ) → (W 1,p (R ε )) and A 0 : W 1,p (0, 1) → (W 1,p (0, 1)) be given by We consider the L 2 -realization of A ε and A 0 , that is, , and D(A 0,2 ) = u ∈ W 1,p (0, 1) : A 0 u ∈ L 2 (0, 1) ) , Then, for any p 2, λ > 0 and forcing terms f ε ∈ L 2 (R ε ), we can consider the following problems (I + λA ε )u ε = f ε , (7.2) and (I + λA 0 )u =f , (7.3) which are well posed (existence and uniqueness of solutions) by the Minty-Browder's theorem. Notice that here, we are using the dual products ·, · ε and ·, · 0 from W 1,p (R ε ) and W 1,p (0, 1) respectively to set the equations (7.2) and (7.3). Hence, with the additional conditions | f ε | L 2 (R ε ) uniformly bounded and M ε f ε f weakly in L 2 (0, 1), it follows from theorem 2.1 that the family of solutions defined by (7.2) converges to the solution of (7.3) as ε → 0. Consequently, we obtain the convergence of the resolvent operators defined by the equation (1.2). In fact, we have for any λ > 0 that In the next, let us obtain the convergence of the semigroup associated to the equations (7.2) and (7.3). As we will see, it is a consequence of [8, theorem 4.2, p 120]. First, let us write the resolvent operators convergence in appropriate spaces. For this purpose, we use the unfolding operator. We have where W = (0, 1) × (0, L) × (0, G 1 ) and ·, · is the dual product in W 1,p (W ). Next, Notice that It remains to observe that wich holds due to theorem 2.1. Therefore, thanks to Neveu-Trotter-Kato theorem, the semigroup S ε (t) associated to −B ε satisfies where S(t) is the semigroup associated to −B 0 . We have the following theorem. Theorem 7.1. Assume p 2 and consider the operators A ε and B ε defined respectively by (7.1) and (7.4). Then, (a) For any f ε ∈ L 2 (R ε ) with | f ε | L 2 (R ε ) uniformly bounded and M ε f ε f weakly in L 2 (0, 1), we have

where S(t) is the semigroup associated
with B 0 given by (7.5).

Appendix A
In the proof of the main result, we used q δ → q uniformly to obtain (6.7). Recall that q δ and q are given by (6.4) and (6.5) respectively. Here we prove such convergence. For this sake, let us first set Hence, for anyḠ ∈ A(M), we can consider the problem and we are looking for solutionsv such that (v − y 1 ) ∈ W 1,p #,0 (Y * G ). Now, for anyḠ, G ∈ A(M), let us consider the following transformation The Jacobian matrix for L is with det(JL) = F. Also, we can consider It is not difficult to see that B = L T L. Then, we can use the change of variables given by L to rewrite (A.2) in the region Y * G as Notice that this problem still has unique solutionv ∈ By the coercivity of (A.3), we get which means that the solutions are uniformly bounded by a constant independent onḠ and G. Now, let us compare the solutions of (A.2) forḠ = G and (A.3). We need to analyse Notice that L(1, 0) = (1, 0). We will distribute the terms finding estimative for each one. First, observe that for any test function ϕ ∈ W 1,p #,0 (Y * G ) in (A.3), we have  On the other side, if 1 < p < 2, we get from Hölder's inequality, proposition 3.1 and (A.13), that Therefore, for 1 < p < ∞, we have where α = 1/2 if 1 < p < 2 and α = 1/p if p 2. Finally, since L∇v − ∇v L p (Y * G ) + L∇v − ∇v L p (Y * G ) , we conclude by (A.14) and (A.12) that We have the following lemma: Remark A.1. We remark that the result of the lemma above, works in a more general framework, that is, the functions do not need to be in A(M). On the other hand, to perform the discretization of the domain in the locally periodic case, in the previous section, we need the hypothesis of A(M) functions defining the domains.