Quasilinear problems with nonlinear boundary conditions in higher-dimensional thin domains with corrugated boundaries

1 Introduction

In this work, we are interested in analyzing the asymptotic behavior of the solutions of a quasilinear elliptic equation defined in the following class of thin domains:

where ω ⊂ R N , N ≥ 1 , is an open, bounded, connected, and regular set with g : R N → R satisfying:

g is a lower semicontinuous function in L ∞ ( R N ) , strictly positive, and L -periodic (i.e., there exists L ∈ R N , L = ( L 1 , … , L N ) , such that

(H1) g ( y + L i e i ) = g ( y ) , for all y ∈ R N , and i = 1 , … , N ,

where { e 1 , … , e N } ⊂ R N denotes the canonical basis of R N ). Also, we set

g 0 = min y ∈ R N g ( y ) > 0 , and g 1 = max y ∈ R N g ( y ) .

The parameter α in (1.1) is assumed to be positive, and establishes the roughness on the upper boundary of R ε . ∂ l R ε is the lateral boundary of R ε given by:

The upper boundary of ∂ R ε plays an important role in this work and is denoted by:

Note that, according to [4], R ε is a purely periodic thin domain in R N + 1 since it exhibits a periodic structure set by the L -periodic function g . Its representative cell is the open set

where Y ⊂ R N is the R N -rectangle

Indeed, R ε can be seen as the union of the cell Y * appropriately rescaled in the vertical and horizontal directions by the terms ε and ε α , respectively.

We deal with the following quasilinear problem with nonlinear boundary condition defined in R ε :

where η ε is the outward unit normal to ∂ R ε , f ε ∈ L p ′ ( R ε ) , H ε ∈ L p ′ ( Γ ε ) , and p − 1 + ( p ′ ) − 1 = 1 . The functions a and b are Carathéodory functions that satisfy monotone and usual p -growth conditions in the third variable (see Section 2). We set homogeneous Dirichlet boundary condition on the lateral borders. On the top, we have a type of Robin’s nonlinear boundary condition (which models the reaction catalyzed by the upper wall), and on the button, we have homogeneous Neumann boundary condition. The term ε β , set by the parameter β , is a reaction coefficient term that acts on the nonlinear boundary condition on Γ ε and depends on the squeezing of the open set R ε . Such term can be used to model many reaction-diffusion processes, which naturally arises, for instance, in chemical engineering, since one shall consider the effects of Newton’s cooling law. Here, as one can see in the following results, depending on the reaction coefficient ε β , the cooling from the outside through the upper wall determines how the limit behavior will be. In particular, it is related to microfluidic applications (see [40] for more details). For simplicity of our arguments, we will assume

with h ∈ W 0 1 , p ( ω ) .

Under these conditions, we know that the variational formulation of (1.3) is given by:

where

is a Sobolev space equipped with the norm:

Furthermore, for each fixed ε > 0 , the existence and uniqueness of solutions are guaranteed by Minty-Browder’s theorem. Here, we are interested in analyzing the asymptotic behavior of the solutions u ε as ε → 0 . We determine the effective problem of (1.3), as the domain R ε becomes thinner and thinner, although with a high oscillating boundary at the top and different order of reactions.

Indeed, the open set R ε ⊂ ω × ( 0 , ε g 1 ) for all ε > 0 , and then, it degenerates to the open set ω as the parameter ε goes to zero. Hence, due to the thickness of R ε at ε ≈ 0 , it is expected that the sequence of solutions u ε will converge to a function depending only on the variable x ∈ ω and that this function will satisfy an equation of the same type as (1.3) but in ω ⊂ R N . Here, we will determine this equation that will depend on the geometry and roughness of the thin domain, as well as, the reaction term on the border.

The parameters α and β define, respectively, the intensity of the roughness of the top boundary and the effect of the flux given by the nonlinear reactions on the border. As we have mentioned, the homogenized limit equation will depend tightly on these numbers. Concerning the parameter α , we will analyze three distinct cases. We will consider the weak oscillatory case ( 0 < α < 1 ), the resonant or critical case ( α = 1 ), and the high oscillatory one ( α > 1 ). We will obtain different limit problems according to these three cases and β varying in R . We will see that β = 1 is also a kind of critical value. When α > 0 and β ≥ 1 , we are able to analyze (1.3) in a satisfactory way. But when β < 1 and p ∈ ( 1 , 2 ) , we need to add some conditions that will depend on the values of α , β , and p ∈ ( 1 , 2 ) (see Proposition 4.1). In particular, we will treat α > 0 , p ≥ 2 , and β ∈ R improving the recent results obtained in [27] for N = 1 , p = 2 , α > 0 , and β ∈ R . In fact, our main goal here is to generalize previous works from bidimensional oscillating thin domains to ( N + 1 ) -dimensional ones for a much more general class of elliptic equations, which also includes nonlinear boundary conditions.

We will combine techniques involving the analysis of monotone operators and the so-called unfolding operator method from homogenization theory. It is worth noting that the unfolding operator was initially developed as an effective method to deal with homogenization problems in partial differential equations (see, for instance, [14,15]). See also the recent monograph on the subject [16] in order to have a nice and broad perspective of this technique. In [6], this method was adapted to bidimensional thin domains with locally periodic oscillatory boundaries, and in [7] with very mild regularity assumptions. In Section 3, we recall such results that can be directly adapted to R N + 1 .

As one will see, if f ε and H ε converge, in a certain sense, to functions f ¯ and H ¯ , respectively, the homogenized equations of (1.3) can be formally described as follows. Let us first assume β ≥ 1 . Hence, if α = 1 , the so-called resonant oscillating case is obtained by Theorem 4.3 and is given by:

where A and B are the following monotone operators defined for z ∈ R N :

X z is an auxiliary function that is defined for each z ∈ R N . It is the unique solution of

with ∫ Y * X z d y 1 d y 2 = 0 , where W # 1 , p ( Y * ) ⊂ W 1 , p ( Y * ) is the Sobolev space of L -periodic functions on variable y 1 , which is given by:

The existence and uniqueness of X z , for each z ∈ R N , are guaranteed by the Minty-Browder’s theorem.

Also, the forcing terms f ¯ and H ¯ are given by:

where f ˆ is the limit of the unfolding operator applied to the sequence f ε (see Section 3 and Theorem 4.3). The reaction term υ in (1.7) depends on the parameter β and is given by:

Thus, under the conditions α = 1 and β ≥ 1 , the nonlinear boundary condition will be captured by the homogenized limit equation, only if β = 1 .

Next, if α ∈ ( 0 , 1 ) , then we are in the weakly oscillatory case (see Theorem 4.4); the limit equation of (1.3) is the same one given by (1.7), but now, with the monotone operator A set by a different auxiliary function. Now, the auxiliary function X z is the unique solution of:

where

Note that A ˜ involves a kind of average of a , and then, it can be seen as a nonlocal monotone operator.

Now, let us suppose α > 1 . Here, see in Theorem 4.7, we still have to split the analysis in two other cases: 1 < α ≤ β and 1 ≤ β < α . If 1 < α ≤ β , the limit equation also assumes the form (1.7), but in the other side, the operators A and B are given by:

where

See that Y − * ⊂ Y * and it is associated with the non-rugged part of the thin domain R ε . Indeed, the expressions obtained for the operators A and B here are in agreement with the results of the previous work [5]. Since the roughness on the top of R ε is too high, the diffusion in this part must vanish setting diffusion coefficients just defined in Y − * . We also obtain a distinguished forcing term H ¯ as a different coefficient υ . They are set by

Note that now, the limit equation (1.7) will capture the nonlinear boundary condition only as β = α . Also, we point out the dependence of the terms H ¯ and B with respect to ∇ y 1 g . It emphasizes the effect of the profile of the thin domain in the limit problem even in this case.

On the other hand, if 1 ≤ β < α , the family of solutions u ε will converge to the function h , which sets the nonlinear boundary condition (1.4). Due to the geometry of the thin domain, we can extend the function h from W 0 1 , p ( ω ) into W 0 l 1 , p ( R ε ) obtaining

Finally, if we have β < 1 , Proposition 4.1 guarantees, for any γ > 0 , that

Thus, we obtain (1.8) for some appropriate combinations of α and p with β < 1 . In particular, (1.8) holds if α > 0 and p ≥ 2 . Under these conditions, the thin domain perturbation affects the solutions in such way that the nonhomogeneous boundary condition, given on the border, will establish the asymptotic behavior of the solutions at ε = 0 . As we have pointed out, it is in agreement with [27], and it is now accomplished for a larger class of quasilinear elliptic equations.

We note that somehow, the homogenized limit operators A and B , reproduce the properties of the operators a and b set in Section 2, which guarantees the existence and uniqueness of the homogenized solution (1.7) in each mentioned case (see Proposition A.1). Also, we point out the dependence of the auxiliary functions X on: ( i ) the function a , which sets the quasilinear equations; ( i i ) the geometry of the thin domain, given by the function g ; and ( i i i ) the intensity of the roughness established by the parameter α . This way we obtain the explicit dependence of the homogenized equation on the perturbed and original Problem (1.3).

2 Settings of the problem and notations

In this section, we introduce some notations setting the necessary conditions that will be needed to introduce the unfolding operators and prove our results. First, we denote by c , c 1 , c 2 , c 3 , … positive constants that independent of ε > 0 . Next, we establish an appropriated partition of the open set ω ⊂ R N . Since g is L -periodic, for some L = ( L 1 , … , L N ) ∈ R N , we can consider the following rescaled rectangular blocks Y k ε ⊂ R N for each k ∈ Z N

Here, we basically rescale the box Y ⊂ R N by ε α and shift it by an integer vector also multiplied by ε α . Note that it is in agreement with the classical unfolding operator techniques developed to fixed domains, for instance, in [14]. Also, we introduce the following open sets illustrated in Figure 1.

Figure 1

Partition of the domain ω .

Now, let us rewrite each x ∈ ω in an appropriated form. For each x ∈ ω , there is a unique diagonal matrix with integer entries, which we denote by x ε α , and a unique x ε α ∈ Y , such that, for each ε > 0 ,

We also denote

Next, we describe the conditions on the Carathéodory functions a and b used to set our perturbed Problem (1.3). For that, let us set the following class of monotone functions:

If p ≥ 2 , we assume

(H3) ⟨ A ( x , y 1 , y 2 , z 1 ) − A ( x , y 1 , y 2 , z 2 ) , z 1 − z 2 ⟩ ≥ c ∣ z 1 − z 2 ∣ p , ∣ A ( x , y 1 , y 2 , z 1 ) − A ( x , y 1 , y 2 , z 2 ) ∣ ≤ c ∣ z 1 − z 2 ∣ ( ∣ z 1 ∣ + ∣ z 2 ∣ ) p − 2 ≤ c ∣ z 1 − z 2 ∣ ( 1 + ∣ z 1 ∣ + ∣ z 2 ∣ ) p − 2

a.e. ( x , y 1 , y 2 ) ∈ R N × R N × R , for some constant c > 0 independent of x , y 1 , y 2 and z i , i = 1 , 2 .