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ASYMPTOTIC APPROXIMATION OF THE SOLUTION OF THE LAPLACE EQUATION IN A DOMAIN WITH HIGHLY OSCILLATING BOUNDARY∗ Y. AMIRAT† , O. BODART‡ , U. DE MAIO§ , AND A. GAUDIELLO¶

R´ esum´ e On ´etudie le comportement asymptotique de la solution de l’´equation de Laplace dans un domaine dont une partie de la fronti`ere est fortement oscillante. La motivation de ce travail est l’´etude d’un ´ecoulement longitudinal dans un domaine infini born´e inf´erieurement par une paroi et sup´erieurement par une paroi rugueuse. Cette derni`ere est un plan recouvert d’asp´erit´es p´eriodiques dont la taille d´epend d’un petit param`etre ε. On fait l’hypoth`ese de rugosit´e forte, ` a savoir que la hauteur des asp´erit´es reste constante. A l’aide d’un correcteur de couche limite, on obtient une approximation non oscillante de la solution qui est d’ordre ε3/2 ) en norme H 1 .

Abstract We study the asymptotic behavior of the solution of the Laplace equation in a domain, a part of which boundary is highly oscillating. The motivation comes from the study of a longitudinal flow in an infinite horizontal domain bounded at the bottom by a wall and at the top by a rugose wall. The latter is a plane covered with periodic asperities which size depends on a small parameter ε > 0. The assumption of sharp asperities is made, that is the height of the asperities is fixed. Using a boundary layer corrector, we derive and analyze a nonoscillating approximation of the solution at order O(ε3/2 ) for the H 1 -norm.

Classification MSC : 35B40, 35C20, 35J05

∗ This

work is partially supported by the C.N.R. Project “Agenzia 2000”. de Math´ ematiques Appliqu´ ees, CNRS (UMR 6620), Universit´ e Blaise Pascal (Clermont-Ferrand 2), 63177 Aubi` ere cedex,France ([email protected]). ‡ Laboratoire de Math´ ematiques Appliqu´ ees, CNRS (UMR 6620), Universit´ e Blaise Pascal (Clermont-Ferrand 2), 63177 Aubi` ere cedex,France ([email protected]). § Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Universit` a di Napoli “Federico II”, Complesso Monte S. Angelo, via Cintia, 80126 Napoli, Italy ([email protected]). ¶ Dipartimento di Automazione, Elettromagnetismo, Ingegneria dell’Informazione e Matematica Industriale, Universit` a di Cassino, via G. Di Biasio 43, 03043 Cassino (FR), Italy, ([email protected] ). † Laboratoire

1

1. Introduction. Boundary-value problems involving oscillating boundaries or interfaces frequently arise when modelling problems in industrial applications, such as flows over rough walls, electromagnetic waves in a region containing a rough interface, elastic bodies containing a rough interface. The mathematical analysis of these problems consists in studying the large scale behavior of the solution. The goal is to determine effective boundary conditions, or to construct accurate and numerically implementable asymptotic approximations. The main difficulty comes from the presence of boundary layers near the rough region, which effects on correctors or error estimates have to be taken into account. In the present paper we consider a boundaryvalue problem for the Laplace equation, arising from the study of a laminar flow over a rough wall. Let us consider a viscous fluid in an infinite horizontal domain limited at the bottom by a wall P and at the top by a rough wall Rε . We assume that P moves at a constant horizontal velocity γ = (γ 0 , 0), γ 0 ∈ R2 , and that Rε is at rest. The latter is assumed to consist of a plane wall covered with periodic asperities which size depends on a small parameter ε > 0, and with a fixed height. Let 0 < ai < bi < li , e and ηε be the i = 1, 2, S = (0, l1 ) × (0, l2 ), Se = (a1 , b1 ) × (a2 , b2 ), Sε = εS, Seε = εS, Sε -periodic function defined on Sε by ( l3 if x0 ∈ Sε \Seε , 0 ηε (x ) = if x0 ∈ Seε , l30 with l30 > l3 > 0, and x0 = (x1 , x2 ) . The domain of the flow is © ª Oε = (x0 , x3 ) ∈ R3 : x0 ∈ R2 , b(x0 ) < x3 < ηε (x0 ) , where b is a smooth and S-periodic function on R2 such that b(x0 ) < l3 in R2 . The domain Oε is bounded at the bottom by © ª P = (x0 , x3 ) ∈ R3 : x0 ∈ R2 , x3 = b (x0 ) , and at the top by Rε = ∂Oε \O, where ∂Oε denotes the boundary of Oε . The profile of the asperities is then comb shaped. Throughout the paper, we will assume that 1/ε ∈ N so that ηε is also periodic with period l1 . Thus, Oε can be viewed as generated by periodic translations of the bounded domain © ª x ∈ R3 : x0 ∈ S, b(x0 ) < x3 < ηε (x0 ) . The velocity vε = (vε1 , vε2 , vε3 ) and the pressure pε of the fluid satisfy the stationary Navier-Stokes equations  −ν∆vε + (vε · ∇) vε + ∇pε = 0 in Oε ,    ∇ · vε = 0 in Oε , (1.1) vε = 0 on Rε ,    vε = γ on P, and they are assumed to be periodic with respect to x1 and x2 , with periods l1 and l2 , respectively. Here ν > 0 denotes the viscosity. Suppose now that γ = (0, g, 0) and that the functions b and ηε do not depend on x2 , i.e. b = b(x1 ) and ηε = ηε (x1 ), b being a smooth function on R, l1 -periodic, and ηε being the εl1 -periodic function defined on (0, εl1 ) by ½ 0 l3 if x1 ∈ (εa1, εb1 ), ηε (x1 ) = (1.2) l3 if x1 ∈ (0, εl1 )\(εa1, εb1 ). 2

Then, a particular solution (vε , pε ) of (1.1) is in the form vε = (0, uε , 0) , pε = 0, provided that uε satisfies the Laplace equation  ∆uε = 0 in Ωε ,    uε = 0 on Rε , (1.3) uε = g on P,    uε l1 -periodic with respect to x1, for a.e. x3 ∈ (b (0) , l3 ) , where Ωε is the bi-dimensional section (see figure) © ª Ωε = x = (x1, x3 ) ∈ R2 : 0 < x1 < l1 , b(x1 ) < x3 < ηε (x1 ) , and © ª P = x = (x1 , x3 ) ∈ R2 : 0 < x1 < l1 , x3 = b(x1 ) , © ª L = x = (x1 , x3 ) ∈ R2 : x1 = 0, b(0) < x3 < η (0) © ª ∪ x = (x1 , x3 ) ∈ R2 : x1 = l1 , b(l1 ) < x3 < η (l1 ) , and Rε = ∂Ωε \ (P ∪ L) , ∂Ωε being the boundary of Ωε . Our aim is to study the asymptotic behavior, as ε goes to 0, of the solution uε of (1.3). The main difficulty is due to the fact that the amplitude of the oscillations of the boundary is large. The case where b = 0 is studied in [4]. The assumption b = 0 allows to consider solutions uε of problem (1.3) that are εl1 -periodic, and then to construct an approximation of uε , up to an exponentially small error, by a nonoscillating explicit function. Here we consider the situation where the function b is not constant, hence the corresponding solution uε is not εl1 -periodic with respect to x1 . Problems involving rough boundaries, in the case where the frequency and the amplitude of the oscillations of the boundary are of the same order ³ 3 ´ε, have been addressed by many authors. In [1], an approximation at order O ε 2 for the H 1 norm is derived and analyzed for the Laplace equation, using a domain decomposition argument. In [2], Y. Achdou, O. Pironneau and F. Valentin consider a laminar flow over a rough wall with periodic roughness elements. Using asymptotic expansions and corresponding boundary layer correctors, the authors derive first and second order effective boundary conditions. ³ 3 ´ In [3], G. Allaire and M. Amar give a nonoscillating approximation at order O ε 2 for the H 1 -norm for the Laplace equation. In [22], W. J¨ager and A. Mikeli´c consider the Laplace equation on a bounded domain consisting 3

in a porous medium, a nonperforated domain and an interface between them. Using boundary layers describing the interaction between the two media, the authors derive asymptotic approximations and establish L2 estimates. In [5], for a flow governed by (1.1) in a domain Oε corresponding to the case where b = 0 and the frequency and the amplitude of the oscillations of the boundary are of the same order ε, it is proved that, outside a neighborhood of the rugose zone, the flow behaves asymptotically as a Couette flow, up to an exponentially small error. The Laplace equation in a domain with very rapidly oscillating locally periodic boundary, the amplitude of the oscillations being ε and the frequency εα (α > 1), is considered by G.A. Chechkin, A Friedman, and A.L. Piatniski in [12]. In this paper, the authors analyze a first order approximation in the H 1 -norm. Asymptotic limits of boundary-value problems in oscillating domains, in the case where the amplitude of the oscillations does not vanish as ε → 0, are studied in [9], [11], [15, 16, 17, 18] and [27, 28, 29]. Problems in domains with fragmented boundaries are treated in [21] and [26]. For general references about homogenization, we refer to [6, 7, 8, 10, 13, 14, 23, 30, 31]. Let © ª Dε = x = (x1 , x3 ) ∈ R2 : x1 ∈ R, b(x1 ) < x3 < ηε (x1 ) , and let, for m ≥ 0, the space m m (Dε ) ; u ∈ H m (Ωε ), u(x1 +l1 , x3 ) = u(x1 , x3 ), x3 ∈ (b(0), l3 )}, (Ωε ) = {u ∈ Hloc Hper

endowed with the norm of H m (Ωε ). In the present paper, we consider the slight generalization of problem (1.3)  −∆uε = f in Ωε ,    uε = 0 on Rε ,  uε = g on P,   1 uε ∈ Hper (Ωε ),

(1.4)

where f is a smooth function and g is a given constant. The paper is organized as follows. In Section 2, we establish a convergence result for the sequence {uε }ε : denoting Ω = {(x1 , x3 ) ; 0 < x1 < l1 , b(x1 ) < x3 < l30 }, and f uε being the zero extension of uε to Ω, we prove (Proposition 2.1) the convergence of {f uε }ε in H 1 (Ω). Section 3 is devoted to decay estimates at infinity for the solution of the Laplace equation in an infinite vertical domain of R2 (Proposition 3.1). These estimates play a key role in the subsequent analysis. Section 4 contains the main result of the paper (Theorem 4.1). Using boundary layer correctors, we construct a nonoscillating approximation of uε in Ωε and an outer boundary layer of height 2ε. We show that this approximation is of order O(ε3/2 ) in the H 1 -norm. This generalizes the result in [1] (see also [3]) which deals with the case where the frequency and the amplitude of the oscillations of the boundary are of same order ε. 2. A convergence result. Let uε be the solution of (1.4). We denote M = max b, 4

(2.1)

and  + Ωε = {(x1, x3 ) ∈ Ωε : l3 < x3 < l30 } ,         Ω+ = (0, l1 ) × (l3 , l30 ) ,      © ª Ω− = (x1, x3 ) ∈ R2 : x1 ∈ (0, l1 ) , b (x1 ) < x3 < l3 ,       Σ = (0, l1 ) × {l3 } ,        Ω = Ω− ∪ Ω+ ∪ Σ. m m In the sequel we will use the spaces Hper (Ω) and Hper (Ω− ) (for m ≥ 0), the definition m is similar to that of Hper (Ωε ) given in the previous section. The only regularity assumptions we make here are that b is Lipschitz-continuous and f ∈ L2 (Ω). Remark that

χΩ+ * ε

¡ ¢ weakly- ? in L∞ Ω+ ,

b1 − a1 l1

(2.2)

denotes the characteristic function of Ω+ where χΩ+ ε . To describe the limit problem, ε as ε → 0, of problem (1.4), we introduce the function ½ 0 in Ω+ , (2.3) u= u− in Ω− , where u− is the unique solution of the following problem:  −∆u− = f in Ω− ,    − u = 0 on Σ, u− = g on P,    − 1 u ∈ Hper (Ω− ) .

(2.4)

Let f uε be the zero extension to Ω of uε . The following convergence result holds: Proposition 2.1. Let uε be the solution of problem (1.4) and let u be the function defined in (2.3), (2.4). Then uε → u f

strongly in H 1 (Ω).

Proof. Let s ∈ C 2 (R) be such that      0 s(t) =

    1

if t >

M + l3 , 2

if t
0, where c is a constant independent of ε and α, and consequently, µZ 2

¶ 12

|∇uε |

dx ≤ c,

Ω²

where c is a constant independent of ε. Then, by virtue of the Poincar´e inequality, the sequence {f uε }ε is bounded in H 1 (Ω). Therefore, up to a subsequence, not relabelled 1 for convenience, there exists a function u ∈ Hper (Ω) (possibly depending on the subsequence) such that u = g on P and ½

u fε * u weakly in H 1 (Ω) , uε → u strongly in L2 (Ω) . f

(2.7)

Moreover, since u fε = f uε χΩ+ ε

in Ω+ ,

from (2.7) and (2.2), it follows that in Ω+ .

u=0

(2.8)

On the other hand, letting ε go to 0 in (1.4) with test functions ϕ ∈ C ∞ (Ω− ) such that ϕ is l1 -periodic with respect to x1 for a.e. x3 ∈ (b (0) , l3 ), ϕ|P = 0 and ϕ|Σ = 0, it follows from (2.7) that u|Ω− solves problem (2.4). Since this problem admits a unique solution, convergences (2.7) hold for the whole sequence. To obtain the strong convergence (2.5), by virtue of (2.7) and (2.8), it is enough to prove that lim k∇f uε k(L2 (Ω))2 = k∇uk(L2 (Ω− ))2 .

ε→0

(2.9)

Relations (2.6), (2.7) and (2.8) provide that µZ

Z 2

lim

ε→0

|∇f uε | dx = lim

∇f uε ∇h dx +

ε→0

Ω

Z

¶ f (f uε − h) dx

Z

Ω

Ω

Z

=

∇u∇h dx + Ω−

(2.10)

f (u − h) dx. Ω−

Finally, choosing u − h as a test function in problem (2.4), it follows that Z

Z

Z

2

|∇u| dx = Ω−

∇u∇h dx + Ω−

f (u − h) dx. Ω−

The convergence (2.9) is then obtained by comparing (2.10) with (2.11). 6

(2.11)

3. Decay estimates. The asymptotic approximation of uε will involve the solution of the Laplace equation in an infinite vertical domain of R2 . Let Λ+ = (a1 , b1 ) × (0, +∞), Λ− = (0, l1 ) × (−∞, 0) and ψ ± be the functions defined by ½ + ψ ∈ H 1 (Λ+ ), (3.1) 1 ψ − ∈ Hloc,per (Λ− ), ∇ψ − ∈ L2 (Λ− ),  ∆ψ ± = 0     ψ+ = 0    − ψ =0  ψ+ = ψ−    ∂ψ −  ∂ψ +  = +1  ∂y3 ∂y3

in Λ± , on ∂Λ+ \Γ, on ((0, a1 ) ∪ (b1, l1 )) × {0} , on Γ,

(3.2)

on Γ,

1 1 where Γ = (a1 , b1 ) × {0}. Here ψ − ∈ Hloc,per (Λ− ) means ψ − ∈ Hper (Λ0 ) for any bounded domain Λ0 ⊂ Λ− . We denote by β the mean of ψ − over an horizontal section of Λ− : Z 1 l1 − β= ψ (y1 , −δ) dy1 , ∀δ ∈ (0, +∞). (3.3) l1 0

The following result is proved in [4]: Proposition 3.1. Problem (3.1), (3.2) admits a unique solution. Moreover: (i) the constant β is independent of δ ; (ii) for any α ∈ N 2 and for any δ ∈ (0, +∞), there exists a constant cα,δ such that |∂ α ψ + (y1 , y3 )| ≤ cα,δ e−cy3 , ∀ (y1 , y3 ) ∈ (a1 , b1 ) × (δ, +∞) ; 2

(iii) for any α ∈ N and for any δ ∈ (0, +∞), there exists a constant cα,δ such that |∂ α (ψ − − β)(y1 , y3 )| ≤ cα,δ ecy3 , ∀ (y1 , y3 ) ∈ (0, l1 ) × (−∞, −δ). The above estimates are of the so-called de Saint-Venant type. The first is proved by means of Tartar’s lemma (see [25], pp. 49–58) ; see also [24]. The second is proved by adapting the proof of the first one. Let us remark that (3.1) and Proposition 3.1 1 provide that ψ − − β ∈ Hper (Λ− ). Proposition 3.1 implies the following result: Corollary 3.2. Let ψ ± be the functions satisfying (3.1), (3.2). Then, there exists two positive constants c and C, independent of ε, such that ¶¯ µ Z ¯ ¯ + x1 x3 − l3 ¯2 ¯ dx ≤ C ε, ¯ψ , ¯ ¯ ε ε Ω+ ε Z Ω

¯2 ¯ ¶ µ ¯ ¯ − x1 x3 − l3 ¯ψ , − β ¯¯ dx ≤ C ε, ¯ ε ε −

Z Ωε \Bε

¯ µ µ ¶¶¯2 ¯ ¯ ¯∇ ψ x1 , x3 − l3 ¯ dx ≤ C e− εc , ¯ ¯ ε ε

where Bε = (0, l1 ) × (l3 − ε, l3 + ε), and ψ is the function defined by ψ = ψ − in Λ− and ψ = ψ + in Λ+ . 7

4. A corrector result. To build a corrector for the solution uε of problem (1.4), we need more regularity on the solution u− of problem (2.4). We then assume the following regularity for f and b: 4 6 f ∈ Hper (Ω− ) ∩ L2 (Ω), b ∈ Hper (0, l1 ). (4.1) © ª Let O− = (x1 , x3 ) ∈ R2 : x1 ∈ R, b(x1 ) < x3 < l3 . The extension of u− to O− by 1 l1 -periodicity is a solution in Hper (Ω− ) of   −∆u− = f in O− , u− = 0 on R × {l3 },  − u =g on {(x1 , b(x1 )) : x1 ∈ R} .

Using standard regularity results, see [19, 20], we then have ³ ´ ¡ −¢ 6 u− ∈ Hper Ω ⊂ C 4 Ω− . Let now w be the function defined by ½ 0 w= w−

in Ω+ , in Ω− ,

(4.2)

(4.3)

1 where w− is the unique solution in Hper (Ω− ) of  ∆w− = 0 in Ω− ,    − ∂u on Σ, w− = β    w− = 0 ∂x3 on P,

(4.4)

where u− is the solution of problem (2.4) and β is defined by (3.3). Let us point out that, due to assumptions (4.1), w− ∈ C 3 (Ω− ).

(4.5)

Indeed, the functions w− and u− being extended by l1 -periodicity to O− , it follows 1 that w− is the solution in Hper (Ω− ) of  −∆w− = 0 in O− ,    − ∂u w− = β on R × {l3 },    w− = 0 ∂x3 on {(x1 , b(x1 )) : x1 ∈ R}. ∂u− 5 5 ∈ Hper (Ω− ), and consequently, w− ∈ Hper (Ω− ) ⊂ C 3 (Ω− ). ∂x3 This section is devoted to prove the following result. Theorem 4.1. Assume (4.1). Let uε be the solution of problem (1.4), u be defined by (2.3), (2.4) and w by (4.3), (4.4). Then, there exists a positive constant c, independent of ε, such that Due to (4.2), we have

ku² − ukH 1 (Ωε \B² ) ≤ c ε,

(4.6)

for ε small enough, where Bε = (0, l1 ) × (l3 − ε, l3 + ε). If in addition f = 0 in Ω+ , there exists a positive constant c, independent of ε, such that 3

ku² − u − εwkH 1 (Ωε \Bε ) ≤ c ε 2 , 8

(4.7)

for ε small enough. To prove Theorem 4.1 we need to introduce some auxiliary functions. Let τε be the function defined by  µ ¶ x1 x3 − l3 ∂u−  + + +  (x1 , l3 ) ψ , in Ω+  τε = uε − εwε − ε ε ,   ∂x3 ε ε τε = µ µ ¶ ¶   ∂u− x1 x3 − l3  − − − −  (x1 , l3 ) ψ , −β in Ω− ,  τε = uε − u − εwε −ε ∂x3 ε ε (4.8) and ρε be the function defined by  µ ¶ x1 x3 − l3 ∂w−  + + +  (x1 , l3 ) ψ , in Ω+  ρε = wε − ε ε ,   ∂x3 ε ε ρε = (4.9) µ ¶  −  ∂w x x − l  1 3 3 − −  , (x1 , l3 ) ψ − in Ω− ,  ρ− ε = wε − w −ε ∂x3 ε ε where uε is the solution of problem (1.4), u− is the solution of problem (2.4), w− the solution of problem (4.4), ψ ± are the functions defined by (3.1), (3.2) and wε± are 1 functions in H 1 (Ω+ ) and Hper (Ω− ) respectively, satisfying  ∆wε+ = 0 in Ω+  ε ,   −  ∆wε = 0 in Ω− ,     wε+ = 0 on Rε \Σ,     − ∂u−   wε = β on Rε ∩ Σ, ∂x3 (4.10) wε− = 0 on P,    −  ∂u    on Σ\Rε , wε+ = wε− − β   ∂x3   + −    ∂wε = ∂wε on Σ\Rε , ∂x3 ∂x3 β being defined by (3.3). Theorem 4.1 will be an immediate consequence of the two following propositions. Proposition 4.2. Assume (4.1). Let τε be the function defined by (4.8). Then, there exists a positive constant c, independent of ε, such that kτ² kH 1 (Ωε ) ≤ c ε,

(4.11)

and, if f = 0 in Ω+ , 3

kτ² kH 1 (Ωε ) ≤ c ε 2 ,

(4.12)

for ε small enough. Proposition 4.3. Assume (4.1). Let ρε be the function defined by (4.9). Then, there exists a positive constant c, independent of ε, such that kρ² kH 1 (Ωε ) ≤ c ε,

(4.13)

for ε small enough. − 1 − Proof of Proposition 4.2. Obviously, τε+ ∈ H 1 (Ω+ ε ) and τε ∈ H (Ω ). Due to ± ± + − the boundary conditions of uε , u, ψ and wε , the functions τε and τε have the same 9

1 − trace on Ω+ ² ∩Ω . Consequently, τε ∈ H (Ωε ) . Moreover, τε is l1 -periodic with respect to x1 for a.e. x3 ∈ (b (0) , l3 ) and τε = 0 on Rε \ {(x1 , l30 ) : x1 ∈ (0, l1 )} . Furthermore, from the jump conditions in (3.2) and in (4.10) it follows that the normal derivatives ³ ´ + + + − − 12 − − Ω² ∩ Ω . Consequently, of τε and τε on Ω² ∩Ω are opposite as elements of H ∆τε is weakly defined in Ωε by

∆τε =

µ ¶  x1 x3 − l3 ∂ 3 u−  +  + (x , l ) ψ , −ε  1 3  ∂x21 ∂x3 ε ε       µ µ ¶¶   ∂ 2 u− ∂ x1 x3 − l3  +  −2ε , (x , l ) ψ −f  1 3   ∂x1 ∂x3 x1 ε ε

in Ω+ ε ,

µ µ ¶ ¶   x1 x3 − l3 ∂ 3 u−  −  −ε (x , l ) ψ , − β +  1 3   ∂x21 ∂x3 ε ε      µ µ ¶ ¶   ∂ x1 x3 − l3 ∂ 2 u−    −2ε (x1 , l3 ) ψ− , −β ∂x1 ∂x3 x1 ε ε

in Ω− . (4.14)

Since τε|P

τε |R

µ µ ¶ ¶ ∂u− x1 b (x1 ) − l3 − = −ε (x1 , l3 ) ψ , −β , ∂x3 ε ε

0 ε ∩((0,l1 )×l3 )

= −ε

∂u− (x1 , l3 ) ψ + ∂x3

µ

x1 l30 − l3 , ε ε

¶ ,

setting τε1 (x1 , x3 ) = −ε

µ µ ¶ ¶ ∂u− x1 b (x1 ) − l3 (x1 , l3 ) ψ − , − β m1 (x3 ) in Ωε , ∂x3 ε ε

τε2 (x1 , x3 ) = −ε

∂u− (x1 , l3 ) ψ + ∂x3

µ

x1 l30 − l3 , ε ε

¶ m2 (x3 ) in Ωε ,

where m1 , m2 ∈ C 2 (R; [0, 1]) and      0 m1 (t) =

    1

M + l3 if t > , 2 3M + l3 if t < , 4

     1 ,

m2 (t) =

    0

if t >

l3 + l30 , 2

3l3 + l30 if t < , 4

(4.15)

1 M being defined by (2.1), it results that τ² − τε1 − τε2 ∈ Hper (Ωε ), and vanishes on

10

Rε ∪ P. Then, multiplying (4.14) by τ² − τε1 − τε2 and integrating on Ωε , it follows Z 2 |∇τ² | dx ΩZ ε Z Z ¡ ¢ = ∇τ² ∇τ²2 dx − ∇τ² ∇τ²1 dx + ∆τ² τ² − τε1 − τ²2 dx Ωε ε ZΩ− ZΩ+ 1 = ∇τ² ∇τ² dx + ∇τ² ∇τ²2 dx Ω+ εµ µ ¶ ¶ ZΩ− ¡ ¢ x1 x3 − l3 ∂ 3 u− − , − β τ² − τε1 dx +ε 2 ∂x (x1 , l3 ) ψ ∂x ε ε − 3 Ω 1 µ µ ¶ ¶ Z (4.16) ¡ ¢ ∂ 2 u− ∂ x1 x3 − l3 − (x1 , l3 ) ψ , − β τ² − τε1 dx +2ε x1µ ε ¶ε 1 ∂x3 Z Ω− ∂x ¢ x − l3 ¡ ∂ 3 u− x 3 1 (x1 , l3 ) ψ + , τ² − τ²2 dx +ε + ∂x2 ∂x3 ε ε Ωε 1 µ µ ¶¶ Z ¢ ¡ ∂ x1 x3 − l3 ∂ 2 u− +2ε (x1 , l3 ) ψ+ , τ² − τ²2 dx ∂x1 ∂x3 x1 ε ε Z Ω+ ε ¡ ¢ 2 + f τ² − τ² dx. Ω+ ε

Let us estimate each term in the right-hand side of (4.16). We first compute the derivatives of τ²1 and τ²2 : µ µ ¶ ¶ ∂τ²1 ∂ 2 u− x1 b (x1 ) − l3 − (x1 , x3 ) = −ε (x1 , l3 ) ψ , − β m1 (x3 ) ∂x1 ∂x1 ∂x3 ε ε µ ¶ ∂u− ∂ψ − x1 b (x1 ) − l3 − (x1 , l3 ) , m1 (x3 ) ∂x3 ∂y1 ε ε µ ¶ ∂ψ − x1 b (x1 ) − l3 db ∂u− (x1 , l3 ) , (x1 ) m1 (x3 ) in Ω− , − ∂x3 ∂y3 ε ε dx1 µ µ ¶ ¶ ∂τ²1 ∂u− x1 b (x1 ) − l3 dm1 (x1 , x3 ) = −ε (x1 , l3 ) ψ − , −β (x3 ) ∂x3 ∂x3 ε ε dx3 Then, from (4.2) and Proposition 3.1, ¯ 1¯ ¯ ∂τ² ¯ − εc ¯ ¯ ¯ ∂x1 ¯ ≤ Ce ,

we have ¯ 1¯ ¯ ∂τ² ¯ − εc ¯ ¯ ¯ ∂x3 ¯ ≤ Ce

in Ω− ,

in Ω− .

(4.17)

for ε small enough. Here and in the sequel C and c denote positive constants independent of ε. Similarly, ¯ 2¯ ¯ 2¯ ¯ ∂τ² ¯ ¯ ¯ ¯ ¯ ≤ Ce− εc , ¯ ∂τ² ¯ ≤ Ce− εc in Ω+ (4.18) ε , ¯ ∂x1 ¯ ¯ ∂x3 ¯ for ε small enough. For the first two terms in the right-hand side of (4.16), from the Cauchy-Schwarz inequality, (4.17) and (4.18), it follows ¯Z ¯ Z ¯ ¯ 1 2 ¯ ¯ ≤ Ce− εc k∇τ² k 2 (4.19) ∇τ ∇τ dx + ∇τ ∇τ dx ² ² ² ² (L (Ωε ))2 , ¯ ¯ Ω−

Ω+ ε

for ε small enough. For the third and fifth terms in the right-hand side of (4.16), Corollary 3.2, the Cauchy-Schwarz inequality, the Poincar´e inequality, (4.2), (4.17) 11

and (4.18) give ¯ Z ¯ ¯ε ¯

¶ µ µ ¶ ¡ ¢ ∂ 3 u− x1 x3 − l3 − (x , l ) ψ , − β τ² − τε1 dx 1 3 2 ε ¶ Ω ¯ µ ε Z− ∂x1 3∂x−3 ¢ ¯ ∂ u x1 x3 − l3 ¡ + 2 ¯ +ε τ − τ dx (x , l ) ψ , ² 1 3 ² ¯ ∂x21 ∂x3 ε ε Ω+ ε³ ´ ° ° ° ° 3 ≤ Cε 2 °τ² − τ²1 °L2 (Ω− ) + °τ² − τ²2 °L2 (Ω+ ) ³° ¡ ´ ° ¡ ε ¢° ¢° 3 ≤ Cε 2 °∇ τ² − τ²1 °(L2 (Ω− ))2 + °∇ τ² − τ²2 °(L2 (Ω+ ))2 ε ³ ´ ° 1° ° ° 3 ° ° ≤ Cε 2 k∇τ² k(L2 (Ω− ))2 + ∇τ² (L2 (Ω− ))2 + k∇τ² k(L2 (Ω+ ))2 + °∇τ²2 °(L2 (Ω+ ))2 ε ε ³ 3 ´ − εc 2 ≤ C ε k∇τ² k(L2 (Ωε ))2 + e , (4.20) for ε small enough. Integrating by parts the fourth and sixth terms in the right-hand side of (4.16), it follows that µ µ ¶ ¶ ¢ ¡ ∂ 2 u− ∂ x1 x3 − l3 (x1 , l3 ) ψ− , − β τ² − τε1 dx x1 µ ε ΩZ− ∂x1 ∂x3 µε ¶¶ ¡ ¢ ∂ 2 u− ∂ x1 x3 − l3 + +2ε (x1 , l3 ) ψ , τ² − τ²2 dx ∂x x1 µ ε Ω+ ε µε ¶ ¶ Z 1 ∂x3 2 − ¢ ∂ u x x ∂ ¡ 1 3 − l3 − = −2ε (x1 , l3 ) ψ , −β τ² − τε1 dx ∂x ∂x ε ε x µ µ ¶ ¶ 1 ZΩ− 31 − 3 ¡ ¢ ∂ u x1 x3 − l3 − −2ε (x1 , l3 ) ψ , − β τ² − τε1 dx 2 ε ¶ 3 µ ε ZΩ− ∂x21 ∂x ¢ ∂ u− x1 x3 − l3 ∂ ¡ + −2ε (x1 , l3 ) ψ , τ² − τε2 dx + ∂x1 ∂x3 ε ¶ x1 µε ZΩε ¢ ∂ 3 u− x1 x3 − l3 ¡ + −2ε (x1 , l3 ) ψ , τ² − τε2 dx. 2 ∂x1 ∂x3 ε ε Ω+ ε Z

2ε

Consequently, Corollary 3.2, the Cauchy-Schwarz inequality, the Poincar´e inequality, (4.2), (4.17) and (4.18) imply ¯ Z ¯ ¯2ε ¯

µ µ ¶ ¶ ¡ ¢ ∂ 2 u− ∂ x1 x3 − l3 (x1 , l3 ) ψ− , − β τ² − τε1 dx x1 µ ε Ω ¯ µε ¶¶ Z− ∂x1 2∂x−3 ¡ ¢ ¯ ∂ u ∂ x1 x3 − l3 + +2ε (x1 , l3 ) ψ , τ² − τ²2 dx¯¯ + ∂x1 ∂x3 x1 ε ε Ω³ ε ° ¡ ° ° ¢° 3 1 1 ° ° ° ≤ Cε 2 ∇ τ² − τ² (L2 (Ω− ))2 + τ² − τ² °L2 (Ω− ) ´ ° ¡ ° ° ¢° + °∇ τ² − τ²2 °(L2 (Ω+ ))2 + °τ² − τ²2 °L2 (Ω+ ) ε ε ³° ¡ ´ ° ¡ ¢° ¢° 3 ≤ Cε 2 °∇ τ² − τ²1 °(L2 (Ω− ))2 + °∇ τ² − τ²2 °(L2 (Ω+ ))2 ε ³ ´ ° 1° ° ° 3 ° ° 2 ≤ Cε k∇τ² k(L2 (Ω− ))2 + ∇τ² (L2 (Ω− ))2 + k∇τ² k(L2 (Ω+ ))2 + °∇τ²2 °(L2 (Ω+ ))2 ε ε ³ 3 ´ c ≤ C ε 2 k∇τ² k(L2 (Ωε ))2 + e− ε , (4.21) for ε enough small. For the last term in the right-hand side of (4.16), we observe, 12

using the Cauchy-Schwarz inequality, that Z Ω+ ε

1/ε−1 X ¯ ¯ ¯τ² (x1 , x3 ) − τ²2 (x1 , x3 )¯2 dx = k=0

Z

l30

l3

Z

ε(b1 +kl1 ) ε(a1 +kl1 )

¯ ¯ ¯τ² (x1 , x3 ) − τ²2 (x1 , x3 )¯2 dx

¯ ¯2 ¡ ¢ 1/ε−1 Z l0 Z ε(b1 +kl1 ) ¯Z x1 ¯ X 3 ∂ τ² − τ²2 ¯ ¯ = (t, x3 ) dt¯ dx ¯ ¯ ¯ ∂t l ε(a +kl ) εa +k 1 1 1 3 k=0 ¯ ¯ ¡ ¢ 2 1/ε−1 Z l0 Z ε(b1 +kl1 ) ¯ 2 ¯ X 3 ¯ ∂ τ² − τ² ¯ 2 ≤ ε2 (b1 − a1 ) ¯ dx ¯ ¯ ∂x1 l3 ε(a1 +kl1 ) ¯ k=0 ¯ ¯ ¡ ¢ 2 Z ¯ 2 ¯ ¯ ∂ τ² − τ² ¯ 2 ≤ ε2 (b1 − a1 ) ¯ dx. ¯ ¯ ¯ ∂x1 Ω+ ε From (4.18) it then follows ¯Z ¯ ¯ ¯

Ω+ ε

¯ ³ ´ ¡ ¢ ¯ c f τ² − τ²2 dx¯¯ ≤ C ε k∇τ² k(L2 (Ωε ))2 + e− ε ,

(4.22)

for ε small enough. Combining (4.16) with (4.19) ÷ (4.22), we have ³ ´ c 2 k∇τ² k(L2 (Ωε ))2 ≤ C ε k∇τ² k(L2 (Ωε ))2 + e− ε , for ε small enough. Therefore k∇τ² k(L2 (Ωε ))2 ≤ Cε,

(4.23)

for ε small enough. Estimate (4.11) follows from (4.23), using the Poincar´e inequality. If f = 0 in Ω+ , from identity (4.16) combined with (4.19) ÷ (4.21), we have ³ 3 ´ c 2 k∇τ² k(L2 (Ωε ))2 ≤ C ε 2 k∇τ² k(L2 (Ωε ))2 + e− ε , for ε small enough. Therefore 3

k∇τ² k(L2 (Ωε ))2 ≤ Cε 2 , for ε small enough, from which estimate (4.12) follows, due to the Poincar´e inequality. The proof of Proposition 4.3 has the same framework than the one of Proposition 4.2. 1 + − 1 − Proof of Proposition 4.3. Obviously, ρ+ ε ∈ H (Ωε ) and ρε ∈ H (Ω ). Due − to the boundary conditions of wε± , w− and ψ ± , the functions ρ+ and ρ ε ε have the + 1 − same trace on Ω² ∩ Ω . Consequently, ρε ∈ H (Ωε ). Moreover, ρε is l1 -periodic with respect to x1 for a.e. x3 ∈ (b (0) , l3 ) and ρε = 0 on Rε \ {(x1 , l30 ) : x1 ∈ (0, l1 )} . Furthermore, from the jump condition in (3.2) and in (4.10), it follows that ³ the normal ´ + + − − 12 − − . derivatives of ρε and ρε on Ω² ∩ Ω are opposite as elements of H Ω+ ² ∩Ω

13

Consequently, ∆ρε is weakly defined in Ωε and satisfies

∆ρε =

µ ¶  ∂ 3 w− x1 x3 − l3  +  −ε (x , l ) ψ , +  1 3  ∂x21 ∂x3 ε ε       µ µ ¶¶   ∂ 2 w− ∂ x1 x3 − l3  +  −2ε (x1 , l3 ) ψ ,    ∂x1 ∂x3 x1 ε ε

in Ω+ ε ,

µ ¶   x1 x3 − l3 ∂ 3 w−  −  (x1 , l3 ) ψ , + −ε 2    ∂x1 ∂x3 ε ε      µ µ ¶¶   ∂ 2 w− x1 x3 − l3 ∂  −   −2ε (x1 , l3 ) ψ , ∂x1 ∂x3 x1 ε ε

in Ω− . (4.24)

We have ρε | P

ρε |R

∂w− (x1 , l3 ) ψ − = −ε ∂x3

0 ε ∩((0,l1 )×l3 )

= −ε

µ

x1 b (x1 ) − l3 , ε ε

∂w− (x1 , l3 ) ψ + ∂x3

µ

¶ ,

x1 l30 − l3 , ε ε

¶ ,

and then, setting ρ1ε (x1 , x3 )

∂w− = −ε (x1 , l3 ) ψ − ∂x3

ρ2ε (x1 , x3 )

µ

∂w− = −ε (x1 , l3 ) ψ + ∂x3

x1 b (x1 ) − l3 , ε ε µ

x1 l30 − l3 , ε ε

¶ m1 (x3 ) in Ωε ,

¶ m2 (x3 ) in Ωε ,

1 (Ωε ) the functions m1 and m2 being defined by (4.15), it follows that ρ² −ρ1ε −ρ2ε ∈ Hper 1 2 and vanishes on Rε ∪ P . Then, multiplying (4.24) by ρ² − ρε − ρε and integrating on Ωε , we find Z 2 |∇ρ² | dx ΩZ ε Z Z ¡ ¢ = ∇ρ² ∇ρ1² dx + ∇ρ² ∇ρ2² dx − ∆ρ² ρ² − ρ1ε − ρ2² dx Ωε ZΩ− ZΩ+ ε 1 2 = ∇ρ² ∇ρ² dx + ∇ρ² ∇ρ² dx Ω+ ε ¶ µ ZΩ− ¢ ∂ 3 w− x1 x3 − l3 ¡ − +ε , (x , l ) ψ ρ² − ρ1ε dx (4.25) 1 3 2 3 µε µ ε ¶¶ ZΩ− ∂x12∂x− ¡ ¢ ∂ x1 x3 − l3 ∂ w (x1 , l3 ) ψ− , ρ² − ρ1ε dx +2ε ∂x ∂x x ε ε − 1µ ¶ Z Ω 3 1− 3 ¢ x1 x3 − l3 ¡ ∂ w + , +ε (x1 , l3 ) ψ ρ² − ρ2² dx + ∂x2 ∂x3 ε ε 1 µ µ ¶¶ ZΩε ¡ ¢ ∂ x1 x3 − l3 ∂ 2 w− + (x1 , l3 ) ψ , ρ² − ρ2² dx. +2ε + ∂x1 ∂x3 x1 ε ε Ωε

14

Let us estimate each term in the right-hand side of (4.25). First, the derivatives of ρ1² and ρ2² are µ ¶ ∂ρ1² ∂ 2 w− x1 b (x1 ) − l3 (x1 , x3 ) = −ε (x1 , l3 ) ψ − , m1 (x3 ) ∂x1 ∂x1 ∂x3 ε ε µ ¶ ∂w− ∂ψ − x1 b (x1 ) − l3 m1 (x3 ) − (x1 , l3 ) , ∂x3 ∂y1 ε ε µ ¶ ∂w− ∂ψ − x1 b (x1 ) − l3 db − , (x1 , l3 ) (x1 ) m1 (x3 ) in Ω− , ∂x3 ∂y3 ε ε dx1

∂ρ1² ∂w− (x1 , x3 ) = −ε (x1 , l3 ) ψ − ∂x3 ∂x3

µ

x1 b (x1 ) − l3 , ε ε

¶

dm1 (x3 ) dx3

in Ω− .

Then, Proposition 3.1 and (4.5) imply ¯ 1¯ ¯ ∂ρ² ¯ ¯ ¯ ¯ ∂x1 ¯ ≤ C ε,

¯ 1¯ ¯ ∂ρ² ¯ ¯ ¯ ¯ ∂x3 ¯ ≤ C ε

in Ω− ,

(4.26)

for ε small enough. Similarly, ¯ 2¯ ¯ ∂ρ² ¯ − εc ¯ ¯ ¯ ∂x1 ¯ ≤ Ce ,

¯ 2¯ ¯ ∂ρ² ¯ − εc ¯ ¯ ¯ ∂x3 ¯ ≤ Ce

in Ω+ ε ,

(4.27)

for ε small enough. From the Cauchy-Schwarz inequality, (4.26) and (4.27), the first two terms in the right-hand side of (4.25) satisfy ¯Z ¯ ¯ ¯

Ω−

Z ∇ρ² ∇ρ1² dx +

Ω+ ε

¯ ¯ ∇ρ² ∇ρ2² dx¯¯ ≤ C ε k∇ρ² k(L2 (Ωε ))2 ,

(4.28)

for ε small enough. For the third and fifth terms in the right-hand side of (4.25), Corollary 3.2, the Cauchy-Schwarz inequality, the Poincar´e inequality, (4.5), (4.26) and (4.27) give ¯ Z ¯ ¯ε ¯

µ ¶ ¢ x1 x3 − l3 ¡ ∂ 3 w− − (x , l ) ψ , ρ² − ρ1ε dx 1 3 2 ε 3 Ω ¯ ¶ µε Z− ∂x13∂x− ¢ ¯ ∂ w x1 x3 − l3 ¡ + 2 +ε , (x1 , l3 ) ψ ρ² − ρ² dx¯¯ + ∂x2 ∂x3 ε ε Ω³ 1 ε ´ ° ° ° ° ≤ Cε °ρ² − ρ1² °L2 (Ω− ) + °ρ² − ρ2² °L2 (Ω+ ) ³° ¡ ´ ° ¡ ε ¢° ¢° ≤ Cε °∇ ρ² − ρ1² °(L2 (Ω− ))2 + °∇ ρ² − ρ2² °(L2 (Ω+ ))2 ε ³ ´ ° 1° ° ° ° ° ≤ Cε k∇ρ² k(L2 (Ω− ))2 + ∇ρ² (L2 (Ω− ))2 + k∇ρ² k(L2 (Ω+ ))2 + °∇ρ2² °(L2 (Ω+ ))2 ε ε ´ ³ ≤ C ε k∇ρ² k(L2 (Ωε ))2 + ε2 ,

(4.29) for ε small enough. Integrating by parts the fourth and sixth terms in the right-hand 15

side of (4.25), it follows that µ µ ¶¶ ¡ ¢ ∂ x1 x3 − l3 ∂ 2 w− (x1 , l3 ) ψ− , ρ² − ρ1ε dx x1 µ ε ΩZ− ∂x1 ∂x3 µε ¶¶ ¡ ¢ ∂ 2 w− ∂ x1 x3 − l3 + (x1 , l3 ) ψ , ρ² − ρ2² dx +2ε ∂x x1 ε Ω+ ε µε ¶ Z 1 ∂x3 2 − ¢ x1 x3 − l3 ∂ ¡ ∂ w − (x1 , l3 ) ψ ρ² − ρ1ε dx = −2ε , ε ¶ x1 3 µε ZΩ− ∂x31 ∂x ¢ ∂ w− x1 x3 − l3 ¡ − −2ε (x1 , l3 ) ψ , ρ² − ρ1ε dx 2 ε ¶ 3 µ ε ZΩ− ∂x21 ∂x ¢ x − l3 ∂ ¡ ∂ w− x 3 1 , ρ² − ρ2ε dx −2ε (x1 , l3 ) ψ + + ∂x1 ∂x3 ε ε x µ ¶ 1 ZΩε ¢ ∂ 3 w− x1 x3 − l3 ¡ + −2ε (x1 , l3 ) ψ , ρ² − ρ2ε dx. + ∂x2 ∂x3 ε ε Ωε 1 Z

2ε

Then, Corollary 3.2, the Cauchy-Schwarz inequality, the Poincar´e inequality, (4.5), (4.26) and (4.27) imply ¯ Z ¯ ¯2ε ¯

µ µ ¶¶ ¡ ¢ ∂ x1 x3 − l3 ∂ 2 w− − (x1 , l3 ) ψ , ρ² − ρ1ε dx x1 ε µ εµ Ω− ∂x1 ∂x ¯ ¶¶ Z3 ¡ ¢ ¯ ∂ 2 w− ∂ x1 x3 − l3 + 2 (x1 , l3 ) ψ , ρ² − ρ² dx¯¯ + 2ε + ∂x1 ∂x3 x ε ε 1 Ω ε ³° ¡ ° ° ¢° ≤ Cε °∇ ρ² − ρ1² °(L2 (Ω− ))2 + °ρ² − ρ1² °L2 (Ω− ) ´ ° ¡ ° ° ¢° + °∇ ρ² − ρ2² °(L2 (Ω+ ))2 + °ρ² − ρ2² °L2 (Ω+ ) ε ε ³° ¡ ´ ° ¡ ¢° ¢° ≤ Cε °∇ ρ² − ρ1² °(L2 (Ω− ))2 + °∇ ρ² − ρ2² °(L2 (Ω+ ))2 ε ´ ³ ° ° ° ° ≤ Cε k∇ρ² k(L2 (Ω− ))2 + °∇ρ1² °(L2 (Ω− ))2 + k∇ρ² k(L2 (Ω+ ))2 + °∇ρ2² °(L2 (Ω+ ))2 ε ε ³ ´ 2 ≤ C ε k∇ρ² k(L2 (Ωε ))2 + ε , (4.30) for ε enough small. Combining (4.25) with (4.28) ÷ (4.30), we obtain ³ ´ 2 k∇ρ² k(L2 (Ωε ))2 ≤ C ε k∇ρ² k(L2 (Ωε ))2 + ε2 , for ε small enough. Therefore k∇ρ² k(L2 (Ωε ))2 ≤ Cε,

(4.31)

for ε small enough. Finally, making use of the Poincar´e inequality, estimate (4.13) follows from (4.31). Proof of Theorem 4.1. Let ψ ± be the functions satisfying (3.1), (3.2), and τε and ρε be the functions defined in (4.8) and (4.9), respectively. Since u² − u − εw = τε + ερε + gε 16

in Ωε ,

where

gε =

 µ ¶ ∂u− x1 x3 − l3  +  ε (x , l ) ψ ,  1 3   ∂x3 εµ ε  ¶  −  x ∂w x3 − l3  1 2 +  (x , l ) ψ , +ε  1 3   ∂x3 ε ε µ ¶ ¶ µ   ∂u− x1 x3 − l3  −  ε (x , l ) ψ , − β  1 3   ∂x3 ε  µε ¶  −  x − l3 ∂w x  3 1 2 −  , (x1 , l3 ) ψ  +ε ∂x3 ε ε

in Ω+ ε ,

in Ω− ,

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18