Analysis of Mixed Elliptic and Parabolic Boundary Layers with Corners

Abstract

We study the asymptotic behavior at small diffusivity of the solutions, 𝑢^𝜀, to a convection-diffusion equation in a rectangular domain Ω. The diffusive equation is supplemented with a Dirichlet boundary condition, which is smooth along the edges and continuous at the corners. To resolve the discrepancy, on 𝜕Ω, between 𝑢^𝜀 and the corresponding limit solution, 𝑢⁰, we propose asymptotic expansions of 𝑢^𝜀 at any arbitrary, but fixed, order. In order to manage some singular effects near the four corners of Ω, the so-called elliptic and ordinary corner correctors are added in the asymptotic expansions as well as the parabolic and classical boundary layer functions. Then, performing the energy estimates on the difference of 𝑢^𝜀 and the proposed expansions, the validity of our asymptotic expansions is established in suitable Sobolev spaces.

1. Introduction

We consider a singularly perturbed convection-diffusion equation in a rectangular domain Ω=(0,1)×(0,1): 𝐿_𝜀𝑢^𝜀∶=−𝜀Δ𝑢^𝜀−𝜕_𝑥𝑢^𝜀𝑢=𝑓inΩ,_𝜀=𝑔on𝜕Ω. (1) Here 𝜀 is a small but strictly positive diffusivity parameter, and 𝑓=𝑓(𝑥,𝑦) is a given smooth data with ‖𝐷^𝛼𝑓‖_𝐿2(Ω)≤𝜅_𝛼, independent of 𝜀, for some 𝛼's as needed in the analysis below. The function 𝑔=𝑔(𝑥,𝑦) is assumed to be continuous on 𝜕Ω and smooth on each edge of 𝜕Ω. Namely, defining the restriction of 𝑔 to the edges of 𝜕Ω as follows: 𝑔_𝑥=0=𝑔₁𝑦,𝑔_𝑥=1=𝑔₂𝑦,𝑔_𝑦=0=𝑔₃(𝑥),𝑔_𝑦=1=𝑔₄(𝑥), (2) we assume that 𝑔₁(0)=𝑔₃(0),𝑔₁(1)=𝑔₄(0),𝑔₂(0)=𝑔₃(1),𝑔₂(1)=𝑔₄(1),𝑔_𝑖issmoothfor1≤𝑖≤4 (3) If these compatibility conditions were not satisfied, some additional considerations would be necessary, which we do not address here.

In what follows, we study the asymptotic behavior of the solutions to (1) at small diffusivity 𝜀.

In a very nice related earlier work, [1], the asymptotic behavior of the solutions of a convection-diffusion equation similar to (1) was discussed. More precisely, in a rectangular domain Ω^∼=(0,𝑙₀)×(0,𝑙₁), the authors considered −𝜀^′Δ𝑢^𝜀′−𝑏𝜕_𝑥𝑢^𝜀′+𝑐𝑢^𝜀′=𝑓^∼in^∼𝑢Ω,_𝜀′=𝑔^∼on𝜕Ω^∼, (4) where 𝑏>0, 𝑐≥0 are constants and 𝑓^∼ is a given smooth function in Ω^∼. The function 𝑔^∼, satisfying the analogue version of (3) in Ω^∼, is assumed to be continuous and piecewise smooth on 𝜕Ω^∼. By constructing, in [1], asymptotic expansions of 𝑢^𝜀′ with respect to a small diffusivity 𝜀^′, the boundary layers of (4) were analyzed in a systematic way. The validity of their asymptotic expansions was established using the maximum principle.

Using a simple change of variables which maps Ω^∼ to Ω, and setting 𝑢^𝜀′=𝑢^𝜀𝑒^𝜆𝑥 with a suitable 𝜆, one can transform (4) to (1) where the two diffusivity parameters 𝜀 and 𝜀^′, respectively, in (1) and (4) are compatible; that is, 𝜀/𝜀^′ is of order one. Hence, via this transformation, our analysis of (1) in this paper is applicable to (4) as well. Our motivations for conducting the present study appear below. In particular we significantly simplify the proofs of [1] and make the study suitable for more general equations or systems, in particular, those which do not satisfy the maximum principle.

In the boundary layer analysis associated with the convection-diffusion equation (1) (or (4)), one of the most important and difficult points is to resolve any possible singularity near the four corners of the rectangular domain Ω. Towards this end, we simplify and improve the method used in [1]. More precisely, concerning the asymptotic expansion of 𝑢^𝜀, we introduce the so-called elliptic boundary layer corrector, appearing in Section 3.2, near the inflow boundary at 𝑥=1 only, while in [1], it was used near both the inflow and outflow boundaries at 𝑥=1 and 𝑥=0. This simplification in the construction of the asymptotic expansions is mainly based on an observation that the corner singularities at (0,0) and (0,1) (where the flows go out) can be handled by the ordinary corner corrector, appearing in Section 3.4 (which is much easier to analyze than the elliptic corrector). Using the energy estimates rather than the maximum principle as in [1], we prove the validity of our proposed asymptotic expansions in suitable Sobolev spaces. Here we make use of the Hardy inequality (see, e.g., [2]) in the estimates. As we said above, our energy estimation approach can be easily extended to some higher order equations or systems which do not admit a maximum principle.

This paper is organized as follows. In Section 2, we introduce a formal expansion of 𝑢^𝜀 as the sum of the outer and inner expansions. The outer expansion (outside of the boundary layers), which is easy to obtain, appears in Section 2. Then, in Section 3, by performing the boundary layer analysis in a systematic way, we construct the inner expansion (inside the boundary layers). In Section 4, we state and prove the main convergence result, Theorem 9, concerning the difference between 𝑢^𝜀 and the proposed asymptotic expansion at a given order. In addition, in the appendix, we prove Lemma 1 that contains some delicate pointwise estimates on the parabolic boundary layer correctors introduced in Section 3.1. The study of the elliptic corrector and its approximation also appears in the appendix.

Throughout this paper, we use the notation conventions below.

Notation 1. 𝜅∶=𝜅(𝑓,𝑔,Ω,𝑛) is a constant depending on the data, but independent of 𝜀; here 𝑛≥0 is the fixed (but arbitrary) order of the asymptotic expansion, as it appears in (6).

Notation 2. An e.s.t. is a function (or a constant) whose norm in all Sobolev spaces 𝐻^𝑠 (and thus spaces 𝐶^𝑠) is exponentially small with, for each 𝑠, a bound of the form 𝑐₁𝑒^{−𝑐2/𝜀𝛾},𝑐₁,𝑐₂,𝛾>0, 𝑐_𝑖, 𝛾 depending possibly on 𝑠.

Notation 3. For a fixed 𝜀>0, we define the energy norm of 𝐻¹(Ω), ‖⋅‖_𝜀∶=√𝜀‖⋅‖_𝐻¹_(Ω)+‖⋅‖_𝐿²_(Ω). (5)

2. Asymptotic Expansions

To study the asymptotic behavior of 𝑢^𝜀, solution of (1), we propose an asymptotic expansion 𝑢^𝜀 of the following type: 𝑢^𝜀≅^𝑛_𝑗=0𝜀^𝑗𝑢^𝑗+Θ^𝑗. (6) Here, at each order of 𝜀^𝑗, 𝑗≥0, 𝑢^𝑗 corresponds to the outer expansion (outside of the boundary layer) of 𝑢^𝜀. To balance the discrepancy on the boundary 𝜕Ω of 𝑢^𝜀 and of the 𝑢^𝑗, 𝑗≥0, we introduce the correctors Θ^𝑗, 𝑗≥0, which will contribute mainly inside of the boundary layer: Θ^𝑗 will be itself the sum of several boundary layer functions as we will see below. The asymptotic expansion (6) is made at order 𝑛≥0. As we will see below, the expansion itself depends on 𝑛, but 𝑛, which is set at the beginning of the study, can be chosen arbitrary large.

To determine the asymptotic expansion (6), we start with the outer expansion for 𝑢^𝜀, 𝑢^𝜀≅∑^𝑛_𝑗=0𝜀^𝑗𝑢^𝑗. Inserting this expansion into (1), we formally find the equations for the 𝑢^𝑗: −𝜕_𝑥𝑢⁰=𝑓,−𝜕_𝑥𝑢^𝑗=Δ𝑢^𝑗−1,1≤𝑗≤𝑛. (7) We supplement these equations with the following inflow boundary conditions (see (1)–(3)): 𝑢⁰=𝑔₂𝑦𝑢at𝑥=1,_𝑗=0at𝑥=1,1≤𝑗≤𝑛. (8) While (7) are “natural”, the choice of the boundary conditions in (8) is not obvious; it will be eventually justified by the convergence theorem below. Then, by integrating the equations in 𝑥, we find the smooth outer solutions 𝑢^𝑗 in the following form: 𝑢⁰=¹_𝑥𝑓𝑥₁,𝑦𝑑𝑥₁+𝑔₂𝑦,𝑢^𝑗=¹_𝑥Δ𝑢^𝑗−1𝑥₁,𝑦𝑑𝑥₁,1≤𝑗≤𝑛. (9)(10)

Under this construction of the outer expansion, we notice that 𝑢^𝜀≅∑^𝑛_𝑗=0𝜀^𝑗𝑢^𝑗 satisfies the boundary condition (1)₂ along the edge 𝑥=1 only, and not on the three other sides of 𝜕Ω, 𝑦=0, 𝑦=1, and 𝑥=0 (in general). Hence, we expect boundary layers to occur near those edges, and the so-called correctors (corresponding to the Θ^𝑗) will be necessary to account for these discrepancies.

In Section 3, performing the boundary layer analysis, we define, for each 0≤𝑗≤𝑛, the corrector Θ^𝑗 appearing in (11) below. Then, in Section 4, using (6), (9), (10), and (11), we prove our main result which is stated as Theorem 9.

3. Construction of the Correctors: Boundary Layers Analysis

We want the boundary value of 𝑢^𝜀 to match that of its approximation. In general, the boundary values of the diffusive solution 𝑢^𝜀 and the outer expansion ∑^𝑛_𝑗=0𝜀^𝑗𝑢^𝑗 match only at 𝑥=1, but not on the other sides of 𝜕Ω, 𝑦=0, 𝑦=1, and 𝑥=0. Hence, we expect that some boundary layers will occur near those three edges. To resolve this inconsistency, we will define a number of boundary layer functions on the sides and corners of 𝜕Ω, following an analysis reminiscent of the theory of the Prandtl boundary layer in [3–6].

For the moment, comparing 𝑢^𝜀 with a finite sum ∑^𝑛_𝑗=0𝜀^𝑗𝑢^𝑗, we infer from the definition of the 𝑢^𝑗 in (7)–(10) that the values of these functions coincide on the segment 𝑥=1. In general, they will not coincide anywhere else on 𝜕Ω. First, to correct the inconsistency on the sides 𝑦=0,1, we will define the so-called parabolic correctors 𝜑^𝑗=𝜑^𝑗_𝑙+𝜑^𝑗_𝑢. Then, to correct the inconsistency at 𝑥=0, we will introduce the (classical) corrector 𝜃^𝑗. However, this will not be enough because additional inconsistencies appear at the corners of Ω, and, as we shall see below, it will be necessary to introduce additional correctors 𝜉^𝑗, 𝜁^𝑗, and 𝜂^𝑗 to handle these inconsistencies.

In summary, in the subsections below, we will define the Θ^𝑗 in the form, Θ^𝑗=𝜑^𝑗+𝜉^𝑗+𝜃^𝑗+𝜁^𝑗+𝜂^𝑗,0≤𝑗≤𝑛, (11) where the role and location of each corrector are explained, respectively, in Box 1 and Figure 1. The stretched variables of the corrector functions, which will be introduced and used in the following subsections, are indicated in the following equation: 𝜑^𝑗_𝑙=𝜑^𝑗_𝑙𝑦𝑥,√𝜀,𝜑^𝑗_𝑢=𝜑^𝑗_𝑢𝑥,1−𝑦√𝜀,𝜉^𝑗_𝑙=𝜉^𝑗_𝑙(1−𝑥)𝜀,𝑦𝜀,𝜉^𝑗_𝑢=𝜉^𝑗_𝑢(1−𝑥)𝜀,1−𝑦𝜀,𝜃^𝑗=𝜃^𝑗𝑥𝜀,𝜁,𝑦_𝑗𝑙=𝜁^𝑗_𝑙𝑥𝜀,𝑦√𝜀,𝜁^𝑗_𝑢=𝜁^𝑗_𝑢𝑥𝜀,1−𝑦𝜀,𝜂^𝑗_𝑙=𝜂^𝑗_𝑙𝑥𝜀,𝑦,𝜂^𝑗_𝑢=𝜂^𝑗_𝑢𝑥𝜀,𝑦. (12)

Box 1 

The different correctors.

Figure 1 

Location of the outer solution 𝑢𝑗 and the boundary layer correctors.

3.1. Parabolic Boundary Layers (PBL)

At the bottom (or top) boundary, that is, at 𝑦=0 (or 𝑦=1), to balance the discrepancy of 𝑢^𝜀 and the outer expansion ∑^𝑛_𝑗=0𝜀^𝑗𝑢^𝑗, (see the 𝑢^𝑗 as defined in (9) and (10)), we construct below the parabolic boundary layer correctors 𝜑^𝑗, 0≤𝑗≤𝑛.

We formally insert the asymptotic expansion 𝑢^𝜀≅∑^𝑛_𝑗=0𝜀^𝑗𝜑^𝑗 into (1). Using the arguments similar to those of the Prandtl theory, we see that the thickness of the boundary layer near 𝑦=0 or 𝑦=1 should be of order 𝜀^1/2. Then, by collecting the terms at the same order of 𝜀^𝑗, 0≤𝑗≤𝑛, we find the equations of the parabolic boundary layer correctors: −𝜀𝜕²_𝑦𝜑^𝑗−𝜕_𝑥𝜑^𝑗=𝜕²_𝑥𝜑^𝑗−1inΩ,0≤𝑗≤𝑛, (13) where we set 𝜑⁻¹=0.

Along the boundaries 𝑦=0, 𝑦=1, and 𝑥=1, comparing (1)₂ and the boundary values of the functions in (9) and (10), we find the boundary conditions of 𝜑^𝑗, 0≤𝑗≤𝑛, which read. 𝜑^𝑗𝜑=0at𝑥=1,0≤𝑗≤𝑛,₀=𝑔₃(𝑥)−𝑢⁰()𝜑𝑥,0at𝑦=0,_𝑗=−𝑢^𝑗()𝜑𝑥,0at𝑦=0,1≤𝑗≤𝑛,₀=𝑔₄(𝑥)−𝑢⁰()𝜑𝑥,1at𝑦=1,_𝑗=−𝑢^𝑗(𝑥,1)at𝑦=1,1≤𝑗≤𝑛. (14)

We will show below how to actually construct 𝜑^𝑗. For the moment, comparing the functions 𝑢^𝜀 and ∑^𝑛_𝑗=0𝜀^𝑗(𝑢^𝑗−𝜑^𝑗) for some 𝑛≥0, we see that the boundary values of these functions coincide at 𝑥=1 and 𝑦=0,1. However this has been achieved at the price of introducing some new inconsistencies. Indeed we notice, in (14), that, for each 𝑗≥0, the boundary conditions at 𝑦=0 and 𝑦=1 are not consistent with that at 𝑥=1. That is, in general, for any 𝑖≥1, 𝜕^𝑖_𝑥𝑢⁰(1,0)−𝜕^𝑖_𝑥𝑔₃(1)≠0,𝜕^𝑖_𝑥𝑢⁰(1,1)−𝜕^𝑖_𝑥𝑔₄(1)𝜕≠0,_𝑖𝑥𝑢^𝑗(1,0)≠0,𝜕^𝑖_𝑥𝑢^𝑗(1,1)≠0,𝑗≥1. (15) Due to this inconsistency of the boundary data, if we define 𝜑^𝑗, 0≤𝑗≤𝑛, as a solution of (13) with (14), some derivatives of 𝜑^𝑗, 0≤𝑗≤𝑛 get singular at the two corners (1,𝑘), 𝑘=0,1. Therefore, to define smooth (smoother) correctors 𝜑^𝑗, 0≤𝑗≤𝑛, we will modify the boundary conditions (14) as introduced in (23).

3.1.1. On Modification of Boundary Conditions for Smooth Correctors at 𝑦=0,1

Equation (13) with initial and boundary conditions (14) is a parabolic problem (heat equation), in which −𝑥 is the (positive) time like variable so that the boundary conditions at 𝑥=1 are the analogue of the initial conditions for an evolution problem, and the possible inconsistencies in (15) correspond to the (analogue of) the compatibility conditions between the initial and boundary values for a parabolic equation; see [7–9] and the references therein. Due to the lack of consistency at the corners (1,0) and (1,1), the correctors constructed as in (13) and (14) might get singular derivatives at those corners. We will overcome this difficulty by considering some corner functions at 𝑥=1 and 𝑦=0,1, similar to what is done in [10, 11] or [12] for the compatibility issue in parabolic problems; see [13] as well.

Defining 𝛿(𝑥) as a smooth cut-off function, independent of 𝜀, such that 𝛿(𝑥)=10for0≤𝑥≤2,31for𝑥≥4, (16) we set 𝛾^𝑗_𝑘(𝑥)=𝛾^∼𝑗_𝑘(𝑥)𝛿(𝑥),𝑘=0,1, (17) where 𝛾^∼0_𝑘(𝑥)=^2𝑛+1_𝑖=1(𝑥−1)^𝑖𝜕𝑖!^𝑖𝑥𝑔_3+𝑘(1)−𝜕^𝑖_𝑥𝑢⁰(1,𝑘)_∼,𝛾^𝑗_𝑘(𝑥)=−^{2𝑛+1−2𝑗}_𝑖=1(𝑥−1)^𝑖𝜕𝑖!^𝑖𝑥𝑢^𝑗(1,𝑘),1≤𝑗≤𝑛. (18) Note that the 𝛾^∼0_𝑘, 𝑘=0,1, depend on 𝑛, but this dependency is not made explicit to make the notations (slightly) simpler. Note also (comparing to (15)) that, at 𝑥=1, for each 𝑘=0,1, 𝛾^∼𝑗_𝑘(1)𝜕=0,0≤𝑗≤𝑛,_𝑖𝑥∼𝛾⁰_𝑘(1)=𝜕^𝑖_𝑥𝑔_3+𝑘(1)−𝜕^𝑖_𝑥𝑢⁰()𝜕1,𝑘,1≤𝑖≤2𝑛+1,_𝑖𝑥∼𝛾^𝑗_𝑘(1)=−𝜕^𝑖_𝑥𝑢^𝑗(1,𝑘),1≤𝑗≤𝑛,1≤𝑖≤2𝑛+1−𝑗. (19)

We introduce the stretched variables, 𝑦𝑦=√𝜀,𝑦^∼=1−𝑦√𝜀. (20) Then, using (13), (14), (17), and (20), we define the parabolic corrector, 𝜑^𝑗, 0≤𝑗≤𝑛, as 𝜑^𝑗=𝜑^𝑗_𝑙+𝜑^𝑗_𝑢=𝜑^𝑗_𝑙𝑥,𝑦+𝜑^𝑗_𝑢𝑥,𝑦^∼, (21) where 𝜑^𝑗_𝑙 and 𝜑^𝑗_𝑢 (lower and upper parts of 𝜑^𝑗, resp.) are defined as the solutions of the 1D heat equations below.

For 0≤𝑗≤𝑛, −𝜕²_𝑦𝜑^𝑗_𝑙−𝜕_𝑥𝜑^𝑗_𝑙=𝜕²_𝑥𝜑_𝑙^𝑗−1for𝑦,𝑥∈ℝ⁺×(),𝜑0,1_𝑗𝑙=ℎ^𝑗₀(𝑥)at𝜑𝑦=0,_𝑗𝑙⟶0as𝜑𝑦⟶∞,_𝑗𝑙=0at𝑥=1,−𝜕^2∼_𝑦𝜑^𝑗_𝑢−𝜕_𝑥𝜑^𝑗_𝑢=𝜕²_𝑥𝜑_𝑢^𝑗−1for𝑦^∼,𝑥∈ℝ⁺×(),𝜑0,1_𝑗𝑢=ℎ^𝑗₁(𝑥)at𝑦^∼𝜑=0,_𝑗𝑢⟶0as𝑦^∼𝜑⟶∞,_𝑗𝑢=0at𝑥=1. (22) Here the ℎ^𝑗_𝑘(𝑥), 0≤𝑗≤𝑛, 𝑘=0,1, are defined by ℎ⁰₀(𝑥)∶=𝑔₃(𝑥)−𝑢⁰(𝑥,0)−𝛾⁰₀(𝑥),ℎ^𝑗₀(𝑥)∶=−𝑢^𝑗(𝑥,0)−𝛾^𝑗₀(𝑥)ℎ,1≤𝑗≤𝑛,₀₁(𝑥)∶=𝑔₄(𝑥)−𝑢⁰(𝑥,1)−𝛾⁰₁(𝑥),ℎ^𝑗₁(𝑥)∶=−𝑢^𝑗(𝑥,1)−𝛾^𝑗₁(𝑥),1≤𝑗≤𝑛. (23) Note that the smooth boundary conditions ℎ^𝑗_𝑘(𝑥), 0≤𝑗≤𝑛, 𝑘=0,1, along 𝑦 or 𝑦^∼=0 are compatible with the other 0 boundary (initial) conditions at 𝑥=1 in the sense that, for each 0≤𝑗≤𝑛, 𝜕^𝑖_𝑥ℎ^𝑗_𝑘(1)=0,𝑘=0,1,0≤𝑖≤2𝑛+1−2𝑗. (24)

From [14–16] or [1], we recall the explicit expressions of 𝜑^𝑗_𝑙=𝜑^𝑗_𝑙(𝑥,𝑦), 0≤𝑗≤𝑛, 𝜑⁰_𝑙=2𝜋^∞_{𝑦/2()1−𝑥}−𝑦exp²¹2ℎ⁰₀𝑥+𝑦²2𝑦²₁𝑑𝑦₁,𝜑^𝑗_𝑙=2𝜋^∞_{𝑦/2()1−𝑥}−𝑦exp²¹2ℎ^𝑗₀𝑥+𝑦²2𝑦²₁𝑑𝑦₁+12√𝜋₀^1−𝑥∞₀1√𝑥₁exp−𝑦−𝑦₁²4𝑥₁−exp−𝑦+𝑦₁²4𝑥₁×𝜕²_𝑥𝜑_𝑙^𝑗−1𝑥+𝑥₁,𝑦₁𝑑𝑦₁𝑑𝑥₁. (25)(26) The expressions of 𝜑^𝑗_𝑢=𝜑^𝑗_𝑢(𝑥,𝑦^∼), 0≤𝑗≤𝑛, are identical to (25) and (26) with ℎ⁰₀,  ℎ^𝑗₀, and 𝑦, respectively, replaced by ℎ⁰₁,   ℎ^𝑗₁ and 𝑦^∼.

3.1.2. Estimates on the Parabolic Boundary Layers

We now state some pointwise estimates for the 𝜑^𝑗_𝑙 and 𝜑^𝑗_𝑢, 0≤𝑗≤𝑛, which are proved in the appendix.

Lemma 1. For each 0≤𝑗≤𝑛 and 𝑠≥0, one has the following pointwise estimates: 𝑦^𝑠𝜕^𝑖_𝑥𝜕^𝑚_𝑦𝜑^𝑗_𝑙≤𝜅𝜀^{(𝑠−𝑚)/2}𝑦exp−𝑐√𝜀,𝑦0≤𝑖+𝑚≤2𝑛+2−2𝑗,_𝑠𝜕^𝑖_𝑥𝜕^𝑚_𝑦𝜑^𝑗_𝑢≤𝜅𝜀^{(𝑠−𝑚)/2}exp−𝑐1−𝑦√𝜀,0≤𝑖+𝑚≤2𝑛+2−2𝑗, (27)(28) for a generic constant 𝑐>0 independent of 𝑥, 𝑦, and 𝜀.

From Lemma 1, it is easy to deduce the following 𝐿^𝑝 estimates.

Lemma 2. For each 0≤𝑗≤𝑛 and 𝑠≥0, one has, for 0≤𝑖+𝑚≤2𝑛+2−2𝑗, 𝑦^𝑠𝜕^𝑖_𝑥𝜕^𝑚_𝑦𝜑^𝑗_{𝑙𝐿𝑝(Ω)}+𝑦^𝑠𝜕^𝑖_𝑥𝜕^𝑚_𝑦𝜑^𝑗_{𝑢𝐿𝑝(Ω)}𝜅𝜀^{(𝑠−𝑚)/2+1/2𝑝}. (29)

Using (21) and (22), we find the equations for 𝜑^𝑗=𝜑^𝑗_𝑙+𝜑^𝑗_𝑢, 0≤𝑗≤𝑛: −𝜀𝜕²_𝑦𝜑^𝑗−𝜕_𝑥𝜑^𝑗=𝜕²_𝑥𝜑^𝑗−1𝜑inΩ,_𝑗=ℎ^𝑗₀(𝑥)+𝜑^𝑗_𝑢𝑦=0𝜑at𝑦=0,_𝑗=ℎ^𝑗₁(𝑥)+𝜑^𝑗_𝑙𝑦=1𝜑at𝑦=1,_𝑗=0at𝑥=1. (30) Thanks to Lemma 1, we notice that 𝜑^𝑗_𝑢𝑦=0=e.s.t.,𝜑^𝑗_𝑙𝑦=1=e.s.t., (31) where e.s.t. denotes an exponentially small term with respect to the (small) parameter 𝜀 as defined in Notation 2.

3.2. Elliptic Boundary Layers (EBL)

In Section 3.1, to construct the consistent parabolic boundary layer correctors 𝜑^𝑗, 0≤𝑗≤𝑛, we considered the modified boundary conditions (23), including 𝛾^𝑗_𝑘, instead of the natural boundary conditions (14). Here, on the two sides 𝑦=0 and 𝑦=1 of Ω, to cancel 𝛾^𝑗_𝑘 which is exactly the difference of (14) and (23), we introduce elliptic boundary layer correctors 𝜉^𝑗, 0≤𝑗≤𝑛, which, like 𝛾^𝑗_𝑘 or 𝛾^∼𝑗_𝑘, depend on the order 𝑛 of the asymptotic expansion in (6).

Using (17), we define 𝜉^𝑗=𝜉^𝑗_𝑙+𝜉^𝑗_𝑢,0≤𝑗≤𝑛, (32) where 𝜉^𝑗_𝑙 and 𝜉^𝑗_𝑢 are the solutions of the elliptic systems below.

For 0≤𝑗≤𝑛, −𝜀𝜕²_𝑥𝜉^𝑗_𝑙−𝜀𝜕²_𝑦𝜉^𝑗_𝑙−𝜕_𝑥𝜉^𝑗_𝑙𝜉=0inΩ,_𝑗𝑙=𝛾^𝑗₀(𝑥)𝜉at𝑦=0,_𝑗𝑙=0at𝑥=0,1,or𝑦=1,−𝜀𝜕²_𝑥𝜉^𝑗_𝑢−𝜀𝜕²_𝑦𝜉^𝑗_𝑢−𝜕_𝑥𝜉^𝑗_𝑢𝜉=0inΩ,_𝑗𝑢=𝛾^𝑗₁(𝑥)𝜉at𝑦=1,_𝑗𝑢=0at𝑥=0,1,or𝑦=0. (33) The systems (33) being elliptic equations, their well-posedness are easy to verify, and hence, we omit the proof here.

The equations satisfied by the elliptic correctors 𝜉^𝑗=𝜉^𝑗_𝑙+𝜉^𝑗_𝑢 read as follows.

For 0≤𝑗≤𝑛, −𝜀𝜕²_𝑥𝜉^𝑗−𝜀𝜕²_𝑦𝜉^𝑗−𝜕_𝑥𝜉^𝑗𝜉=0inΩ,_𝑗=𝛾^𝑗_𝑘(𝑥)𝜉at𝑦=𝑘,𝑘=0,1,_𝑗=0at𝑥=0,1. (34)

In the error analysis below in Section 4, we will see that no extra terms related to the 𝜉^𝑗 appear; see (65) below. This is because the 𝜉^𝑗, as appearing in (34), satisfies the same equation as for 𝑢^𝜀, supplemented with the (exact) boundary conditions that we need. From this observation, as it will be justified in Section 4, we notice that our main result in Theorem 9 does not require any estimate on the 𝜉^𝑗.

As an extra information which might be useful elsewhere, but not in this paper, by performing the energy estimate on (34), we find that 𝜉^𝑗_𝜀≤𝜅𝜀^1/4,0≤𝑗≤𝑛. (35) In the appendix, we introduce an explicit approximate 𝜉 of 𝜉. Then, thanks to the estimates on 𝜉, one can obtain some pointwise estimates on 𝜉 as well; see Appendix B.

At this stage, the functions 𝑢^𝜀 and ∑^𝑛_𝑗=0𝜀^𝑗(𝑢^𝑗+𝜑^𝑗+𝜉^𝑗) match along the sides 𝑥=1 and 𝑦=0,1 (with singular derivatives of high orders for the 𝜑^𝑗). More precisely, from (1), (8), (30), (31), and (34), we see that 𝑢^𝜀−^𝑛_𝑗=0𝜀^𝑗𝑢^𝑗+𝜑^𝑗+𝜉^𝑗=𝑔₁+∑^𝑛_𝑗=0𝜀^𝑗𝑢^𝑗+𝜑^𝑗at𝑥=0,0at𝑥=1,_𝑛𝑗∑₌₀𝜀^𝑗𝜑^𝑗_𝑢=e.s.t.at𝑦=0,_𝑛𝑗∑₌₀𝜀^𝑗𝜑^𝑗_𝑙=e.s.t.at𝑦=1. (36)

We will now deal with the side 𝑥=0 and construct the ordinary boundary layer 𝜃^𝑗 and the ordinary corner layer functions (correctors) 𝜁^𝑗.

3.3. Ordinary Boundary Layers (OBL)

To handle the discrepancy of 𝑢^𝜀 and ∑^𝑛_𝑗=0𝜀^𝑗𝑢^𝑗 at 𝑥=0, which is a part of 𝑢^𝜀−∑^𝑛_𝑗=0𝜀^𝑗(𝑢^𝑗+𝜑^𝑗+𝜉^𝑗) appearing in (36), the so-called ordinary boundary layer corrector 𝜃^𝑗 is introduced in this section. We insert a formal expansion 𝑢^𝜀≅∑^∞_𝑗=0𝜀^𝑗𝜃^𝑗 into the diffusive equation (1). Then, using the stretched variable 𝑥=𝑥/𝜀, we find the equations of 𝜃^𝑗: −𝜀𝜕²_𝑥𝜃^𝑗−𝜕_𝑥𝜃^𝑗=𝜀⁻¹𝜕²_𝑦𝜃^𝑗−2inΩ,0≤𝑗≤𝑛, (37) where we set 𝜃⁻¹=𝜃⁻²=0.

The main role of the 𝜃^𝑗 is, at each order of 𝜀^𝑗, to cancel the error 𝑢^𝜀−∑^𝑛_𝑗=0𝜀^𝑗𝑢^𝑗 at 𝑥=0. Hence, we supplement (37) with the boundary conditions 𝜃⁰=𝑔₁𝑦−𝑢⁰𝜃0,𝑦at𝑥=0,_𝑗=−𝑢^𝑗𝜃0,𝑦at𝑥=0,1≤𝑗≤𝑛,_𝑗⟶0as𝑥⟶∞,0≤𝑗≤𝑛. (38) The explicit expressions of 𝜃^𝑗, 0≤𝑗≤𝑛, are inductively shown to be of the form 𝜃^𝑗=𝑃^𝑘𝑥𝜀,𝑦exp−𝑥𝜀,𝑗=2𝑘,2𝑘+1, (39) where 𝑃^𝑘(𝑥/𝜀,𝑦) is a polynomial in 𝑥/𝜀 of degree 𝑘 whose coefficients, independent of 𝜀, are linear combinations of the 𝜕^𝑠𝑢^𝑗−𝑠(0,𝑦)/𝜕𝑦^𝑠, 0≤𝑠≤𝑗, and 𝑃^𝑘(0,𝑦)=−𝑢^𝑗(0,𝑦).

According to (39), we deduce the following lemmas.

Lemma 3. For each 0≤𝑗≤𝑛, one has the pointwise estimates: 𝑥^𝑠𝜕^𝑖_𝑥𝜕^𝑚_𝑦𝜃^𝑗≤𝜅𝜀^𝑠−𝑖𝑥exp−𝑐𝜀,𝑠,𝑖,𝑚≥0, (40) for a constant 𝑐 independent of 𝑥, 𝑦, and 𝜀.

Lemma 4. For each 0≤𝑗≤𝑛, one has 𝑥^𝑠𝜕^𝑖_𝑥𝜕^𝑚_𝑦𝜃^𝑗_𝐿𝑝(Ω)≤𝜅𝜀^{𝑠−𝑖+(1/𝑝)},𝑠,𝑖,𝑚≥0. (41)

Thanks to Lemma 3, we notice that the effect of 𝜃^𝑗 near the boundary 𝑥=1 is exponentially small: 𝜃^𝑗1,𝑦=e.s.t.,0≤𝑗≤𝑛. (42)

From (36), (38), and (42), we infer that 𝑢^𝜀−^𝑛_𝑗=0𝜀^𝑗𝑢^𝑗+𝜑^𝑗+𝜉^𝑗+𝜃^𝑗=^𝑛_𝑗∑₌₀𝜀^𝑗𝜑^𝑗at𝑥=0,_𝑛𝑗∑₌₀𝜀^𝑗𝜃^𝑗=e.s.t.at𝑥=1,_𝑛𝑗∑₌₀𝜀^𝑗𝜑^𝑗_𝑢+𝜃^𝑗=∑^𝑛_𝑗=0𝜀^𝑗𝜃^𝑗+e.s.t.at𝑦=0,_𝑛𝑗∑₌₀𝜀^𝑗𝜑^𝑗_𝑙+𝜃^𝑗=∑^𝑛_𝑗=0𝜀^𝑗𝜃^𝑗+e.s.t.at𝑦=1. (43)

3.4. Ordinary Corner Layers (OCL)

To account for the value ∑^𝑛_𝑗=0𝜀^𝑗𝜑^𝑗 at 𝑥=0, which is exactly the difference of 𝑢^𝜀 and ∑^𝑛_𝑗=0𝜀^𝑗(𝑢^𝑗+𝜑^𝑗+𝜉^𝑗+𝜃^𝑗) at 𝑥=0 (see (43)), we introduce the ordinary corner layer correctors 𝜁^𝑗, 0≤𝑗≤𝑛, in the form 𝜁^𝑗=𝜁^𝑗_𝑙+𝜁^𝑗_𝑢, (44) where the 𝜁^𝑗_𝑙 (or 𝜁^𝑗_𝑢) are the correctors near the corner (0,0) (or (0,1)) as constructed below.

To define 𝜁^𝑗_𝑙 (or 𝜁^𝑗_𝑢), we insert a formal expansion 𝑢^𝜀≅∑^∞_𝑗=0𝜀^𝑗𝜁^𝑗 into (1). Then, using the stretched variables 𝑥=𝑥/𝜀 and 𝑦=𝑦/√𝜀 near (0,0) and using 𝑥 and 𝑦^∼=(1−𝑦)/√𝜀 near (1,1), we collect the terms of order 𝜀^𝑗, 0≤𝑗≤𝑛. As a result, we find the equations for 𝜁^𝑗_𝑙 (or 𝜁^𝑗_𝑢): −𝜀𝜕²_𝑥𝜁^𝑗_𝑘−𝜕_𝑥𝜁^𝑗_𝑘=𝜕²_𝑦𝜁_𝑘^𝑗−1inΩ,0≤𝑗≤𝑛,𝑘=𝑢or𝑙. (45) Here, we set 𝜁_𝑘⁻¹=0 for 𝑘=𝑙,𝑢.

At each order of 𝜀^𝑗, 0≤𝑗≤𝑛, to cancel the error 𝜑^𝑗 near the corner (0,0) or (0,1), we supplement (45) with the boundary conditions 𝜁^𝑗_𝑘=−𝜑^𝑗_𝑘𝜁0,𝑦at𝑥=0,0≤𝑗≤𝑛,𝑘=𝑙,𝑢,_𝑗𝑘⟶0as𝑥⟶∞,0≤𝑗≤𝑛,𝑘=𝑙,𝑢. (46)

The explicit solutions 𝜁^𝑗_𝑙 of (45) and (46), 0≤𝑗≤𝑛, are inductively found to be of the following form: 𝜁^𝑗_𝑙=𝑃^𝑗𝑥𝜀,𝑦√𝜀exp−𝑥𝜀, (47) where 𝑃^𝑗(𝑥/𝜀,𝑦/√𝜀) is a polynomial in 𝑥/𝜀 of degree 𝑗 whose coefficients, independent of 𝜀, are linear combinations of the quantities 𝜀^𝑠𝜕^2𝑠𝜑_𝑙^𝑗−𝑠(0,𝑦/√𝜀)/𝜕𝑦^2𝑠, 0≤𝑠≤𝑗. In the same manner, the explicit forms of the 𝜁^𝑗_𝑢, 0≤𝑗≤𝑛 can be found as well.

Using Lemma 1 and (47), we obtain the lemmas below.

Lemma 5. For each 0≤𝑗≤𝑛 and 𝑘,𝑠≥0, one has the pointwise estimates: 𝑥^𝑘𝑦^𝑠𝜕^𝑖_𝑥𝜕^𝑚_𝑦𝜁^𝑗_𝑙≤𝜅𝜀^{𝑘−𝑖+((𝑠−𝑚)/2)}𝑥exp−𝑐𝜀+𝑦√𝜀,𝑥0≤𝑖+𝑚≤2𝑛+2−2𝑗,_𝑘𝑦^𝑠𝜕^𝑖_𝑥𝜕^𝑚_𝑦𝜁^𝑗_𝑢≤𝜅𝜀^{𝑘−𝑖+((𝑠−𝑚)/2)}𝑥exp−𝑐𝜀+1−𝑦√𝜀,0≤𝑖+𝑚≤2𝑛+2−2𝑗, (48) for a generic constant 𝑐>0 independent of 𝑥, 𝑦, and 𝜀.

Lemma 6. For each 0≤𝑗≤𝑛 and 𝑘,𝑠≥0, one has, for 0≤𝑖+𝑚≤2𝑛+2−2𝑗, 𝑥^𝑘𝑦^𝑠𝜕^𝑖_𝑥𝜕^𝑚_𝑦𝜁^𝑗_{𝑙𝐿𝑝(Ω)}+𝑥^𝑘𝑦^𝑠𝜕^𝑖_𝑥𝜕^𝑚_𝑦𝜁^𝑗_{𝑢𝐿𝑝(Ω)}≤𝜅𝜀^{𝑘−𝑖+((𝑠−𝑚)/2)+(3/2𝑝)}. (49)

Thanks to (46) and Lemma 5, we estimate the values of  𝜁^𝑗, 0≤𝑗≤𝑛, along the sides 𝑥=1, 𝑦=0, and 𝑦=1: 𝜁^𝑗=𝜁e.s.t.at𝑥=1,_𝑗𝑙()𝜁𝑥,0+e.s.t.at𝑦=0,_𝑗𝑢(𝑥,0)+e.s.t.at𝑦=1. (50)

Now from (43), (46), and (50), we observe that 𝑢^𝜀−^𝑛_𝑗=0𝜀^𝑗𝑢^𝑗+𝜑^𝑗+𝜉^𝑗+𝜃^𝑗+𝜁^𝑗=0at𝑥=0,_𝑛𝑗∑₌₀𝜀^𝑗𝜃^𝑗+𝜁^𝑗=e.s.t.at𝑥=1,_𝑛𝑗∑₌₀𝜀^𝑗𝜑^𝑗_𝑢+𝜃^𝑗+𝜁^𝑗=∑^𝑛_𝑗=0𝜀^𝑗𝜃^𝑗+𝜁^𝑗_𝑙+e.s.t.at𝑦=0,_𝑛𝑗∑₌₀𝜀^𝑗𝜑^𝑗_𝑙+𝜃^𝑗+𝜁^𝑗=∑^𝑛_𝑗=0𝜀^𝑗𝜃^𝑗+𝜁^𝑗_𝑢+e.s.t.at𝑦=1. (51)

3.5. Supplementary Correctors

Now, our task is to handle, at each order 0≤𝑗≤𝑛, the effects of 𝜃^𝑗 and 𝜁^𝑗 along the top and bottom boundaries 𝑦=0,1. To this end, using (51), we construct the supplementary correctors 𝜂^𝑗 in the form 𝜂^𝑗=𝜂^𝑗_𝑙+𝜂^𝑗_𝑢,0≤𝑗≤𝑛, (52) where 𝜂^𝑗_𝑙=−1−𝑦𝜃^𝑗+𝜁^𝑗_𝑙,𝜂𝑥,𝑦=0_𝑗𝑢=−𝑦𝜃^𝑗+𝜁^𝑗_𝑢𝑥,𝑦=1. (53)

From Lemmas 3 and 5, it is easy to verify the lemmas below.

Lemma 7. For each 0≤𝑗≤𝑛 and 𝑠≥0, one has the pointwise estimates: 𝑥^𝑠𝜕^𝑖_𝑥𝜕^𝑚_𝑦𝜂^𝑗≤𝜅𝜀^𝑠−𝑖𝑥exp−𝑐𝜀,0≤𝑖+𝑚≤2𝑛+2−2𝑗, (54) for a constant 𝑐 independent of 𝑥, 𝑦, and 𝜀.

Lemma 8. For each 0≤𝑗≤𝑛 and 𝑠≥0, one has 𝑥^𝑠𝜕^𝑖_𝑥𝜕^𝑚_𝑦𝜂^𝑗_𝐿𝑝(Ω)≤𝜀^{𝑠−𝑖+(1/𝑝)},0≤𝑖+𝑚≤2𝑛+2−2𝑗. (55)

Using (3), (14), (17), (22), (23), (38), and (46), we find that, near the corner (0, 0), 𝜃⁰+𝜁⁰_𝑙(0,0)=𝑔₁(0)−𝑢⁰(0,0)−𝜑⁰_𝑙(0,0)=𝑔₁(0)−𝑢⁰(0,0)−𝑔₃(0)−𝑢⁰(0,0)=0. (56) More generally, one can verify that 𝜃^𝑗+𝜁^𝑗_𝑙(0,0)=0,𝜃^𝑗+𝜁^𝑗_𝑢(0,1)=0,0≤𝑗≤𝑛. (57)

Using (42), (50), and (57), we infer from (52) and (53) that, for 0≤𝑗≤𝑛, 𝜂^𝑗=−𝜃0at𝑥=0,e.s.t.at𝑥=1,_𝑗+𝜁^𝑗_𝑙−𝜃at𝑦=0,_𝑗+𝜁^𝑗_𝑢at𝑦=1. (58)

Now, we finally obtain from (51) and (58) that 𝑢^𝜀−^𝑛_𝑗=0𝜀^𝑗𝑢^𝑗+𝜑^𝑗+𝜉^𝑗+𝜃^𝑗+𝜁^𝑗+𝜂^𝑗=0at𝑥=0,_𝑛𝑗∑₌₀𝜀^𝑗𝜃^𝑗+𝜁^𝑗+𝜂^𝑗=e.s.t.at𝑥=1,_𝑛𝑗∑₌₀𝜀^𝑗𝜑^𝑗_𝑢+𝜃^𝑗+𝜁^𝑗+𝜂^𝑗=∑^𝑛_𝑗=0𝜀^𝑗𝜃^𝑗+𝜁^𝑗_𝑙+𝜂^𝑗+e.s.t.=e.s.t.at𝑦=0,_𝑛𝑗∑₌₀𝜀^𝑗𝜑^𝑗_𝑙+𝜃^𝑗+𝜁^𝑗+𝜂^𝑗=∑^𝑛_𝑗=0𝜀^𝑗𝜃^𝑗+𝜁^𝑗_𝑢+𝜂^𝑗+e.s.t.=e.s.t.at𝑦=1. (59) The e.s.t. on the right hand side of (59)_2,3,4 are caused, respectively, by (𝜃^𝑗+𝜁^𝑗+𝜂^𝑗) at 𝑥=1, (𝜑^𝑗_𝑢+𝜁^𝑗_𝑢) at 𝑦=0, and (𝜑^𝑗_𝑙+𝜁^𝑗_𝑙) at 𝑦=1.

4. Asymptotic Error Analysis: The Main Result

We define 𝜌∶=𝑢^∼𝜀−^𝑛_𝑗=0𝜀^𝑗𝑢^𝑗+Θ^𝑗_𝜕Ω=theright-handsideof(59), (60) where Θ^𝑗=𝜑^𝑗+𝜉^𝑗+𝜃^𝑗+𝜁^𝑗+𝜂^𝑗 as defined in (11).

We construct an extension 𝜌 of 𝜌^∼ in the form 𝜌𝑥,𝑦∶=𝑟+(1−𝑥)𝜌^∼−𝑟_𝑥=0+𝑥𝜌^∼−𝑟_𝑥=1+1−𝑦𝜌^∼−𝑟_𝑦=0+𝑦𝜌^∼−𝑟_𝑦=1, (61) where 𝑟𝑥,𝑦∶=(1−𝑥)1−𝑦𝜌^∼_𝑥=𝑦=0+𝑥1−𝑦𝜌^∼_{𝑥=1,𝑦=0}+(1−𝑥)𝑦𝜌^∼_{𝑥=0,𝑦=1}+𝑥𝑦𝜌^∼_𝑥=𝑦=1. (62) Then one can verify that 𝜌|_𝜕Ω=𝜌^∼ and 𝜌isane.s.t.inΩ. (63)

We set 𝑤_𝜀𝑛∶=𝑢^𝜀−^𝑛_𝑗=0𝜀^𝑗𝑢^𝑗+Θ^𝑗−𝜌. (64) Then, using (1), (7), (11), (30), (34), (37), (45), (53), and (60), the equation of 𝑤_𝜀𝑛 reads 𝐿_𝜀𝑤_𝜀𝑛=𝜀^𝑛+1Δ𝑢^𝑛+𝜕²_𝑥𝜑^𝑛+𝜕²_𝑦𝜃^𝑛+Ξ^𝑛₁+𝜀^𝑛𝜕²_𝑦𝜃^𝑛−1+𝜀𝜕²_𝑦𝜁^𝑛+Ξ^𝑛₂𝑤+e.s.t.inΩ,_𝜀𝑛=0on𝜕Ω, (65) where Ξ₁=1−𝑦𝜕²_𝑦𝜃^𝑛𝑥,𝑦=0+𝑦𝜕²_𝑦𝜃^𝑛,Ξ𝑥,𝑦=1₂=1−𝑦𝜕²_𝑦𝜃^𝑛−1+𝜀𝜕²_𝑦𝜁^𝑛_𝑙𝜕𝑥,𝑦=0+𝑦²_𝑦𝜃^𝑛−1+𝜀𝜕²_𝑦𝜁^𝑛_𝑢𝑥,𝑦=1. (66)

We multiply (65) by 𝑒^𝑥𝑤_𝜀𝑛 and integrate over Ω. Then, using Lemmas 1, 3, and 5 and the Hardy inequality (see, e.g., [2]), we find that 𝜀𝑤_𝜀𝑛²_𝐻1+1−𝜀2𝑤_𝜀𝑛²_𝐿2≤𝜅𝜀^𝑛+1_ΩΔ𝑢^𝑛+𝜕²_𝑥𝜑^𝑛+𝜕²_𝑦𝜃^𝑛+Ξ^𝑛₁𝑤_𝜀𝑛𝑑𝑥𝑑𝑦+𝜅𝜀^𝑛_Ω𝑥𝜕²_𝑦𝜃^𝑛−1+𝜀𝜕²_𝑦𝜁^𝑛+Ξ^𝑛₂𝑤_𝜀𝑛𝑥𝑑𝑥𝑑𝑦≤𝜅𝜀^2𝑛+2+14𝑤_𝜀𝑛²_𝐿2+𝜀2𝑤_𝜀𝑛²_𝐻1. (67)

Thanks to the elliptic regularity theory and (67), we also find that 𝑤_{𝜀𝑛𝐻2(Ω)}≤𝜅Δ𝑤_{𝜀𝑛𝐿2(Ω)}≤𝜅𝜀⁻¹𝜕_𝑥𝑤_𝜀𝑛+lowerorderterms_𝐿2(Ω)≤𝜅𝜀^𝑛−(1/2). (68)

Finally, from (63), (64), (67), and (68), we obtain the convergence result on the difference of 𝑢^𝜀 and the proposed asymptotic expansions that we summarize in Theorem 9.

Theorem 9. For each fixed 𝑛≥0, as the diffusivity parameter 𝜀 vanishes, the difference between the solution 𝑢^𝜀 of (1) and its asymptotic expansion (6) converges to zero in the following sense: 𝑢^𝜀−^𝑛_𝑗=0𝜀^𝑗𝑢^𝑗+Θ^𝑗_𝜀≤𝜅𝜀^𝑛+1,𝑢^𝜀−^𝑛_𝑗=0𝜀^𝑗(𝑢^𝑗+Θ^𝑗)_𝐻2(Ω)≤𝜅𝜀^𝑛−(1/2). (69)(70)

Remark 10. Due to the continuity of 𝑔 as appearing in (3), 𝜉⁰ can be omitted in (11) for 𝑗=0 at order 𝜀⁰. Then one can verify that 𝑢^𝜀−𝑢⁰+𝜑⁰+𝜃⁰+𝜁⁰+𝜂⁰_𝜀≤𝜅𝜀^3/4, (71) which gives a convergence estimate that a factor of 𝜀^1/4 is worse than what is stated in (69) for 𝑛=0.

Appendix

A. Proof of Lemma 1

To prove (27), for 𝑦=𝑦/√𝜀, since 𝑦^𝑠exp(−𝑐𝑦)≤𝜅𝜀^𝑠/2exp(−𝑐₀𝑦), for all 𝑦, 0<𝑐₀<𝑐, it suffices to prove that 𝜕^𝑖_𝑥𝜕^𝑚_𝑦𝜑^𝑗_𝑙≤𝜅exp−𝑐𝑦√,1−𝑥0≤𝑖+𝑚≤2𝑛+2−2𝑗,𝑖,𝑚≥0. (A.1) Note that (28) would follow similarly. Since ℎ^𝑗(𝑥)=ℎ^𝑗₀(𝑥)=−𝑢^𝑗(𝑥,𝑖)−𝛾^𝑗₀(𝑥) for 𝑗≥1 and 𝑔₃(𝑥)−𝑢⁰(𝑥,0)−𝛾⁰₀(𝑥) for 𝑗=0; thanks to (3), we note that 𝜕^𝑘_𝑥ℎ^𝑗(1)=0 for 0≤𝑘≤2𝑛+1−2𝑗.

Now, to prove (A.1), we use an induction on 𝑗.(1) We begin with 𝑗=0.

From (25), since 𝜕^𝑘_𝑥ℎ⁰(1)=0 for 0≤𝑘≤2𝑛+1, differentiating 𝜑⁰_𝑙 in 𝑥 we find that 𝜕^𝑖_𝑥𝜑⁰_𝑙=2𝜋^∞_{𝑦/2(1−𝑥)}−𝑦exp²¹2𝜕^𝑖_𝑥ℎ⁰𝑥+𝑦²2𝑦²₁𝑑𝑦₁≤𝜅^∞_{𝑦/2(1−𝑥)}−𝑦exp²¹2𝑑𝑦₁,0≤𝑖≤2𝑛+2. (A.2)

Using exp(−𝑦²₁/2)≤𝜅exp(−𝑐𝑦₁), for all 𝑦₁≥0, for some 𝑐>0, (A.1) for 𝑚=0 is proved. Since −𝜑⁰_𝑙𝑦𝑦−𝜑⁰_𝑙𝑥=0, for 𝑚=2(𝑘+1),  𝑘≥0, from (A.2) we easily find that 𝜕^𝑖_𝑥𝜕^2(𝑘+1)_𝑦𝜑⁰_𝑙=𝜕_𝑥^𝑖+𝑘+1𝜑⁰_𝑙≤𝜅exp−𝑐𝑦√,1−𝑥0≤𝑖+𝑘+1≤2𝑛+2; (A.3) thus (A.3) holds for 0≤𝑖+𝑚=𝑖+2(𝑘+1)≤𝑘+1+2𝑛+2, and this implies that (A.1) holds for 𝑚=2(𝑘+1), 𝑘≥0, 𝑗=0.

Since 𝜕_𝑥^𝑖+1ℎ⁰(1)=0,  0≤𝑖+1≤2𝑛+1, we now notice that 𝜕^𝑖_𝑥𝜕_𝑦𝜑⁰_𝑙=2𝜋^∞_{𝑦/2()1−𝑥}−𝑦exp²¹2𝜕_𝑥^𝑖+1ℎ⁰×𝑥+𝑦²2𝑦²₁𝑦𝑦²₁𝑑𝑦₁≤𝜅𝑦^∞_{𝑦/2(1−𝑥)}−𝑦exp²¹2𝑑−𝑦₁⁻¹,0≤𝑖+1≤2𝑛+2. (A.4) Integrating by parts in the last integral, we deduce that 𝜕^𝑖_𝑥𝜕_𝑦𝜑⁰_𝑙≤𝜅√1−𝑥exp−𝑦²4()+1−𝑥𝑦^∞_{𝑦/2()1−𝑥}−𝑦exp²¹2𝑑𝑦₁≤𝜅exp−𝑐𝑦√+1−𝑥𝑦exp−𝑐𝑦√≤()1−𝑥since𝑡exp−𝑐𝑡≤𝜅exp−𝑐₀𝑡,∀𝑡≥0,forsome0<𝑐₀<𝑐≤𝜅exp−𝑐₀𝑦√1−𝑥,0≤𝑖+1≤2𝑛+2. (A.5) Hence, for 𝑚=2𝑘+1, 𝑘≥0, using −𝜑⁰_𝑙𝑦𝑦−𝜑⁰_𝑙𝑥=0 again, we note from (A.5) that 𝜕^𝑖_𝑥𝜕^2𝑘+1_𝑦𝜑⁰_𝑙=𝜕_𝑥^𝑖+𝑘𝜕_𝑦𝜑⁰_𝑙≤𝜅exp−𝑐𝑦√,1−𝑥0≤𝑖+𝑘+1≤2𝑛+2; (A.6) thus (A.6) holds for 0≤𝑖+𝑚=𝑖+2𝑘+1≤𝑘+2𝑛+2, and this proves (A.1) for 𝑚=2𝑘+1, 𝑘≥0, 𝑗=0. Hence, (A.1) is proved for 𝑗=0.(2) We now assume that (A.1) holds at any order less than or equal to 𝑗−1, and we want to prove it at the order 𝑗.

At order 𝑗≥1, we perform the analysis as follows.

We first note that the homogeneous solution of 𝜑^𝑗_𝑙, that is the first integral of (26) for 𝜑^𝑗_𝑙, can be similarly estimated. We just replace ℎ⁰₀(𝑥) by ℎ^𝑗₀(𝑥) in the above analysis.

Since 𝜕^𝑘_𝑥ℎ^𝑗(1)=0 for 0≤𝑘≤2𝑛+1−2𝑗, the first integral is then estimated as in (A.1) for 𝑗=0. We just replace the above 2𝑛+2 by 2𝑛+2−2𝑗. We now estimate the particular solution of 𝜑^𝑗_𝑙, that is, the second integral of (26) denoted by 𝐼. For simplicity in the analysis below, we write that 𝐽=exp−𝑦−𝑦₁²4𝑥₁−exp−𝑦+𝑦₁²4𝑥₁. (A.7) Using the induction assumption at order 𝑗−1, from (A.1) for 𝑗−1, we note that for 0≤2+𝑖≤2𝑛+2−2(𝑗−1), that is, 0≤𝑖≤2𝑛+2−2𝑗, 𝜕_𝑥^2+𝑖𝜑_𝑙^𝑗−11⁻,𝑦₁𝜕=0,_𝑥2+𝑖𝜑_𝑙^𝑗−1𝑥+𝑥₁,𝑦₁𝑦≤𝜅exp−𝑐¹1−𝑥−𝑥₁𝑦≤𝜅exp−𝑐¹√.1−𝑥 (A.8)(A.9) Thanks to (A.8), differentiating 𝐼, the second integral of (26), we find that 𝜕^𝑖_𝑥1𝐼=2√𝜋₀^1−𝑥∞₀𝐽√𝑥₁𝜕_𝑥^2+𝑖𝜑_𝑙^𝑗−1𝑥+𝑥₁,𝑦₁𝑑𝑦₁𝑑𝑥₁,0≤𝑖≤2𝑛+2−2𝑗. (A.10) Since 0≤𝑥₁≤1−𝑥≤1, |𝐽|≤2exp(−(𝑦−𝑦₁)²/4𝑥₁)≤2exp(−(𝑦−𝑦₁)²/4(1−𝑥)), from (A.9) we find that for 0≤𝑖≤2𝑛+2−2𝑗, 𝜕^𝑖_𝑥𝐼≤𝜅₀^1−𝑥1√𝑥₁𝑑𝑥₁×^∞₀exp−𝑦−𝑦₁²4()𝑦1−𝑥exp−𝑐¹√1−𝑥𝑑𝑦₁≤setting𝛽=2𝑐√1−𝑥≤𝜅^∞₀−𝑦exp¹−𝑦+𝛽²4()−𝛽1−𝑥2(1−𝑥)𝛽𝑦+²4(1−𝑥)𝑑𝑦₁≤𝜅exp−𝑐𝑦√1−𝑥^∞0−𝑦exp¹−𝑦+𝛽²4(1−𝑥)𝑑𝑦₁≤𝜅exp−𝑐𝑦√.1−𝑥 (A.11) This proves (A.1) for 𝑚=0, 𝑗≥1.

For 𝑚=2(𝑘+1), 𝑘≥0, thanks to the induction assumption on 𝑗−1, from (A.1) we note that for 0≤𝑖+𝑠+2+2(𝑘−𝑠)≤2𝑛+2−2(𝑗−1), 𝜕_𝑥^𝑖+𝑠+2𝜕^{2(𝑘−𝑠)}_𝑦𝜑_𝑙^𝑗−1≤𝜅exp−𝑐𝑦√,1−𝑥 (A.12) from (A.11) we also note that 𝜕_𝑥^𝑖+𝑘+1𝐼≤𝜅exp−𝑐𝑦√,1−𝑥for0≤𝑖+𝑘+1≤2𝑛+2−2𝑗. (A.13) Since −𝐼_𝑦𝑦−𝐼_𝑥=𝜑^𝑗−1_𝑙𝑥𝑥, we thus find that 𝜕^𝑖_𝑥𝜕^2(𝑘+1)_𝑦𝐼≤𝜕_𝑥^𝑖+𝑘+1𝐼+^𝑘_𝑠=0𝜕_𝑥^𝑖+𝑠+2𝜕^{2(𝑘−𝑠)}_𝑦𝜑_𝑙^𝑗−1≤𝜅exp−𝑐𝑦√,()1−𝑥for0≤𝑖+2𝑘+1≤2𝑛+2−2𝑗. (A.14) This proves (A.1) for 𝑚=2(𝑘+1), 𝑘≥0, 𝑗≥1.

Since 𝜕_𝑦exp(−(𝑦−𝑦₁)²/4𝑥₁)=−𝜕_𝑦1exp(−(𝑦−𝑦₁)²/4𝑥₁), differentiating (A.10) in 𝑦 we observe that 𝜕^𝑖_𝑥𝜕_𝑦𝐼≤𝜅₀^1−𝑥1√𝑥₁^∞₀𝜕_𝑦𝐽⋅𝜕_𝑥^2+𝑖𝜑_𝑙^𝑗−1𝑥+𝑥₁,𝑦₁𝑑𝑦₁𝑑𝑥₁=𝜅₀^1−𝑥1√𝑥₁^∞₀𝜕𝐽_𝑦1^∼⋅𝜕_𝑥^2+𝑖𝜑_𝑙^𝑗−1𝑥+𝑥₁,𝑦₁𝑑𝑦₁𝑑𝑥₁≤integratingbypartsin𝑦₁≤𝜅₀^1−𝑥1√𝑥₁exp−𝑦²4𝑥₁𝜕_𝑥^2+𝑖𝜑_𝑙^𝑗−1𝑥+𝑥₁,0𝑑𝑥₁+𝜅₀^1−𝑥1√𝑥₁^∞₀^∼𝐽⋅𝜕_𝑥^2+𝑖𝜕_𝑦1𝜑_𝑙^𝑗−1𝑥+𝑥₁,𝑦₁𝑑𝑦₁𝑑𝑥₁, (A.15) where 𝐽^∼=exp(−(𝑦−𝑦₁)²/4𝑥₁)+exp(−(𝑦+𝑦₁)²/4𝑥₁)≤2exp(−(𝑦−𝑦₁)²/4(1−𝑥)). Here, we note that 𝜕_𝑦𝐽=−𝜕_𝑦1𝐽. Thanks to the induction assumption on 𝑗−1, from (A.1) we thus note that for 0≤2+𝑖+1≤2𝑛+2−2(𝑗−1), that is, 0≤𝑖≤2𝑛+1−2𝑗, 𝜕_𝑥^2+𝑖𝜕_𝑦1𝜑_𝑙^𝑗−1𝑥+𝑥₁,𝑦₁𝑦≤𝜅exp−𝑐¹√,1−𝑥 (A.16) and for 0≤2+𝑖≤2𝑛+2−2(𝑗−1), that is, 0≤𝑖≤2𝑛+2−2𝑗, 𝜕_𝑥^2+𝑖𝜕_𝑦1𝜑_𝑙^𝑗−1𝑥+𝑥₁,0≤𝜅. (A.17) As in (A.11) we similarly find that for 0≤𝑖≤2𝑛+1−2𝑗, 𝜕^𝑖_𝑥𝜕_𝑦𝐼≤𝜅exp−𝑐𝑦√.1−𝑥 (A.18) Hence, for 𝑚=2𝑘+1, 𝑘≥0, using the induction assumption at order 𝑗−1, from (A.1) we note that for 0≤𝑖+𝑠+2+2(𝑘−𝑠)−1≤2𝑛+2−2(𝑗−1), 𝜕_𝑥^𝑖+𝑠+2𝜕^{2(𝑘−𝑠)−1}_𝑦𝜑_𝑙^𝑗−1≤𝜅exp−𝑐𝑦√,1−𝑥 (A.19) from (A.18) we also note that 𝜕_𝑥^𝑖+𝑘𝜕_𝑦𝐼≤𝜅exp−𝑐𝑦√,1−𝑥for0≤𝑖+𝑘≤2𝑛+1−2𝑗. (A.20) Using −𝐼_𝑦𝑦−𝐼_𝑥=𝜑^𝑗−1_𝑙𝑥𝑥 again we thus find that 𝜕^𝑖_𝑥𝜕^2𝑘+1_𝑦𝐼≤𝜕_𝑥^𝑖+𝑘𝜕_𝑦𝐼+^𝑘−1_𝑠=0𝜕_𝑥^𝑖+𝑠+2𝜕^{2(𝑘−𝑠)−1}_𝑦𝜑_𝑙^𝑗−1≤𝜅exp−𝑐𝑦√1−𝑥,for0≤𝑖+2𝑘+1≤2𝑛+2−2𝑗. (A.21) This proves (A.1) for 𝑚=2𝑘+1, 𝑘≥0, 𝑗≥1. Hence, the lemma is proved.

B. Study of the Elliptic Corner Correctors 𝜉^𝑗=𝜉^𝑗_𝑙+𝜉^𝑗_𝑢

For each 0≤𝑗≤𝑛, using the stretched variables 𝑋=1−𝑥𝑦2𝜀,𝑌=,2𝜀 (B.1) one can define an approximation 𝜉^𝑗_𝑙 of 𝜉^𝑗_𝑙 as a solution of the system, −𝜕²_𝑋𝜉^𝑗_𝑙−𝜕²_𝑌𝜉^𝑗_𝑙+2𝜕_𝑋𝜉^𝑗_𝑙=0,𝜉^𝑗_𝑙=0at𝑋=0,𝜉^𝑗_𝑙=𝛾^𝑗₀(1−2𝜀𝑋)at𝑌=0,𝜉^𝑗_𝑙⟶0at𝑋²+𝑌²⟶∞,𝑌>0. (B.2) The explicit expression of 𝜉^𝑗 is given as (see [1]) 𝜉^𝑗_𝑙=𝑌𝜋^∞₀𝐾₁𝑠₁𝑠₁−𝐾₁𝑠₂𝑠₂×𝛾^𝑗₀(1−2𝜀𝑠)exp(−(𝑠−𝑋))𝑑𝑠, (B.3) where 𝐾₁ is the modified Bessel function of the second kind of the first order, and 𝑠₁=(𝑋−𝑠)²+𝑌²,𝑠₂=(𝑋+𝑠)²+𝑌². (B.4) Using an appropriate barrier function and the maximum principle (see [1]), one can verify that 𝜉^𝑗_𝑙(𝑋,𝑌)≤𝜅exp−𝑐𝑋²+𝑌²−𝑋, (B.5) for a constant 𝑐 independent of 𝜀. An approximation 𝜉^𝑗_𝑢 of 𝜉^𝑗_𝑢 can be constructed in the same manner.

Using (33) and (B.5), we notice that 𝜉^𝑗−(𝜉^𝑗_𝑙+𝜉^𝑗_𝑢)=e.s.t. at 𝑦=0,1. We deduce from (17), (18), and (B.2)₂ that 𝛾^𝑗_𝑘(1)=0, and hence 𝜉^𝑗−(𝜉^𝑗_𝑙+𝜉^𝑗_𝑢)=0 at 𝑥=1. Moreover, using Lemma B.1 below, we see that 𝜉^𝑗−(𝜉^𝑗_𝑙+𝜉^𝑗_𝑢)=0 at 𝑥=0. Then, using these observations, thanks to the maximum principle, we find that 𝜉^𝑗−𝜉^𝑗_𝑙+𝜉^𝑗_𝑢=e.s.t.,0≤𝑗≤𝑛, (B.6) which implies that the elliptic corrector 𝜉^𝑗, 0≤𝑗≤𝑛, satisfies the pointwise estimates similar to (B.5) as for its approximation 𝜉^𝑗=𝜉^𝑗_𝑙+𝜉^𝑗_𝑢.

Lemma B.1. There exists a positive constant 𝜅, independent of 𝜀, such that 𝜉^𝑗(𝑋,𝑌)≤𝜅min{𝑈(𝑋),𝑈(𝑌)},0≤𝑗≤𝑛, (B.7) where 𝑈(𝑍)=()1𝑖𝑓0≤𝑍≤𝑎,1−4𝑍−𝑎²𝑖𝑓𝑎<𝑍≤𝑏,0𝑖𝑓𝑍>𝑏, (B.8) and 𝑎=3/8𝜀, 𝑏=𝑎+(1/2).

To prove Lemma B.1, we will recall below the weak maximum principle (see, e.g., [17]).

We consider an operator 𝐿 of the form, 𝐿𝑢=^𝑚_𝑖,𝑗=1𝐷_𝑖𝑎^𝑖𝑗𝐷_𝑗𝑢+^𝑚_𝑖=1𝐷_𝑖𝑏^𝑖𝑢+𝑐^𝑖𝐷_𝑖𝑢+𝑑𝑢, (B.9) where the coefficients 𝑎^𝑖𝑗,  𝑏^𝑖,  𝑐^𝑖, and  𝑑 are assumed to be measurable functions in a domain Ω_𝑚⊂ℝ^𝑚, and 𝐷_𝑖 denotes 𝜕/𝜕𝑥_𝑖, 1≤𝑖≤𝑚.

We assume that 𝐿 is strictly elliptic in Ω_𝑚, that is, there exists 𝜆>0, such that ^𝑚_𝑖,𝑗=1𝑎^𝑖𝑗𝜉_𝑖𝜉_𝑗≥𝜆𝜉²,∀𝜉∈ℝ^𝑚, (B.10) and that 𝐿 has bounded coefficients, that is, for some constants Λ and 𝜈≥0, we have ^𝑚_𝑖,𝑗=1𝑎^𝑖𝑗2≤Λ²,𝜆^𝑚−2_𝑖=1𝑏^𝑖2+𝑐^𝑖2+𝜆⁻¹|𝑑|≤𝜈². (B.11) We also assume that _Ω𝑚𝑑𝑣−^𝑚_𝑖=1𝑏^𝑖𝐷_𝑖𝑣≤0,∀𝑣≥0,𝑣∈𝒞¹_𝑐Ω_𝑚. (B.12)

We say that a weakly differentiable function 𝑢 satisfies 𝐿𝑢=0, (≥0 or ≤0), in the weak sense, if _Ω𝑚^𝑚_𝑖,𝑗=1𝑎^𝑖𝑗𝐷_𝑗𝑢𝐷_𝑖𝑣+^𝑚_𝑖=1𝑏^𝑖𝑢𝐷_𝑖𝑣−𝑐^𝑖𝐷_𝑖()𝑢𝑣−𝑑𝑢𝑣=0,≤0or≥0 (B.13) for all nonnegative functions 𝑣∈𝒞¹_𝑐(Ω_𝑚), provided that 𝑎^𝑖𝑗𝐷_𝑗𝑢+𝑏^𝑖𝑢 and 𝑐^𝑖𝐷_𝑖𝑢+𝑑𝑢, 1≤𝑖≤𝑚, are locally integrable.

Now we state the following maximum principle.

Lemma B.2. Under the conditions (B.10), (B.11), and (B.12), if 𝑢∈𝐻¹(Ω_𝑚) satisfies 𝐿𝑢≥0, or 𝐿𝑢≤0, in the weak sense, then one has, respectively, sup_Ω𝑚𝑢≤sup_𝜕Ω𝑚𝑢⁺𝑜𝑟inf_Ω𝑚𝑢≥inf_𝜕Ω𝑚𝑢⁻. (B.14)

Thanks to Lemma B.2, we can now prove Lemma B.1.

Proof of Lemma B.1. Given 𝜀₀>0, thanks to (B.2)₄, we may choose a region Ω₀={(𝑋,𝑌)∈[0,∞)×[0,∞)|𝑋²+𝑌²≤𝑅²} with 𝑅 sufficiently large so that |𝜉^𝑗|<𝜀₀ on the circle 𝑋²+𝑌²=𝑅², 𝑋,𝑌>0. We then define the barrier function 𝑈^∼=𝑈^∼(𝑋,𝑌)=𝐶𝑈(𝑋)+𝜀₀,(𝑋,𝑌)∈[0,∞)×[0,∞), (B.15) where 𝑈(𝑋) is given in (B.8). Here, 𝐶 is a positive constant and it will be determined below. Note that 𝑈^∼(𝑋,𝑌)∈𝐻¹(Ω₀).
Going back to the elliptic problem (B.2) and writing the elliptic operator 𝐿 as 𝐿𝜉^𝑗∶=𝜉^𝑗_𝑋𝑋+𝜉^𝑗_𝑌𝑌−2𝜉^𝑗_𝑋, (B.16) we find that the operator 𝐿 satisfies (B.10), (B.11), and (B.12). We also find that for 𝑣∈𝒞¹_𝑐(Ω₀), 𝑣≥0, _Ω0𝑈^∼_𝑋𝑣_𝑋+𝑈^∼_𝑌𝑣_𝑌+2𝑈^∼_𝑋𝑣=^𝑅₀𝐶𝑈_𝑋𝑣_𝑋+2𝑈_𝑋𝑣𝑑𝑋𝑑𝑌≥0. (B.17) Indeed, if 0≤𝑋≤𝑎 or 𝑋>𝑏 (𝑎,𝑏 as in (B.8)), since 𝑈_𝑋=0, we only have to consider the case 𝑎<𝑅≤𝑏. For this 𝑅, since 𝑈∈𝐻²(Ω₀), we note that ∫^𝑅₀(𝑈_𝑋𝑣_𝑋+2𝑈_𝑋𝑣)𝑑𝑋=∫^𝑅₀(−𝑈_𝑋𝑋+2𝑈_𝑋)𝑣𝑑𝑋=∫^𝑅_𝑎(−𝑈_𝑋𝑋+2𝑈_𝑋)𝑣𝑑𝑋≥0 for all non-negative 𝑣∈𝒞¹_𝑐(Ω₀).
Since 𝐿(±𝜉^𝑗)=0, we thus find that 𝐿(𝑈^∼±𝜉^𝑗)≤0 in the weak sense.
Thanks to the cut-off function 𝛿(𝑥), choosing 𝐶=max_{𝑥∈[1/2,1]}|𝛾^∼𝑗(𝑥)|, we find that 𝑈^∼(𝑋,0)=𝐶𝑈(𝑋)+𝜀₀≥𝛾^∼𝑗(𝑥)𝜒_[1/2,1](𝑥)=𝛾^∼𝑗(1−2𝜀𝑋)𝜒_[0,1/4𝜀](𝑋)≥𝑃^𝑗(𝑋=)𝜉^𝑗(𝑋,0)_∼,𝑈(0,𝑌)=𝐶+𝜀₀≥0=𝜉^𝑗(0,𝑌)_∼,𝑈(𝑋,𝑌)≥𝜀₀>𝜉^𝑗(𝑋,𝑌)on𝜕Ω₀with𝑋²+𝑌²=𝑅²,𝑋,𝑌>0. (B.18) Thus, we note that 𝑈^∼±𝜉^𝑗≥0 on 𝜕Ω₀. Using Lemma B.2 we conclude that, for (𝑋,𝑌)∈Ω₀, 𝑈^∼±𝜉^𝑗(𝑋,𝑌)≥inf𝑈_Ω0^∼±𝜉^𝑗≥inf𝑈_𝜕Ω0^∼±𝜉^𝑗−=0. (B.19) Hence, we have 𝜉^𝑗(𝑋,𝑌)≤𝑈^∼(𝑋,𝑌)=𝐶𝑈(𝑋)+𝜀₀. (B.20) Letting 𝑅→∞, we deduce that |𝜉^𝑗(𝑋,𝑌)|≤𝐶𝑈(𝑋). Similarly, we can conclude that |𝜉^𝑗(𝑋,𝑌)|≤𝐶𝑈(𝑌). This proves the lemma.

Acknowledgments

This work was supported in part by the Research Fund of Indiana University, by NSF Grants DMS 0906440, DMS 1206438, and DMS 1212141, and by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012001167).