Point-condensation phenomena and saturation effect for the one-dimensional Gierer–Meinhardt system

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Point-condensation phenomena and saturation effect for the one-dimensional Gierer–Meinhardt system

Kotaro Morimoto

Department of Mathematics and Information Sciences, Tokyo Metropolitan University, 1-1 Minami-Ohsawa, Hachioji, Tokyo 192-0397, Japan Received 1 October 2009; received in revised form 11 December 2009; accepted 11 December 2009

Available online 7 January 2010

Abstract

In this paper, we are concerned with peak solutions to the following one-dimensional Gierer–Meinhardt system with saturation:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

0=ε²A−A+ A²

H (1+κA²)+σ, A >0, x∈(−1,1), 0=DH−H+A², H >0, x∈(−1,1), A(±1)=H(±1)=0,

whereε, D >0,κ0,σ0. The saturation effect of the activator is given by the parameterκ. We will give a sufficient condition ofκfor which point-condensation phenomena emerge. More precisely, for fixedD >0, we will show that the Gierer–Meinhardt system admits a peak solution whenεis sufficiently small under the assumption:κdepends onε, namely,κ=κ(ε), and there exists a limit lim_ε→0κε⁻²=κ₀for certainκ₀∈ [0,∞).

©2010 Elsevier Masson SAS. All rights reserved.

MSC:35K57; 35Q80; 92C15

Keywords:Gierer–Meinhardt system; Saturation effect; Pattern formation; Nonlinear elliptic system

1. Introduction

In this paper, we are concerned with the following system of ordinary differential equations:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

0=ε²A−A+ A²

H (1+κA²)+σ, A >0, x∈(−1,1), 0=DH−H+A², H >0, x∈(−1,1), A(±1)=H(±1)=0,

(1)

E-mail address:[email protected].

1 Supported by Research Fellowships of Japan Society for the Promotion of Science for Young Scientists.

0294-1449/$ – see front matter ©2010 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2010.01.003

where unknowns areA=A(x)andH=H (x).ε >0,D >0,κ0 andσ0 are constants. This system arises as a steady-state problem of the 1-dimensional Gierer–Meinhardt system with saturation which was proposed by A. Gierer and H. Meinhardt [4]. The general Gierer–Meinhardt system is written by

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

A_t=ε²A−A+ A^p

H^q(1+κA^p)+σ, A >0, x∈Ω, t >0, τ Ht=DH−H+ A^r

H^s, H >0, x∈Ω, t >0,

∂A

∂ν =∂H

∂ν =0, x∈∂Ω, t >0,

A(x,0)=A0(x), H (x,0)=H0(x), x∈Ω,

(2)

whereA=A(x, t )andH=H (x, t ),τ >0,is the Laplace operator inR^N,Ωis a bounded smooth domain inR^N, ν is the unit outer normal to∂Ω. The exponents satisfy the conditionsp >1,q, r >0,s0, and 0< (p−1)/q <

r/(s+1). The unknownsA(x, t )andH (x, t )represent the concentrations of an activator and an inhibitor, respectively, atx∈Ω and timet >0.A₀andH₀are their initial data. One of the parameters of (2),κ stands for the degree of a saturation effect to the reaction term of the activator. The termσ0 is the source term.σ represent the source rate of the activator. This system expresses some models of biological pattern formation. It is known that (2) has various kinds of striking solutions whenεis small andDis large. In particular, we are mainly interested in a solution such that the activatorAis concentrated at a finite number of points inΩ. Such a solution is called a “peak solution”. Peak solutions represent point-condensation phenomena of the activator. Whenκ=0 (no saturation case), a lot of methods to construct peak solutions were established by many mathematicians. However, whenκ >0, it is not trivial whether a peak solution also exists or not. Whenκ >0 is fixed independently ofε, due to the bistable nonlinearity, solutions with transition layers may exist. Indeed, M. del Pino [3] showed the existence of solutions with multiple layers when the domainΩ is a ball. See also [1,16,7].

We introduce the shadow system of (2). Dividing the second equation in (2) by Dand taking the limitD→ ∞ formally, we haveH=0 inΩ and ^∂H_∂ν =0 on∂Ω. This means thatH (x, t )does not depend onx, and hence we can regardH (x, t )=ξ(t ). Thus we have the following system which is called the shadow system of (2):

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

At=ε²A−A+ A^p

ξ^q(1+κA^p)+σ, A >0, x∈Ω, t >0, τ ξt= 1

|Ω|

Ω

−ξ+A^r ξ^s

dx, ξ >0, t >0,

∂A

∂ν =0, x∈∂Ω, t >0,

A(x,0)=A0(x), ξ(0)=ξ0, x∈Ω.

(3)

For this shadow system, in the case(p, q, r, s)=(2,1,2,0),σ=0,κ >0, J. Wei and M. Winter [22] showed that the shadow system (3) admits a stationary solution concentrating at one point of the boundary for sufficiently smallε, and the stability was studied. In [8], multi-boundary peak stationary solutions to (3) has been constructed for sufficiently small ε in the case whereΩ ⊂R^N is axially symmetric with respect tox_N-axis,(p, q, r, s)=(2,1,2,0),σ =0, κ >0,N 5. Moreover, multi-boundary peak stationary solutions to the original Gierer–Meinhardt system (2) was constructed near the solution to the shadow system (3) for sufficiently largeDby using the implicit function theorem.

The result was extended to the case σ >0 in [10]. In [22,8,10], it was supposed thatκ0 depends on ε, namely κ =κ(ε), and there exists a limitκε^−2N→κ0∈ [0,∞)asε→0 for certainκ0. This condition is called a “weak saturation” condition. This condition gives one of the sufficient conditions for which peak solutions appear.

The method, namely, to find a stationary solution to (2) near the stationary solution to the shadow system (3) by the implicit function theorem, is one of the methods to construct a solution to the Gierer–Meinhardt system (2), which was developed from the work by W.-M. Ni and I. Takagi [14]. In general, the number D must be large enough in the method. However, the following question arises, “forD >0 given arbitrarily, does the Gierer–Meinhardt system (2) possess a peak solution under the weak saturation condition?”. The purpose of this paper is to construct a 1-peak solution concentrating at x=0 to the 1-dimensional Gierer–Meinhardt system (1) for any fixed finiteD (which is called the strong coupling case) under the weak saturation condition.

We give remarks on other related results. For fixedκ >0, M. Mimura, M. Tabata and Y. Hosono [1] showed the existence of interior transition layers by using the singular perturbation method in the caseN =1. Y. Nishiura [15] showed that, for some 1-dimensional reaction–diffusion systems including the Gierer–Meinhardt system (2), the bifurcating branch emanating from a uniform state continues to exist until it is connected to the singularly perturbed solutions when one of the diffusion constants is sufficiently large. Multi-peak stationary solutions to (2) were first constructed by I. Takagi [17] in the caseκ=0,N=1. Moreover, its stability was discussed in [5]. In the caseκ=0 andN =1, J. Wei and M. Winter [23] studied the existence and stability of symmetric and asymmetric multi-peak stationary solutions to (2), and they showed that multi-peak stationary solutions are generated by exactly two types of peaks if the peaks are separated. In the caseκ=0 andN=2, multi-interior peak stationary solutions were constructed and the stability was discussed in [19–21]. With respect to the stability analysis for the Gierer–Meinhardt system and its shadow system, see [12,7,9], and the references therein. Some a priori estimate for a stationary solutions to (2) were given in [6,2,13]. For other results related to the Gierer–Meinhardt system, see [11,18,24] and the references therein.

Finally, we state remarks on our notation. For a domainΩ⊂R, we use standard Lebesgue spaces and Sobolev spacesL²(Ω),L^∞(Ω),H²(Ω), and so on, with the usual norm. Throughout this paper, unless otherwise stated, we use the symbolsC, C, C, c, c, cas positive constants, but they need not have the same value in each situation.

This paper is composed as follows. In Section 2, we will state our main results, Theorem 1 and Theorem 2. In Section 3, we will prepare some lemmas and state an outline of our construction of a solution. In Section 4, we will give some estimates in order to prove the theorems. In Sections 5 and 6, we will give the proofs of Theorems 1 and 2.

2. Main results

We need some preliminaries to state our main results. We introduce a solution denoted byw_δ to the following problem:

w−w+fδ(w)=0, w >0,inR, w(0)=max

y∈Rw(y), w(y)→0 as|y| → ∞, (4)

f_δ(w):= w²

1+δw². (5)

It is known that, there exists a constantδ_∗>0, the problem (4) has a unique solutionw_δ for eachδ∈ [0, δ_∗), andw_δ is radially symmetric, namely,w_δ(y)=w_δ(−y),y∈R. This fact was established in [22]. The numberδ_∗is given by

δ_∗=sup δ >0: there existsa >0 such that a 0

−t+fδ(t ) dt=0

. For fixedD >0, letG_D(x, z)be Green’s function to

DG_xx(x, z)−G(x, z)= −δ_z(x) in(−1,1),

Gx(±1, z)=0. (6)

G_D(x, z)can be written explicitly G_D(x, z)=

θ

sinh(2θ )cosh[θ (1+x)]cosh[θ (1−z)], −1< x < z,

θ

sinh(2θ )cosh[θ (1−x)]cosh[θ (1+z)], z < x <1, (7)

whereθ:=D^−1/2. We put α_D:= 1

GD(0,0). (8)

Moreover, the non-smooth part ofG_D(x, z)is given by K_D

|x−z|

= 1 2√

De⁻

√1

D|x−z|. (9)

LetH_D(x, z)be the regular part ofG_D(x, z), G_D(x, z)=K_D

|x−z|

−H_D(x, z).

H_D(x, z)isC^∞in bothxandz.

Next, we prepare a cut-off function. Let χ∈C₀^∞(R)be a function such that, 0χ1,χ (x)=0 for|x|<1, χ (x)=1 for|x|>2. Letr₀be a fixed constant such that 0< r₀<1/2, for example,r₀=1/10. We will use a cut-off function in the formχ (_r^x

0). Note thatχ (_r^x

0)=0 for|x|>2r0. We suppose the following assumption on the constantκ in (1).

(A) κ0 depends onε, and there exists a limit

εlim→0κε⁻²=κ₀ (10)

for someκ0∈ [0,∞).

Let us state our main results. We first state a result in the caseσ=0.

Theorem 1.Letσ=0. FixD >0arbitrarily. We suppose(A), and let the valueκ0α²_Dbe sufficiently small. Then, for sufficiently smallε >0,(1)admits a1-peak radially symmetric solution(Aε(x), Hε(x))such thatAε(x)concentrates atx=0. More precisely, there existsδε∈ [0, δ∗)for eachεsufficiently small such thatδε→δ0asε→0for some δ0∈ [0, δ∗)which is decided byκ0andDand satisfies

δ₀

R

w_δ²

0(y) dy 2

=κ₀α²_D, (11)

andA_εtakes the form:

Aε(x)= 1 ε

Rw²_δ

ε

αDwδ_ε

x ε

χ

x r₀

+εφε

x ε

, x∈(−1,1), (12)

where α_D is defined by(8),w_δ is the unique solution to(4), andφ_ε(y)is a radially symmetric function onΩ_ε:=

(−¹_ε,¹_ε)such that

φε H²(Ωε)C (13)

holds for some constantC >0independent ofε.Hεhas the following property:

H_ε(0)= 1 ε

Rw²_δ

ε

α_D+O(ε)

asε→0. (14)

Next, we state a result in the caseσ=0.

Theorem 2.Letσ >0. We assume the same assumption onκas in Theorem1. Then,(1)admits a radially symmetric solution providedεis sufficiently small. More precisely, if we fixσ >0andγ∈(0,1/2), there existsεˆ₁>0such that, for all ε∈(0,εˆ₁)andσ ∈(0, σ ),(1)admits a radially symmetric solution(A_ε,σ(x), H_ε,σ(x)), andA_ε,σ takes the form:

A_ε,σ(x)= 1 ε

Rw_δε

α_Dwδ_ε

x ε

χ

x r₀

+εφ_ε

x ε

+ε^γφε,σ

x ε

+σ, x∈(−1,1), (15) whereδ_εandφ_εare given in Theorem1, andφ_ε,σ(y)is a radially symmetric function onΩ_εsuch that

φ_{ε,σ H}2(Ω_ε)σ (16)

holds, andH_ε,σ satisfies

H_ε,σ(0)= 1 ε

Rw_δ²

ε

α_D+O(ε)+ σ+σ²

O ε^γ

, (17)

asε→0, whereO(ε)andO(ε^γ)are independent ofσ.

Remark 3.The setting of the domain(−1,1)is not essential. For givenk∈N, if we construct a 1-peak solution to (1) on smaller domain in advance, then we can obtain ak-peak symmetric solution to (1) by reflections.

Remark 4.The assumption “κ0α²_Dis sufficiently small” in Theorem 1 is due to some technical reason. See Remark 14 stated later.

3. Basic analysis and preliminaries

In this section, we prepare some lemmas to prove Theorem 1 and state the outline of our construction. We first define some function spaces as follows:

L²_r(R):=

u∈L²(R): u(x)=u(−x), x∈R

, (18)

H_r²(R):=H²(R)∩L²_r(R), (19)

and for a domain(−a, a),a∈(0,∞), L²_r(−a, a):=

u∈L²(−a, a): u(x)=u(−x), x∈(−a, a)

, (20)

H_r²(−a, a):=H²(−a, a)∩L²_r(−a, a), (21)

H_r,ν² (−a, a):=

u∈H_r²(−a, a): u(±a)=0

. (22)

Because we will frequently use rescaling, we introduce the following notations.

Definition 5.PutΩ_ε:=(−¹_ε,¹_ε).

For a functionu:(−1,1)→R, letu(y):=u(εy),y∈Ω_ε.

Inversely, for a functionv:Ω_ε→R, letv(x):=v(^x_ε),x∈(−1,1).

3.1. Basic analysis

For the unique solutionwδto (4), let us state some known facts. After that, we state some new lemmas.

Lemma 6.For eachδ∈ [0, δ_∗), the unique radially symmetric solutionw_δhas the following properties:

(i) w_δ∈C^∞(R).

(ii) Let

Lδ:= d²

dx²−1+f_δ(wδ):H²(R)→L²(R),

wheref_δ(w_δ)=2wδ/(1+δw_δ²). Then,Ker(Lδ)=span{w_δ}.

(iii) If we restrict the domain toDom(Lδ)=H_r²(R), thenL_δhas a bounded inverseL⁻_δ¹:L²_r(R)→H_r²(R).

(iv) If we fixδ∈(0, δ_∗), then there exist constantsC, c >0such that w_δ(y),

dⁿw_δ dyⁿ (y)

Ce^−c|y|, y∈R, n=1,2, (23) holds for anyδ∈ [0, δ].

Proof. (i)–(iii) have been proven in Lemma 2.2 of [22]. (iv) have been proven in Lemma 2.4 of [8]. 2 We state continuity and differentiability ofw_δonδ.

Lemma 7.As aC¹(R)-valued function ofδ,w_δsatisfies the following:

(i) w_δis continuous inδ∈ [0, δ_∗)with respect to theC¹(R)-norm.

(ii) w_δis of classC¹((0, δ_∗), C¹(R)).

Proof. This fact was proven in Lemma 2.3 of [22] (see also Lemma 2.3 of [8]). 2

Let us denote the derivatives of w_δ inx and inδ by w_δ(x)and ^dw_dδ^δ, respectively. Next, we state some useful formulae.

Lemma 8.The following identities hold:

L_δw_δ=f_δ(w_δ)w_δ−f_δ(w_δ), (24)

L_δdw_δ

dδ =f_δ²(w_δ), (25)

Lδ

wδ+2δdw_δ dδ +1

2y·w_δ

=wδ, (26)

Lδ

wδ+2δdw_δ dδ

=fδ(wδ). (27)

Proof. These facts were proven in Lemma 2.3 of [22]. 2

Lemma 9.w_δ→binC_loc² (R)holds asδ→δ_∗, whereb >0is the second positive root of−t+f_δ∗(t )=0,t∈R.

Proof. This fact was proven in Lemma 2.3 of [22]. 2 Lemma 10.For anyδ∈(0, δ_∗), it holds that

d dδ

∞

−∞

w²_δ(y) dy

>0. (28)

Proof. This fact was proven in Lemma 2.6 of [22]. 2

Lemma 11.For fixedδ∈(0, δ_∗), there exists constantC >0such that dw_δ

dδ

H²(R)

C (29)

holds for anyδ∈(0, δ).

Proof. It is easy to see thatL⁻_δ¹is bounded uniformly inδ∈ [0, δ]. By using (25) and Lemma 6(iv), we can estimate by some constantsC, C>0 independent ofδ∈ [0, δ]as follows:

dwδ

dδ

H²(R)

=L⁻_δ¹f_δ²(w_δ)

H²(R)Cf_δ²(w_δ)

L²(R)C. (30)

Hence we complete the proof. 2 Lemma 12.

(i) For eachδ∈ [0, δ_∗), ifφ∈H_r²(R)satisfies the following:

φ−φ+f_δ(wδ)φ−γ

Rw_δφ

Rw_δ² fδ(wδ)=0 inR, (31)

γ=

Rw²_δ

Rw²_δ+2δ

Rw_δ^dw_dδ^δ, (32)

thenφ=0.

(ii) There existsδ₁∈(0, δ_∗)such that, forδ∈ [0, δ1), ifφ∈H_r²(R)satisfies the following:

φ−φ+f_δ(w_δ)φ−γ

Rfδ(wδ)φ

Rw²_δ w_δ=0 inR, (33)

γ=

Rw²_δ

RL⁻_δ¹(w_δ)f_δ(w_δ), (34)

thenφ=0.

Before the proof, we state some remarks. Lemma 10 implies that

Rw_δ^dw_dδ^δ >0 for anyδ∈(0, δ_∗). Hence, we first notice that

0<

Rw_δ²

Rw²_δ+2δ

Rwδdwδ

dδ

1, δ∈ [0, δ_∗). (35)

Secondly, we consider the value of

RL⁻_δ¹(w_δ)f_δ(w_δ). By using (26) and integration by parts, we have

δlim→0

R

L⁻_δ¹(w_δ)f_δ(w_δ)=lim

δ→0

R

w_δ(y)+2δdw_δ dδ (y)+1

2y·w_δ(y)

f_δ w_δ(y)

dy

=

R

w₀³(y)+1

2y·w₀(y)w₀²(y)

dy

=

R

w₀³(y)−1 6w³₀(y)

dy

=5 6

R

w₀³(y) dy >0. (36)

Here, we note thatδ

Rdwδ

dδ f_δ(w_δ) dy→0 asδ→0 by Lemma 11. Moreover, we see that w₀−w₀+w²₀=0,

R

w₀w₀−

R

w₀²+

R

w³₀=0,

R

w₀2

+

R

w²₀=

R

w₀³. (37)

Therefore,

Rw₀³>

Rw²₀. Thus we have

Rw²_δ

RL⁻_δ¹(w_δ)f_δ(w_δ)

δ=0

=

Rw₀²

5 6

Rw³₀ <6

5. (38)

Proof. (i) By using (27), the equation (31) can be written as follows:

Lδφ=γ

Rw_δφ

Rw_δ² fδ(wδ), φ=γ

Rw_δφ

Rw²_δ L⁻_δ¹ fδ(wδ)

,

R

w_δφ=γ

Rwδφ

Rw²_δ

R

w²_δ+2δ

R

w_δdw_δ dδ

. Hence,

Rw_δφ=0 must be hold by (32). Thus we haveL_δφ=0,φ∈H_r²(R), and henceφ=0 by Lemma 6(iii).

(ii) Defineδ₁by δ₁:=sup

δ∈(0, δ_∗):

R

L⁻_δ¹(w_δ)f_δ(w_δ) >0 forδ∈(0, δ)

. (39)

Thisδ₁is well defined by (36). Then we can prove by the same argument as in the proof of (i). 2 Now, we define an operatorLδonL²(R)with Dom(Lδ)=H²(R)by

Lδφ=φ−φ+f_δ(w_δ)φ−2

Rw_δφ

Rw_δ² f_δ(w_δ). (40)

Its conjugate operator is given by L^∗δψ=ψ−ψ+f_δ(w_δ)ψ−2

Rfδ(wδ)ψ

Rw²_δ w_δ, ψ∈H²(R). (41)

Let us defineδ2by δ2=sup

δ∈(0, δ1):

Rw²_δ

RL⁻_δ¹(w_δ)f_δ(w_δ)<2 forδ∈(0, δ)

, (42)

whereδ1is defined by (39). Thisδ2is well defined by (38).

Lemma 13.For the operatorsLδandL^∗_δ, andδ2defined above, there hold that (i) Ker(Lδ)∩H_r²(R)= {0}for anyδ∈ [0, δ_∗),

(ii) Ker(L^∗_δ)∩H_r²(R)= {0}for anyδ∈ [0, δ2).

Proof. This lemma is a consequence of Lemma 12. 2

Remark 14.We do not know whether Ker(L^∗_δ)∩H_r²(R)is trivial or not forδ nearδ_∗. If Ker(L^∗_δ)∩H_r²(R)= {0} holds for allδ∈ [0, δ_∗), then we can remove the assumption “κ0α_D² is sufficiently small” in Theorem 1. However, it seems to be a difficult problem.

3.2. Outline of our construction

We state an outline of our construction. We see by Lemmas 9–11 that there exists uniqueδ_ε∈ [0, δ_∗)such that δε

R

w²_δ

ε

2

=κε⁻²α²_D (43)

holds for eachε >0. By the assumption (A), in the limitε→0, there hold that δ_ε→δ₀, δ₀

R

w_δ²

0

2

=κ₀α²_D, (44)

as ε→0, for someδ0∈ [0, δ_∗). We assume henceforth that κ0α²_D0 is small enough so thatδ0∈ [0, δ2), where δ₂ is given by (42). Then we note that there existsδ∈(0, δ2)such thatδ_ε ∈ [0, δ] holds for allε >0 sufficiently small. Hence, we may assume thatc <

Rw_δ²

ε(y) dy < Cholds for allεsufficiently small, the constantsc, C >0 are independent ofε.

Put

c_ε:= 1 ε

Rw_δ²

ε

. (45)

We consider the following problem foraandh:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

ε²a−a+ a²

h(1+δ_εα_D⁻²a²)+σ_ε=0, a >0, x∈(−1,1), Dh−h+cεa²=0, h >0, x∈(−1,1), a(±1)=h(±1)=0,

(46)

where σ_ε:= σ

c_ε. (47)

If we obtain a solution to (46), then we obtain a solution to (1) by puttingA(x)=c_εa(x)andH (x)=c_εh(x). For U∈H_r,ν² (Ω_ε), letT[U]be a unique solution to the following problem forv:

Dv−v+cεU²=0, x∈(−1,1),

v(±1)=0. (48)

Here, the under-bar and over-bar notation is due to Definition 5. Moreover, we put S[U](y):=U(y)−U (y)+ U²(y)

T[U](y)(1+δ_εα_D⁻²U²(y)), y∈Ω_ε. (49) If we can findU∈H_r,ν² (Ωε)such that,S[U] +σε=0, U >0 inΩε, then we obtain a solution to (46) by putting a(x)=U (x)andh(x)=T[U](x).

Here, we note thatT[U]is written by using Green’s function as follows:

T[U](x)=c_ε 1

−1

G_D(x, z)U²(z) dz, x∈(−1,1), (50)

forU∈L²(Ω_ε). In particular,T[U]is radially symmetric providedU is radially symmetric. Now, let us define an approximate functionw_εas follows:

w_ε(x):=α_Dw_δε x

ε

χ x

r₀

, (51)

whereα_D=G_D(0,0)⁻¹,w_δε is the unique solution to (4) forδ=δ_ε,χis the cut-off function defined in the previous section. We will first consider the case σ =0 and prove Theorem 1 in Section 5. For the purpose, we will seek U∈H_r,ν² (Ω_ε)such thatS[U] =0,U >0 inΩ_εin the formU (y)=w_ε(y)+εφ(y)for someφ∈H_r,ν² (Ω_ε). Next, we will consider the caseσ=0 in Section 6. Note that(σ_ε=)σ/c_εCσ εholds for some constantC >0 independent ofεsufficiently small. Therefore, we can prove Theorem 2 by a perturbation argument.

4. Basic estimates

In this section, we show some basic estimates.

Lemma 15.There existsc₁>0such thatT[w_ε](x)c₁,x∈(−1,1), for allεsufficiently small.

Proof.

T[w_ε](x)=c_ε 1

−1

G_D(x, z)w²_ε(z) dx

=α²_Dc_ε 1

−1

G_D(x, z)w²_δ

ε

z ε

χ² z

r0

dz

=α_D²εc_ε 1/ε

−1/ε

G_D(x, εz)w²_δ

ε(z)χ² ε

r0

z

dz

α_D²

Rw_δ²

ε

θ sinh(2θ )

1/ε

−1/ε

w²_δ

ε(z)χ² ε

r₀z

dz

= α²_D

Rw_δ²

0

θ sinh(2θ )

∞

−∞

w²_δ

0(z) dz+o(1),

asε→0, whereo(1)is uniform inx∈(−1,1). This estimate completes the proof. 2 Next, we show the following elementary inequality.

Lemma 16.For the non-smooth partK_D(|x−z|)ofG_D(x, z), the following estimate holds:

K_D

|x|

−K_D

|y| 1 2√ D

1

√D|x| − |y|+1 2

1

√D 2

|x|²+ |y|²

. (52)

Proof. This lemma is easily verified by (9) and the following elementary inequality:

1− |x|e^−|x|1− |x| +1 2|x|². Thus, we omit the details. 2

Lemma 17.Forw_εdefined by(51), it holds that

T[wε](0)=αD+O(ε), (53)

asε→0.

Proof. Note that T[w_ε](0)=c_ε

1

−1

G_D(0, z)w_ε²(z) dz= α_D²

Rw_δ²

ε

1/ε

−1/ε

G_D(0, εz)w_δ²_ε(z)χ² ε

r0

z

dz, and the following inequality holds:

|z|<^r_ε⁰

G_D(0, εz)w_δ²_ε(z) dz 1/ε

−1/ε

G_D(0, εz)w²_δε(z)χ² ε

r₀z

dz

|z|<^2r_ε⁰

G_D(0, εz)w_δ²_ε(z) dz. (54)

The left-hand side of (54) is written as follows:

(l.h.s.)=G_D(0,0)

|z|<^r_ε⁰

w_δ²

ε(z) dz+

|z|<^r_ε⁰

G_D(0, εz)−G_D(0,0) w²_δ

ε(z) dz≡I+II.

Moreover, notingα_D⁻¹=GD(0,0), we can estimate by Lemma 6(iv) so that I=α⁻_D¹

R

w²_δ

ε(z) dz−

|z|>^r_ε⁰

w_δ²

ε(z) dz

=α_D⁻¹

R

w²_δ

ε(z) dz+e.s.t., (55)

where “e.s.t.” means “exponentially small term”. Next, we can estimate by Lemma 16 and the mean value theorem as follows:

|II|

|z|<^r_ε⁰

K_D ε|z|

−K_D(0)w_δ²

ε(z) dz+

|z|<^r_ε⁰

H_D(0, εz)−H_D(0,0)w_δ²

ε(z) dz

C

|z|<^r_ε⁰

ε|z|w²_δ

ε(z) dz Cε.

Here, we note that,ε²|z|²< ε|z|r₀for|z|< r₀/ε,

R|z|w²_δ

ε(z) dzis bounded uniformly inεsufficiently small since we may assumeδε∈ [0, δ]and we can apply Lemma 6(iv). Hence

l.h.s. of (54)

=α⁻_D¹

R

wδ_ε+O(ε). (56)

We can see that the right-hand side of (54) have the same behavior as (56). Thus we have T[w_ε](0)= α_D²

Rw_δ²

ε

α_D⁻¹

R

w²_δ

ε(z) dz+O(ε)

=α_D+O(ε).

Thus we complete the proof. 2

Lemma 18.For some constantC >0, the following estimate holds:

T[w_ε](εy)−T[w_ε](0)C

ε|y| +ε

, y∈Ω_ε, (57)

for allεsufficiently small.

Proof.

T[wε](εy)−T[wε](0)=cε

1

−1

GD(εy, z)−GD(0, z)

w_ε²(z) dz

= α²_D

Rw²_δ

ε

1/ε

−1/ε

G_D(εy, εz)−G_D(0, εz) w²_δ

ε(z)χ² ε

r₀z

dz

= α²_D

Rw²_δ

ε

1/ε

−1/ε

KD

ε|y−z|

−KD

ε|z|

w²_δε(z)χ² ε

r₀z

dz

− 1/ε

−1/ε

HD(εy, εz)−HD(0, εz) w_δ²

ε(z)χ² ε

r₀z

dz

. Now, by Lemma 16, and notingε|z|1 for|z|1/ε, the following estimate holds:

K_D

ε|y−z|

−K_D

ε|z|Cε|y| − |y−z|+ε²

|y−z|²+ |z|² Cε

|y| + |z|

, y, z∈Ω_ε, for someC, C>0 independent ofε,yandz. Moreover, we can estimate by the Maclaurin expansion as follows:

HD(εy, εz)−HD(0, εz)Cε

|y| + |z|

, y, z∈Ωε, for someC>0 independent ofε,yandz. Thus we have

T[w_ε](εy)−T[w_ε](0)ε α_D²

Rw_δ²

ε

C+C ^1/ε

−1/ε

|y| + |z| w_δ²

ε(z) dzC

ε|y| +ε

, y∈Ω_ε, for someC>0 independent ofεandy. Thus we complete the proof. 2

Lemma 19.There existsC₁>0such that S[w_ε]

L²(Ωε)C₁ε, (58)

for allεsufficiently small.

Proof. It is easily to see thatw_ε−w_ε= −f_δε(w_δε)α_D+e.s.t. inL²(Ω_ε)asε→0. Hence, S[w_ε](y)= −f_δε(w_δε)α_D+ 1

T[w_ε](y)

w²_ε(y)

1+δ_εα⁻_D²w²_ε(y)+e.s.t.

= −f_δε(w_δε)α_D+ 1 T[wε](y)

α²_Dw²_δ

ε(y)χ²(_r^ε

0y) 1+δ_εw_δ²

ε(y)χ²(_r^ε

0y)+e.s.t.

= −f_δε(w_δε)α_D+ α²_D

T[w_ε]f_δε(w_δε)+e.s.t. inL²(Ω_ε).

By Lemma 17, we have

−fδ_ε(wδ_ε)αD+ α²_D

T[wε]fδ_ε(wδ_ε)=fδ_ε(wδ_ε)αD

−1+ α_D

T[w_ε](0)+ α_D

T[wε](y)− α_D T[w_ε](0)

=f_δε(w_δε)α_D

O(ε)+ α_D

T[w_ε](εy)T[w_ε](0)

T[w_ε](0)−T[w_ε](εy) . Moreover, by Lemma 18, the following estimate holds:

f_δ

ε(w_δε)

T[w_ε](0)−T[w_ε](εy)²

L²(Ωε)=

Ωε

w⁴_δ

ε

(1+δ_εw²_δ

ε)²

T[w_ε](0)−T[w_ε](εy)2

dy

C

Ωε

w⁴_δ

ε(y)

ε|y| +ε2

dyCε²,

for some constantsC, C>0 independent ofεsufficiently small. From these estimates and by Lemma 15, we have a conclusion. 2

Next, we give the derivatives ofT andS. The proofs of Lemmas 20, 21 below are uninteresting calculation. So we give their proofs in Appendix A.

Lemma 20. If we regard T as a mapping form L²(−1,1) into L^∞(−1,1), then T is Fréchet differentiable on L²(−1,1), and its derivative atu∈L²(−1,1)is given by

T[u]φ=2cε

1

−1

GD(x, z)u(z)φ(z) dz, φ∈L²(−1,1). (59)

Moreover, for some constantC >0independent ofεsufficiently small, the following estimates hold:

T[u+h] −T[u] −T[u]h

L∞(Ω_ε)C h ²

L²(Ωε), (60)

T[u]h

L^∞(Ω_ε)C u _L2(Ω_ε) h _L2(Ω_ε), (61)

for anyu, h∈L²(Ω_ε).