www.elsevier.com/locate/anihpc
Transition layer for the heterogeneous Allen–Cahn equation
Fethi Mahmoudi
a, Andrea Malchiodi
a,∗, Juncheng Wei
baSISSA, via Beirut 2-4, 34014 Trieste, Italy
bDepartment of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong Received 28 February 2007; accepted 28 March 2007
Available online 15 February 2008
Abstract
We consider the equation ε2u=
u−a(x) u2−1
inΩ, ∂u
∂ν=0 on∂Ω, (1)
whereΩis a smooth and bounded domain inRn,νthe outer unit normal to∂Ω, andaa smooth function satisfying−1< a(x) <1 inΩ. We setK,Ω+andΩ−to be respectively the zero-level set ofa,{a >0}and{a <0}. Assuming∇a=0 onKanda=0 on∂Ω, we show that there exists a sequenceεj→0 such that Eq. (1) has a solutionuεj which converges uniformly to±1 on the compact sets ofΩ±asj→ +∞. This result settles in general dimension a conjecture posed in [P. Fife, M.W. Greenlee, Interior transition layers of elliptic boundary value problem with a small parameter, Russian Math. Surveys 29 (4) (1974) 103–131], proved in [M. del Pino, M. Kowalczyk, J. Wei, Resonance and interior layers in an inhomogeneous phase transition model, SIAM J. Math.
Anal. 38 (5) (2007) 1542–1564] only forn=2.
©2008 Elsevier Masson SAS. All rights reserved.
MSC:35J25; 35J40; 35B34; 35B40
Keywords:Fife–Greenlee problem; Heterogeneous Allen–Cahn equation; Interior transition layers; Spectral gaps
1. Introduction
Given a smooth bounded domainΩofRn(n2), we consider the following problem ε2u=h(x, u) inΩ,
∂u
∂ν =0 on∂Ω, (2)
whereεis a small parameter,ν the unit outer normal vector to∂Ω andha smooth function such that the equation h(x, t )=0 admits two different stable solutionst1=t2for anyx∈Ω. Using matched asymptotics, Fife and Greenlee in [19] proved under some hypothesis onhthe existence of a solution of (2) which converges uniformly toti in the compact subsets ofΩi,i=1,2, whereΩ1andΩ2are two subdomains ofΩ such thatΩ=Ω1∪Ω2.
* Corresponding author.
E-mail addresses:[email protected] (F. Mahmoudi), [email protected] (A. Malchiodi), [email protected] (J. Wei).
0294-1449/$ – see front matter ©2008 Elsevier Masson SAS. All rights reserved.
doi:10.1016/j.anihpc.2007.03.008
In this paper we consider themodelheterogeneous caseh(x, u)=(u−a(x))(u2−1), for a smooth function a satisfying−1< a(x) <1 onΩ and∇a=0 on the setK= {a(x)=0}, withK∩∂Ω= ∅. We prove the existence of a new type of solution of (2) for anyn2 settling in full generality a result previously proved in [15] for the particular casen=2.
Let us describe the result in more detail: in the caseh(x, u)=(u−a(x))(u2−1)problem (2) becomes ε2u=(u−a(x))(u2−1) inΩ,
∂u
∂ν =0 on∂Ω. (3)
In particular, whena≡0, (3) is nothing but the Allen–Cahn equation in material sciences (see [6]) ε2u+u−u3=0 inΩ,
∂u
∂ν =0 on∂Ω. (4)
Here the function u(x)represents a continuous realization of the phase present in a material confined to the region Ω at the pointx. Of particular interest are the solutions which, except for a narrow region, take values close to+1 or −1. Such solutions are calledtransition layers, and have been studied by many authors, see for instance [4,8,20, 23,24,30–32,34,35,37–40,43], and the references therein for these and related issues.
In this paper, we are interested in transition layers for theheterogeneousequation (3). Define K=
x∈Ω: a(x)=0 .
We assume thatKis a smooth closed hypersurface ofΩ which separates the domain into two disjoint components
Ω=Ω−∪K∪Ω+, (5)
with
a(x) <0 inΩ−, a(x) >0 inΩ+, ∇a=0 onK. (6)
We then define the Euler functionalJε(u)associated to (3) inΩ as Jε(u)=ε2
2
Ω
|∇u|2+
Ω
F (x, u) dx, (7)
where
F (x, u):=
u
−1
s−a(x) s2−1
ds.
The solution constructed by Fife and Greenlee in [19] (adapted to our choice of the functionh) consists in adding an interior transition layer correction to expressions of the formti+εti1+ε2ti2, which approximate the solutionuin the regionsΩi (notice that with our choice of the functionh, we haveΩ1=Ω+,Ω2=Ω−,t1≡ −1 andt2≡1). This allowed Fife and Greenlee to construct an approximationUε which yields an exact solution to (11) using a classical implicit function argument. No restrictions onεare required, and the solution satisfies
uε→ −1 inΩ+ and uε→1 inΩ− asε→0. (8)
Super-subsolutions were later used by Angenent, Mallet-Paret and Peletier in the one-dimensional case (see [7]) for construction and classification of stable solutions. Radial solutions were found variationally by Alikakos and Simpson in [5]. These results were extended by del Pino in [12] for general (even non-smooth) interfaces in any dimension, and further constructions have been done recently by Dancer and Yan [11] and Do Nascimento [16]. In particular, it was proved in [11] that solutions with the asymptotic behavior like (8) are typically minimizer ofJε. Related results can be found in [1,2].
On the other hand, a solution exhibiting a transition layer in theopposite direction, namely
uε→ +1 inΩ+, uε→ −1 onΩ− asε→0 (9)
has been believed to exist for many years. Hale and Sakamoto [21] established the existence of this type of solution in the one-dimensional case, while this was done for the radial case in [13], see also [10]. The layer with the asymptotics
in (9) in this scalar problem is meaningful in describing pattern-formation for reaction–diffusion systems such as Gierer–Meinhardt with saturation, see [13,18,36,41,42] and the references therein.
For one-dimensional or radial problems it is possible to use finite-dimensional reductions, which basically consist in determining the location of the transition layer. In this kind of approach, the same technique works for both the asymptotic behaviors in (8) and (9): the only difference is the sign of the small eigenvalue (of orderε) arising from the approximate degeneracy of the equation (when we tilt the solutions perpendicularly to the interface). This makes the former solution stable and the latter unstable.
On the other hand, one faces a dramatically different situation in higher-dimensional, non-symmetric cases. This is clearly seen already linearizing around a spherically symmetric solution of (1) (with profile as in (9)), as bifurcations of non-radial solutions along certain infinite discrete set of values forε→0 take place, as established by Sakamoto in [42]. This reveals that the radial solution has Morse index which changes withε(precisely diverges asε→0, as shown in [17]). This poses a serious difficulty for a general construction. A phenomenon of this type was previously observed in the one-dimensional case by Alikakos, Bates and Fusco [3] in finding solutions with any prescribed Morse index.
In [15], del Pino, Kowalczyk and the third author considered the two-dimensional case, constructing transition layer solutions with asymptotics as in (9), while in this paper we extend that result to any dimension. Our main theorem is the following.
Theorem 1.1.Let Ω be a smooth bounded domain of Rn (n2)and assume that a:Ω →(−1,1) is a smooth function. DefineK,Ω+andΩ−to be respectively the zero-set, the positive set and the negative set ofa. Assume that
∇a=0onKand thatK∩∂Ω= ∅. Then there exists a sequenceεj→0such that problem(3)has a solutionuεj which approach1inΩ+and−1inΩ−. Precisely, parameterizing a pointx nearKbyx=(y,¯ ζ ), with¯ y¯∈Kand ζ¯=d(x, K)(with sign, positive inΩ+),uεj admits the following behavior
uεj(y,¯ ζ )¯ =H ζ¯
εj +Φ(y)¯ +O(εj) asj→ +∞.
HereΦ is a smooth function defined onKandH (ζ )is the unique hetheroclinic solution of
H+H−H3=0, H (0)=0, H (±∞)= ±1. (10)
As in [14,15,26–28,31] and other results for singularly perturbed (or geometric) problems, the existence is proved only along a sequenceεj→0 (actually it can be obtained forεin a sequence of intervals(aj, bj)approaching zero, but not for any smallε). This is caused by a resonance phenomenon we are going to discuss below, explaining the ideas of the proof.
To describe the reasons which cause the main difficulty in proving Theorem 1.1, we first scale problem (3) using the change of variablex→εx, so Eq. (3) becomes
u=(u−a(εx))(u2−1) inΩε,
∂u
∂ν =0 on∂Ωε, (11)
whereΩε=1εΩ. Near the hypersurfaceKε:=1εK, we can choose scaled coordinates(y, ζ )inΩεwithy∈Kε and ζ =dist(x, Kε)(with sign), see Subsection 2.2, and we letu˜ε denote the scaling ofuε toΩε: with these notations we have thatu˜ε(y, ζ )=uε(y, εζ )H (ζ ). The functionH (ζ )=H (dist(x, Kε))forx∈Ωε can be then considered as a first order approximate solutionto (11), so it is natural to use local inversion arguments near this function in order to find true solutions. For this purpose it is necessary to understand the spectrum of the linearization of (11) at approximate solutions.
LettingLε be the linearization of (11) atu˜ε, it turns out thatLεadmits a sequence of small positive eigenvalues of orderε. Using asymptotic expansions (see Section 3, and in particular formula (72)), one can see that this family behaves qualitatively likeε−ε2λj, where theλj’s are the eigenvalues of the Laplace–Beltrami operator ofK. By the Weyl’s asymptotic formula, we have thatλj jn−12 as j → +∞, therefore we have an increasing number of positive eigenvalues, many of which accumulate to zero and sometimes, depending on the value ofε, we even have the presence of a kernel: this clearly causes difficulties if one wants to apply local inversion arguments. Notice that,
by the above qualitative formula, the average spectral gap of resonant eigenvalues is of orderεn+21. For the casen=2 (considered in [15]) this gap is relatively large, so it was possible to show invertibility using direct estimates on the eigenvalues. However in higher dimension this is not possible anymore, and one needs to apply different arguments.
To overcome this problem, we use an approach introduced in [28,29] (see also [25–27]) to handle similar resonance phenomena for another class of singularly perturbed equations. The main idea consists in looking at the eigenvalues (of the linearized problem) as functions of the parameter ε, and estimate their derivatives with respect to ε. This can be rigorously done employing a classical theorem due to T. Kato, see Proposition 3.3, and by characterizing the eigenfunctions corresponding to resonant modes. Using this result we get invertibility along a suitable sequence εj →0, and the norm of the inverse operator along this sequence has an upper bound of orderε−
n+1 2
j (consistently with the above heuristic evaluation of the spectral gaps). This loss of uniform bounds asj→ +∞should be expected, since more and more eigenvalues are accumulating near zero. However, we are able to deal with this further difficulty by choosing approximate solutions with a sufficiently high accuracy.
Fixing an integerk1 and using the coordinates introduced after (11), from the fact thata vanishes onK, one can consider the Taylor expansion
a(εy, εζ )=εζ b(εy)+ k l=2
(εζ )lbl(εy)+ ˜b(y, ζ ) withb(y, ζ )˜ Ck|εζ|k+1, and look at an approximate solution of the form
uk,ε(y, ζ )=H
ζ−Φ(εy) +
k i=1
εihi
εy, ζ−Φ(εy) ,
for a smooth functionΦ(εy)=Φ0(εy)+k−1
i=1εiΦi(εy)defined onKand some correctionshi defined onK×R+. Using similar Taylor expansions of the Laplace–Beltrami operator in the above coordinates, see Section 2.2, the couple (hj, Φj−1)forj1 can be determined via equations of the form
L0h1= −κ(εy)H(s)+(s+Φ0)b(εy)(1−H2(s)),
L0hj=Φj−1b(εy)(1−H2(s))+Fk(s, Φ0, . . . , Φj−2, h1, . . . , hj−1, b1, . . . , bj), forj2, (12) whereL0u=u+(1−3H2)u,Fkis a smooth function of its arguments, ands=ζ−Φ(εy). (12) is always solvable inhj by the Fredholm alternative if we choose properly the functionsΦl.
Such an accurate approximate solution allows us, using the above characterization of the spectrum of the linearized operator and the bound on its inverse, to apply the contraction mapping theorem and find true solutions. Specifically for the homogeneous Allen–Cahn equation, a related method was used in [31] to study the effect of∂Ωon the structure of solutions to (4). Some common arguments are here simplified, and we believe our approach could also be used to handle general non-linearities as in [19].
The paper is organized in the following way: in Section 2 we collect some preliminary results concerning the profileH, we expand the Euclidean metric and the Laplace–Beltrami operator in suitable coordinates nearKε, and recall some well-known spectral results. In Section 3 we first construct approximate solutions, and then derive some spectral properties of the linearized operator characterizing the resonant eigenfunctions: this is a crucial step to apply Kato’s theorem. Finally, Section 4 is devoted to the proof of our main result.
2. Notation and preliminaries
In this section we first collect some notation and conventions. Then, we list some properties of the hetheroclinic solutionH, and we expand the metric and the Laplace–Beltrami operator in a local normal coordinates. Finally we recall some results in spectral theory like the Weyl asymptotic formula.
Notation and conventionWe shall always use the convention that capital letters likeA, B, . . .will vary between 1 and n, while indices likei, j, . . . will run between 1 andn−1. We adopt the standard geometric convention of summing over repeated indices.
(y1, . . . , yn−1) will denote coordinates inRn−1, and they will also be written asy=(y1, . . . , yn−1), while coordi- nates inRnwill be writtenx=(y, ζ )∈Rn−1×R.
The hypersurface K will be parameterized with local coordinates y¯=(y¯1, . . . , yn−1). It will be convenient to define its dilationKε:=1εKwhich will be parameterized by coordinates(y1, . . . , yn−1)related to they¯’s simply by
¯ y=εy.
Derivatives with respect to the variablesy,¯ yorζ will be denoted by∂y¯,∂y,∂ζ and for brevity we shall sometimes use the notations∂i for∂yi. When dealing with functions depending just on the variable ζ we will writeH, h, . . . instead of∂ζH, ∂ζh, . . ..
In a local system of coordinates,g¯ij are the components of the metric onKnaturally induced byRn−1. Similarly,
¯
gAB are the entries of the metric onΩ in a neighborhood of the hypersurfaceK.κij will denote the components of the mean curvature operator ofKintoRn−1.
For a real positive variablerand an integerm, O(rm)(resp. o(rm)) will denote a function for which|O(rrmm)|remains bounded (resp.|o(rrmm)|tends to zero) whenrtends to zero. For brevity, we might also write O(1)(resp. o(1)) for a quantity which stays bounded (resp. tends to zero) asεtends to zero.
2.1. Some analytic properties of the hetheroclinic solutionH
In this subsection we collect some useful properties of the hetheroclinic solutionH to (10). Note first thatH can be explicitly determined by
H (ζ )=tanh √
2
2 ζ , (13)
and moreover the following estimates hold
⎧⎪
⎨
⎪⎩
H (ζ )−1= −A0e−√2|ζ|+O(e−(2√2)|ζ|) forζ → +∞;
H (ζ )+1=A0e−√2|ζ|+O(e−(2√2)|ζ|) forζ → −∞;
H(ζ )=√
2A0e−√2|ζ|+O(e−(2√2)|ζ|) for|ζ| → +∞,
(14)
whereA0>0 is a fixed constant. We have the following well-known result (we refer to Lemma 4.1 in [33] for the proof).
Lemma 2.1.Consider the following eigenvalue problem φ+
1−3H2
φ=Λφ, φ∈H1(R). (15)
Then, letting Λj be the eigenvalues arranged in non-increasing order (counted with multiplicity) and φj be the corresponding eigenfunctions, one has that
Λ1=0, φ1=cH; Λ2<0. (16)
As a consequence(by Fredholm’s alternative), given any functionψ∈L2(R)satisfying
Rψ H=0, the following problem has a unique solutionφ
φ+
1−3H2
φ=ψ inR,
R
Hφ=0. (17)
Furthermore, there exists a positive constantCsuch thatφH1(R)CψL2(R). We collect next some useful formulas: first of all we notice that
H= 1
√2
1−H2
and H= −√
2H H. (18)
Moreover, setting L0u=u+
1−3H2
u, (19)
we have that
L0(H H)= −3√
2H (H)2. (20)
2.2. Geometric background
In this subsection we expand the coefficients of the metric in local normal coordinates. We then derive as a conse- quence an expansion for the Laplace–Beltrami operator. First of all, it is convenient to scale by 1ε the coordinates in Eq. (3) to obtain
u=(u−a(εx))(u2−1) inΩε,
∂u
∂ν =0 on∂Ωε, (21)
where we have setΩε=1εΩ. Following the same notation we also setKε=1εK, and forγ∈(0,1)we define Sε=
x∈Ωε: dist(x, Kε) < ε−γ
; Iε=
−ε−γ, ε−γ .
We parameterize elements x∈Sε using their closest pointy inKε and their distanceζ (with sign, positive in the dilation ofΩ+). Precisely, we choose a system of coordinatesy¯ onK, and denote byn(y)¯ the (unique) unit normal vector to K (at the point with coordinates y) pointing towards¯ Ω−. Choosing also coordinatesy on Kε such that
¯
y=εy, we define the diffeomorphismΓε:Kε×Iε→Sεby
Γε(y, ζ )=y+ζn(εy). (22)
We let the upper-case indicesA, B, C, . . .run from 1 ton, and the lower-case indicesi, j, l, . . .run from 1 ton−1.
Using some local coordinates(yi)i=1,...,n−1onKε, and lettingϕε be the corresponding immersion intoRn, we have
⎧⎨
⎩
∂Γε
∂yi(y, ζ )=∂ϕ∂yεi(y)+εζ∂y∂n
i(εy)=∂ϕ∂yεi(y)+εζ κij(εy)∂ϕ∂yε
j(y), fori=1, . . . , n−1;
∂Γε
∂ζ (y, ζ )=n(εy),
where(κij)are the coefficients of the mean-curvature operator onK. Let alsog¯ij be the coefficients of the metric on Kεin the above coordinatesy. Then, lettingg=gεdenote the metric onΩεinduced byRn, we have
gAB= ∂Γε
∂xA,∂Γε
∂xB =
(gij) 0
0 1 , (23)
where gij=
∂ϕε
∂yi(y)+εζ κik(εy)∂ϕε
∂xk(y),∂ϕε
∂yj(y)+εζ κjl(εy)∂ϕε
∂xl(y)
= ¯gij+εζ
κikg¯kj +κjlg¯il
+ε2ζ2κikκjlg¯kl. Note that also the inverse matrix{gAB}decomposes as
gAB=
(gij) 0
0 1 .
From the above decomposition ofgAB (andgAB) and forudefined onSε, one has gu=gABuAB+ 1
√detg∂A gAB
detg uB
=uζ ζ+gijuij+ 1
√detg∂ζ(
detg )uζ+ 1
√detg∂i gij
detg
uj. (24)
We have, formally detg=det
¯ g−1g
detg¯=(detg)¯
1+εζtr
¯ g−1α
+o(ε), (25)
where
αij=κikg¯kj +κjlg¯il. There holds
g¯−1α
is= ¯gsjαij= ¯gsj
κikg¯kj+κjlg¯il ,
and hence tr
¯ g−1α
= ¯gij
κikg¯kj+κjlg¯il
=2g¯ijκikg¯kj=2κii. (26) We recall that the quantityκii represents the mean curvature ofK, and in particular it is independent of the choice of coordinates.
We note that the metricgAB can be expressed in function of the metricg¯ij, the operatorκji, and the variableεζ. Hence, fixing an integerkand using a Taylor expansion, we can write
√1
detg∂ζ detg=
k =1
εζ−1G˜(εy)+ ˜G(εy, ζ ), (27)
whereG˜:K→Rare smooth functions, andG˜ satisfies G(˜ ·, ζ )
Cm(K)Ck,m|ζ|kεk+1, ζ∈Iε, (28)
whereCk,mis a constant depending only onK,k, andm. Again (and in the following), when we write · we keep the variableζ fixed. In particular, from the above computations it follows that
G˜1(εy)=κ(εy):=κii(εy). (29)
We need now a similar expansion for the operatorg: fixing the variableζ∈Iε, the metricg(y, ζ )=gε(y, ζ )induces a metricgˆε,ζ onKin the following way. Consider the homothetyiε:K→Kε. We definegˆε,ζ to be
ˆ
gε,ζ=ε2iε∗gε(·, ζ ),
whereiε∗ denotes the pull-back operator. Basically, we are freezing the variableζ and lettingy vary. Fixing an in- tegerk, for any smooth functionv:K→Rwe have the expansion below, which follows from (24), reasoning as for (27)
gˆε,ζv= k =0
(εζ )Lv+ ˜Lε,k+1v=Kv+ k =1
(εζ )Lv+εk+1L˜ε,k+1v. (30) Here{Li}i,L˜ε,k+1are linear second-order differential operators acting onyand satisfying
LivCm(K)CmvCm+2(K); ˜Lε,k+1vCm(K)Cm|ζ|k+1vCm+2(K) (31) for all smoothv, whereCmis a constant depending only onK,k, andm.
Consider now a functionu:Sε→Rof the form
u(y, ζ )= ˜u(εy, ζ ), y∈Kε, ζ∈Iε. (32)
Then, scaling in the variabley, we have gεu(y, ζ )= ˜uζ ζ(εy, ζ )+ 1
√detg∂ζ(
detg )u˜ζ(εy, ζ )+ε2gˆε,ζu(εy, ζ ).˜ Using the expansions (27), (30) together with (29), the latter equation becomes
gεu(y, ζ )= ˜uζ ζ(εy, ζ )+
εκ+ k =2
εζ−1G˜
˜
uζ(εy, ζ )+ ˜G(εy, ζ )u˜ζ(εy, ζ )
+ε2Ku˜+ k =1
ε2+ζLu(εy, ζ )˜ +εk+3L˜ε,k+1u(εy, ζ ).˜ (33)
2.3. Spectral analysis
We define the scaled Euler functionalJε(u)inΩεby Jε(u)=1
2
Ωε
|∇u|2+
Ωε
F (εx, u) dx, withF (x, u):=
u
−1
s−a(x) s2−1
ds. (34)
We set for brevity
b(y):=∂na(y,0) (35)
and we notice that by our choice ofn, we haveb >0 onK. Now, we letϕj andλj be the eigenfunctions and the eigenvalues (with weightb) of
−Kϕj=λjb(y)ϕ¯ j.
Theλj’s can be obtained for example using the Rayleigh quotient: precisely ifMjdenotes the family ofj-dimensional subspaces ofH1(K), then one has
λj= inf
M∈Mj
sup
ϕ∈M, ϕ=0
K|∇Kϕ|2
Kb(y)ϕ¯ 2 = sup
M∈Mj−1
ϕ⊥M, ϕinf=0
K|∇Kϕ|2
Kb(y)ϕ¯ 2, (36)
where⊥denotes the orthogonality with respect to theL2scalar product with weightb. We can estimate theλjusing a standard Weyl’s asymptotic formula [9], one has
λjCK,bjn−21 asj → +∞, (37)
for some constantCK,bdepending only onKandb.
3. Asymptotic analysis
This section is devoted to the construction of approximate solutions to (21), and of approximate eigenfunctions (and eigenvalues) in theζ component (see the coordinates introduced in (22)) of the relative linearized equation. Then we characterize, via Fourier analysis, the profile of resonant eigenfunctions in both the variablesyandζ.
3.1. Approximate solutions and eigenfunctions
In this section, given any integerk1, we construct an approximate solutionuk,εto problem (21), which solves the equation up to an error of orderεk+1. Using the above parametrization(y, ζ )inSε, we make the following ansatz
uk,ε(y, ζ )=H
ζ−Φ(εy) +
k i=1
εihi
εy, ζ−Φ(εy)
inSε, (38)
whereHis the hetheroclinic solution of (10) and whereΦ(εy)=Φ0(εy)+k−1
i=1εiΦi(εy)for some smooth functions Φj defined onK. The correctionshi andΦi are to be constructed recursively in the indexi, depending on the Taylor expansion ofa and the geometry ofK. Since all thehi’s will turn out to have an exponential decay inζ,uk,εcan be easily extended (via some cutoff functions) to an approximate solution in the wholeΩε, see (102) below.
We first determineh1by solving the equation up to an error of orderε2. To this aim, we expand the functionain powers ofεas (notice thata(εy,0)≡0)
a(εy, εζ )=εb(εy)ζ+ k =2
(εζ )b(εy)+ ˜b(y, z), (39)
whereb(y, z)˜ is smooth and satisfies b(y, z)˜ Ck|εz|k+1.
Using the above expansion of the metric coefficients and the Laplace–Beltrami operator, see in particular (33), setting s=ζ−Φ, we obtain that the term (formally) of orderεin the equation is identically zero if and only if the correction h1satisfies
L0h1:=(h1)ss+
1−3H (s)2
h1= −κ(εy)H(s)+(s+Φ0)b(εy)
1−H2(s)
. (40)
By the asymptotics in (14), the right-hand side is of classL2inRand, by Lemma 2.1, (40) is solvable provided the latter is orthogonal inL2to the functionH(s). SinceH(s)is even insand sinceb(εy) >0, this is possible choosing Φ0(εy)so that
b(εy)Φ0(εy)=κ(εy)
+∞
−∞(H)2(s) ds +∞
−∞ H(s)(1−H2(s)) ds =
√2
3 κ(εy). (41)
Moreover one can prove, using standard ODE estimates, thath1has the following (regularity properties and) decay at infinity
∂sl∂ymh1(εy, s)Cmεm 1+ |s|
e−
√2|s|, l=0,1,2, m=0,1,2, . . . , (42) whereCmdepends only onm,aandK.
To obtain the other correctionsΦi andhi one can proceed by induction, assuming thatN2, thatΦ0, . . . , ΦN−2 andh1, . . . , hN−1have been determined, and that(hi)iN−1satisfy
∂sl∂ymhi(εy, s)Cmεm
1+ |s|di e−
√2|s|, iN−1, l=0,1,2, m=0,1,2, . . . , (43) whereCmdepends only onm,a,Kanddi only oni. When we expand Eq. (21) foru=uN,εin power series ofε, the couple(hN, ΦN−1)can be found reasoning as for(h1, Φ0): indeed, considering the coefficient ofεNin this expansion, one can easily see thathNsatisfies an equation on the form
L0hN=ΦN−1b(εy)
1−H2(s)
+FN(s, Φ0, . . . , ΦN−2, h1, . . . , hN−1, b1, . . . , bN), (44) whereFNis a smooth function of its argument. Reasoning as forh1this equation is solvable provided the right-hand side isL2-orthogonal to the functionH(s). This is indeed true choosingΦN−1so that
b(εy)ΦN−1(εy)=−+∞
−∞ H(s)FN(s, . . .) ds +∞
−∞ H(s)(1−H2(s)) ds.
Furthermore, one can show thathN satisfies regularity and decay estimates as in (43). Reasoning as in Section 3 of [29] one can check that the above formal estimates can be made rigorous, and that the exponential decay of the corrections yields the following result.
Proposition 3.1.Given any integerk1there exist a functionuk,ε:Sε→Rwhich solves Eq.(21)up to an error of orderεk+1. Precisely, setting
Sε(u)=u−
u−a(εx) u2−1
, (45)
there exist a polynomialPk(ζ )such that Sε
uk,ε(εy, ζ )εk+1Pk(ζ )e−
√2|ζ| inSε. (46)
Moreover, the following estimate holds ∂sl∂ymuk,ε(εy, ζ )CmεmPk(ζ )e−
√2|ζ|, l=0,1,2, m=0,1,2, . . . , (47) whereCmis a constant depending only onm,aandK.
We will look at solutionsuof (21) as small corrections ofuk,ε(suitably extended toΩε via some cutoffs inζ, see (102) below), namely of the form
u=uk,ε+w,
forwsmall in a sense to be specified later. The equationSε(u)=0 is then equivalent to Lε(w)+Sε(uk,ε)+Nε(w)=0,
whereLεis nothing but the linearized operator at the approximate solutionuk,ε
Lεw:=gεw+
1−3u2k,ε
w+2a(εx)uk,εw, (48)
and whereNε is the remainder given by non-linear terms inSε, namely Nε(w):= −
3uk,ε−a(εx)
w2−w3. (49)
It is also convenient to define the following linear operator Lεw:=wζ ζ+ 1
√detg∂ζ(
detg )wζ+
1−3u2k,ε
w+2a(εy, ζ )uk,εw. (50)
In particular, using the expansion (33), the operatorsLεandLεare related by the following formula: settingw(y, ζ )=
˜
w(εy, ζ )one has
Lεw=Lεw˜ +ε2Kw˜ +ε3Lˆ3,εw,˜ (51)
whereLˆ3,εconsists of the last two terms in (33) (replacinguwithw). Precisely,Lˆ3,εis a linear differential operator of second order acting on the variablesy, which for every integer¯ msatisfies
ˆL3,εvCm(K)P (ζ )ˆ vCm+2(K). (52)
HereP (ζ )ˆ is a polynomial inζ with fixed degree, and coefficients depending only onm.
We want next to derive some formal estimates on the following eigenvalue problem
Lεv=μv inIε, (53)
with zero Dirichlet boundary conditions. It follows from Lemma 2.1 that the eigenvalues either stay bounded away from zero, or converge to zero asε→0: we are interested in the latter case. We argue heuristically expanding (53) at first order inε. In the limitε→0, we haveμ=0 with corresponding eigenfunctionH, therefore it is natural to look for approximate eigenfunctions of the form
Ψ =H+εH1, (54)
and eigenvaluesμ=εμ¯ +o(ε). We impose thatH1is orthogonal toHinL2(R). Therefore the approximate eigen- value equation formally becomes
L0H1= −2b(εy)(s+Φ0)H H−Hκ(εy)+6H Hh1+ ¯μH+o(1). (55) As for (40), solvability is guaranteed provided the right-hand side is orthogonal inL2toH. Using the oddness ofH, formulas (18), (20), (40) and the self-adjointness ofL0we find that orthogonality is equivalent to
¯
μ=4b(εy)
RsH (s)(H(s))2ds
R(H(s))2ds =√ 2b(εy).
With this choice ofμ, the function¯ H1is defined as the unique solution of L0H1= −2b(εy)(s+Φ0)H H−Hκ(εy)+6H Hh1+√
2b(εy)H.
From the exponential decay ofHandh1(see (42)) we deduce thatH1satisfies estimates similar to (43).
Using this fact and the rigorous expansions in (27), (39), we then derive the following estimate
Lε(Ψ )=εμH¯ +ε2Rε(εy, ζ ), (56)
where the error termRεsatisfies Rε(εy, ζ )P (ζ )e−
√2|ζ| (57)
for some polynomialP (ζ ). Also, from the regularity and the decay ofH1we have ∂sl∂ymΨ (εy, s)P (ζ )εm+1e−
√2|ζ|, l=0,1,2, m=0,1,2, . . . . (58)
3.2. Characterization of resonant eigenfunctions
We characterize next the eigenfunctions ofLε, see (48), corresponding to small eigenvalues. Let us recall first the definition ofϕj andλj in Section 2.3.
Lemma 3.2.Letλε=O(ε2)be an eigenvalue of the linearized operatorLεinSεwith eigenfunctionφand weightb, namely
Lεφ=λεbφ inSε
(and with zero Dirichlet boundary conditions). Let us write the eigenfunctionφas φ(y, ζ )=ϕ(εy)Ψ (εy, ζ )+φ⊥(y, ζ ),
withΨ defined in(54)and withφ⊥satisfying the following orthogonality condition(we are freezing they variables in the volume element)
Iε
Ψ (εy, ζ )φ⊥(y, ζ ) dVgε(ζ )=0 for everyy∈Kε. (59)
Then one hasφ⊥H1(Sε)=o(ε)φH1(Sε)asεtends to zero.
Proof. We notice first that, sinceΨ =H+o(1)inH1(R), the operatorL0is negative definite onφ⊥by Lemma 2.1.
Therefore, using the estimates on the metricgεin Section 2, we find easily that there exist a constantC >0 such that
Sε
φ⊥(y, ζ )Lεφ⊥(y, ζ ) dVgε(y, ζ )−1 Cφ⊥2
H1(Sε). (60)
Let us write the eigenvalue equationLεφ=λεbφas Lεφ⊥= −Lε(ϕΨ )+λεbφ⊥+λεbϕΨ.
Multiplying byφ⊥, integrating overSεand using (59) we obtain
Sε
φ⊥(y, ζ )Lεφ⊥(y, ζ ) dVgε= −
Sε
φ⊥(y, ζ )Lε
ϕ(εy)Ψ (εy, ζ )
dVgε+λε
Sε
b(εy)φ⊥(y, ζ )2dVgε.
By (60) (and the smallness ofλε) it then follows φ⊥2
H1(Sε)C
Sε
φ⊥(y, ζ )Lε
ϕ(εy)Ψ (εy, ζ ) dVgε
. (61)
Now by (51) and (56) we can writeLε(ϕΨ )as Lε(ϕΨ )=Lε(ϕΨ )+ε2K(ϕΨ )+ε3Lˆ3,ε(ϕΨ )
=ϕ ε√
2b(εy)H+ε2Rε(εy, ζ )
+ε2K(ϕΨ )+ε3Lˆ3,ε(ϕΨ ). (62) Then, again by (59), we have
Sε
φ⊥(y, ζ )Lεϕ(εy)Ψ (εy, ζ )ε2
Sε
φ⊥(y, ζ )Kϕ(εy)Ψ (εy, ζ ) +
Sε
ε2R˜ε(εy, ζ )ϕ(εy)+ε3Lˆ3,ε
ϕ(εy)Ψ (εy, ζ ) φ⊥
, (63)
