PHASE TRANSITION LAYERS FOR FIFE-GREENLEE PROBLEM ON SMOOTH BOUNDED DOMAIN

. We consider the Fife-Greenlee problem where Ω is a bounded domain in R 2 with smooth boundary, (cid:15) > 0 is a small parameter, ν denotes the unit outward normal of ∂ Ω. Let Γ = { y ∈ Ω : a ( y ) = 0 } be a simple smooth curve intersecting orthogonally with ∂ Ω at exactly two points and dividing Ω into two disjoint nonempty components. We assume that − 1 < a ( y ) < 1 on Ω and (cid:79) a (cid:54) = 0 on Γ, and also some admissibility conditions between the curves Γ, ∂ Ω and the inhomogeneity a hold at the connecting points. We can prove that there exists a solution u (cid:15) such that: as (cid:15) → 0, u (cid:15) approaches +1 in one part, while tends to − 1 in the other part, except a small neighborhood of Γ.

where Ω is a bounded domain in R 2 with smooth boundary, > 0 is a small parameter, ν denotes the unit outward normal of ∂Ω. Let Γ = {y ∈ Ω : a(y) = 0} be a simple smooth curve intersecting orthogonally with ∂Ω at exactly two points and dividing Ω into two disjoint nonempty components. We assume that −1 < a(y) < 1 on Ω and a = 0 on Γ, and also some admissibility conditions between the curves Γ, ∂Ω and the inhomogeneity a hold at the connecting points. We can prove that there exists a solution u such that: as → 0, u approaches +1 in one part, while tends to −1 in the other part, except a small neighborhood of Γ.
1. Introduction. We consider the Fife-Greenlee problem 2 u + u − a(y) (1 − u 2 ) = 0 in Ω, ∂u ∂ν = 0 on ∂Ω, (1.1) where Ω is a smooth and bounded domain in R N , −1 < a(y) < 1 on Ω, is a small positive parameter, ν is the unit outer normal to ∂Ω. The function u represents a continuous realization of the phase present in a material confined to the region Ω at the point y which, except for a narrow region, is expected to take values close to +1 or −1. Of interest are of course non-trivial state configurations in which the antiphases coexist.
for which extensive literature on transition layer solution is available, see for instance [3,4,8,21,22,24,25,28,29,30,31,33,34,35,36,39], and the references therein for these and related issues. We first mention some works on problem (1.1). Let us assume thatΓ = {y ∈ Ω : a(y) = 0} is a simple, closed and smooth curve in Ω ⊂ R 2 which separates the domain into two disjoint components such that a(y) < 0 in Ω + , a(y) > 0 in Ω − , ∂a ∂ν 0 > 0 onΓ, (1.4) where ν 0 is the outer normal of ∂Ω + , pointing to the interior of Ω − . By matching asymptotic and bifurcation arguments, Fife and Greenlee [20] constructed a solution u to problem (1.1) with the properties u → +1 in Ω + and u → −1 in Ω − as → 0. (1.5) 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 [11] for general (even non smooth) interfaces in any dimension, and further constructions have been done by Dancer and Yan [10] and Do Nascimento [17]. In particular, it was proved in [10] that solutions with the asymptotic behavior like (1.5) are typically minimizer of the related Euler functional J . Related results can be found in [1], [2]. On the other hand, a solution exhibiting a transition layer in the opposite direction, namely u → −1 in Ω + and u → +1 in Ω − as → 0, (1.6) has been believed to exist for many years. Hale and Sakamoto [23] established the existence of this type of solution in the one-dimensional case, while this was done for the radial case in [12], see also [9]. The layer with the asymptotics in (1.6) in this scalar problem is meaningful in describing pattern-formation for reactiondiffusion systems such as Gierer-Meinhardt with saturation, see [12,19,32,37] and the references therein. Recently this problem has been completely solved by del Pino-Kowalczyk-Wei [14] (in the two dimensional domain case) and Mahmoudi-Malchiodi-Wei [27] (in the higher dimensional case), see also [18].

1.2.
Motivations and results. Note that the results, mentioned in the above, concerned the existence of interior phase transition phenomena (away from ∂Ω) for problem (1.1). On the other hand, phase transition layers, connecting ∂Ω, were found for Allen-Cahn problem (1.2). See [39], [25], [38], [15] and the references therein. Hence, in the present paper, we will construct the phase transition layers connecting ∂Ω for (1.1) on two dimensional smooth bounded domain. It will be shown that the inhomogeneous term a as well as the boundary of Ω will play an important role in the procedure of the construction. This is the reason for the requirement of the admissibility conditions between Γ and ∂Ω, which will be stated in (1.7). Whence we can use suitable methods to decompose the interaction between the layer, the boundary and also the inhomogeneous term a, in such a way that we can construct good local approximate solution and then use the reduction method to get the solutions. More precisely, in the present paper, for the existence of phase transition layers connecting ∂Ω, we consider problem (1.1) and make the following assumptions: (A1). Let Γ = {y ∈ Ω ⊂ R 2 : a(y) = 0} be a simple smooth curve, which is intersecting ∂Ω at exactly two points, saying P 1 , P 2 , and, at these points Γ⊥∂Ω. In the small neighborhoods of P 1 , P 2 , the boundary ∂Ω are two curves, say C 1 and C 2 , which can be represented by the graphs of two functions respectively y 2 = ϕ 1 (y 1 ) with (0, ϕ 1 (0)) = P 1 , Without loss of generality, we can assume Γ has length 1, and then denote k 1 , k 2 , k the signed curvatures of C 1 , C 2 and Γ respectively. (A2). Γ separates the domain into two disjoint nonempty components, where ν 0 is the outer normal of ∂Ω + , pointing to the interior of Ω − . (A4). We also assume the validity of the admissibility conditions where τ is the unit tangent vector of Γ, k denote the derivative of the signed curvatures k of Γ. Remark 1.1. An example will be provided to show the validity of the assumptions (A1)-(A4). We can choose a(y) = y 1 − 1 2 , Ω, Ω + , Ω − and Γ as the following , As Γ is a straight line, the signed curvature k of Γ is zero. Moreover, the derivative of the signed curvature k is zero too. Therefore, the assumption (A4) as well as (A1)-(A3) holds. Let be the unique heteroclinic solution of It is well known that H is odd and enjoys the following behavior , as x → +∞, where A 0 is a universal constant. It is trivial to derive that . (1.9) A constant λ * is defined by (1.10) Our main result is as follows.
Let Ω be a smooth and bounded domain in R 2 . If the assumptions (A1)-(A4) hold, then for given c 0 > 0 there exists 0 such that for all < 0 satisfying the gap condition problem (1.1) has a solution u with the asymptotics where t is the normal coordinate directed along ν 0 , θ is the natural arc length parameter of the curve Γ and f is a bounded function of θ as in (4.4).
Remark 1.2. As in [14] and [27], to deal with the resonance phenomenon, the phase transition layers with asymptotic behavior in (1.12) can be constructed whenever is small and away from the critical numbers λ * j 2 , in the sense that the gap condition (1.11) holds, i.e. (1.14) On the other hand, if the assumptions (A1)-(A4) hold, we can also construct solutionsũ to problem (1.1) with asymptotical behavior where t is the normal coordinate directed along ν 0 andf is also a bounded function of θ. The profile ofũ will be given in Remark 4.1. In this case, the problem does not have resonance phenomenon, see Remark 6.1.
In [39], [25], [38], [15], the phase transition layers connecting the boundary ∂Ω were found for Allen-Cahn equation in (1.2). On the other hand, in the present paper, the new ingredients are the admissibility conditions in (A4) to decompose the interaction among the term a, the phase transition layer and the boundary ∂Ω, see (4.11). In fact, this can be done in local coordinate system, called modified Fermi coordinates from [40] with little bit delicate analysis.
Here are some words for further discussions and the organization of the paper. For the convenience of expression, by the rescaling y = ỹ (1.16) in R 2 , problem (1.1) can be rewritten as where Ω = Ω/ , ν is the unit outer normal of ∂Ω . We also denote the curve Γ = Γ/ and then write down the local form of (1.17), especially the differential operators in Section 2.2. The outline of the proof will be given in Section 3. Gluing procedure from [13] will be applied to deduce the projected form of (1.17), see (3.46)-(3.49). Section 4 is devoted to the constructing of a local approximate solution in such a way that it solves the nonlinear problem locally up to order O( 2 ). To get a real solution, the well-known infinite dimensional reduction method [13] will be needed in Sections 5-6. Note that we also need suitable analysis from [14] to deal with the resonance phenomena in Lemma 6.1.

2.1.
Recalling the coordinates. Recall the assumptions (A1)-(A4) in Section 1 and the notation therein. For basic notions of curves, such as the signed curvature of a plane curve, the reader can refer to the book by do Carmo [16]. In this section, for the convenience of readers, we recall the coordinate system in the neighborhood of Γ from [40], called modified Fermi coordinates.
We choose the normal n of Γ with the same direction of ν 0 , the outer normal of ∂Ω + . There holds the Frenet formula where k is the signed curvature of Γ. Choosing δ 0 > 0 very small, and setting we construct the following mapping Note that H is a diffeomorphism (locally) and H(0,θ) = γ(θ).
This transformation will straighten up the curvesC 1 andC 2 .
Step 3. We define the modified Fermi coordinates for the given small positive constants σ 0 and δ 0 . More precisely, For convenience's sake, in the following, we also denote and so that The asymptotic expressions of this coordinate system will be given by the following basic facts.

Local forms of the operators.
With the aid of the local coordinates (t, θ) in (2.1), the local form of the differential operator ∆ in (1.1) are given in (B.4)-(B.6) of [40], i.e., (2.5) In the above formulas, :=< q 1 , γ >= Θ tt (0, θ), (2.6) and a 1 , · · · , a 5 are smooth functions of the variable t. On the other hand, the local expression of ∂/∂ν can be written in (B.7)-(B.9) of [40]. More precisely, for θ = 0, 8) and the constants b 1 and b 2 are given by On the other hand, for θ = 1, In the above, the functions σ 3 , · · · , σ 6 are smooth functions of t with the properties By recalling the change of coordinates y = ỹ in (1.16), it is useful to locally introduce change of variables It is readily checked that (2.14) (2.15) Similarly, the normal derivative ∂/∂ν can be derived from (2.7) and (2.10). Indeed, if z = 0, it becomes the following boundary operator whereD 0 0 = D 0 0 . And, at z = 1/ , it has the form 3. Outline of the proof. In this section, the strategy to prove Theorem 1.1 will be provided step by step.
3.1. The gluing procedure. Consider any given approximate solution H (to be chosen later in (4.20)) with the properties: where s is the normal coordinate to Γ , 0 and δ are small constants. For a perturbation term Φ defined in Ω , the function u(ỹ) = H(ỹ) + Φ(ỹ) satisfies (1.17) if and only if We further separate Φ in the following form where s is the normal coordinate to Γ . In the above formula, the cut-off function is defined as is also a smooth cut-off function defined as η δ (t) = 1 for 0 ≤ t ≤ δ and η δ (t) = 0 for t > 2δ, (3.5) for any fixed number δ < δ 0 /100, where δ 0 is a constant defined in (2.1). With this definition, Φ is a solution of (3.3)-(3.4) if the pair (φ,ψ) satisfies the following coupled system: and First, given a smallφ, we solve problems (3.8)-(3.9) forψ. The problem has a unique bounded solutionψ due to (3.1) and (3.2), whenever h ∞ < +∞. Moreover, Assume now thatφ satisfies the following decay property where is a very small constant. Note that N has a power-like behavior with power greater than one. A direct application of contraction mapping principle yields that (3.8)-(3.9) has a unique (small) solutionψ =ψ(φ) with where |s| > δ/ denotes the complement of Ω of δ/ -neighborhood of Γ . Moreover, the nonlinear operatorψ satisfies a Lipschitz condition of the form After solving (3.8)-(3.9), we can concern (3.6)-(3.7) as a local nonlinear problem involvingψ =ψ(φ), which can be solved in local coordinates. This is due to the fact that we can decompose the interaction among the boundary and the phase transition layer, and then construct a good approximate solution and also derive the resolution theory of the nonlinear problem by delicate analysis. This is called the gluing procedure in [13].

3.2.
Local formulation of the problem. As described in the above, the next step is to consider (3.6)-(3.7) in the neighbourhood of Γ so that by the relatioñ y = y/ in (1.16) close to Γ , the variables y can be represented by the modified Fermi coordinates (t, θ) in (2.1), which have been prepared in Section 2.
More precisely, in the coordinates (s, z) given by (2.13), the equation in (3.6) becomes where the linear operator is with the expressions ofB 0 andB 1 given in (2.15) and (2.14). The error is now locally expressed in the form The boundary condition in (3.7) can also be expressed precisely in the local coordinates. If z = 0, Similarly, at z = 1/ there holds Introduce the following change of variables for any functionw where f is a parameter to be chosen in (4.4), and then take the Taylor expansion where a 6 (t, θ) is a smooth function. In order to express problem (3.14) with boundary conditions (3.17)-(3.18) in terms of these new coordinates, we derive the relationsw By the above change of coordinates and Taylor expansion, we computẽ This gives a new linear operatorL. The operators are given by and In the coordinates (x, z), the boundary conditions in (3.17) and (3.18) can be recast as follows. For z = 0, and and (3.27) As a conclusion, it is derived that equations (3.6)-(3.7) become, in local coordinates (x, z),  The boundary components of S are One of the left jobs is to find the local form, say v 2 in (4.19), of the approximate solution H. This will be given in Section 4, see (4.19) and (4.20). As we have done for the equation (3.14), E can be locally recast in (x, z) coordinate system by the relation . (3.37) Moreover, the boundary errors can be expressed in coordinates (x, z) as follows. For z = 0, there holds and also for z = 1/ , we have The exact forms of the error terms E, g 0 and g 1 will be given in (4.29) and (4.33). It is of importance that (3.36), (3.38) and (3.39) hold only in a small neighbourhood of Γ . Hence we will make extensions and consider v 2 , E as functions of the variables x and z on S, and also g 0 , g 1 on ∂ 0 S and ∂ 1 S in the sequel. Moreover, the unknown parameter f (θ) in (3.19) will be chosen in the form, (cf. (4.4)) where f 0 (θ) is given in (4.4) and f (θ) is a new parameter satisfying (4.5). Now define an operator on the whole strip S in the form and also the operators 43) where χ(r) is a smooth cut-off function which equals 1 for 0 ≤ r < 10δ that vanishes identically for r > 20δ. For the local form of the nonlinear part, we have (3.44) by the notation where in the right hand side the term ψ is transformed fromψ by the relation (3.19). Rather than solving problem (3.28)-(3.30) directly, we deal with the following projected problem: for each f satisfies the constraint (4.5), finding functions φ ∈ H 2 (S), c( z) ∈ L 2 (0, 1) and constants l 0 , l 1 such that We will prove that this problem has a unique solution φ so thatφ satisfies the constraint (3.11) due to the relation given the change of variables in (3.19). The result reads Moreover φ depends Lipschitz-continuously on f in the sense of estimate Proof. The proof is similar as that for Proposition 5.1 in [14].
After this has been done, our task is to adjust the parameter f such that the function c( z) and the constants l 1 , l 0 are zero, see (5.1)-(5.3). By the estimates in Section 5, it is equivalent to solving a nonlinear second-order differential equation for f with suitable boundary conditions, see (6.1)-(6.2). In Section 6, by using the gap condition (1.11) and delicate analysis we will prove that the reduced problem is solvable. 4. Local approximate solutions. The main objective of this section is to construct the approximate solution and then evaluate its error terms E, g 0 and g 1 in the coordinate system (x, z).

The first approximation solution.
Recall the function H in (1.8) and the relation in (1.9). We take v 1 (x, z) = H(x) as the first trial of local approximation to a real solution. Here is the error in the interior of the region, (4.1) The quantities S 1 , S 3 , S 5 are odd functions of x, while S 2 , S 4 , S 6 are even. In addition, B 6 (H) turns out to be size of O( 3 ). For the first approximate solution H, the boundary errors can be formulated as follows. For z = 0, Similarly, for z = 1 , we have 4.2. The improvements. We now want to construct correction terms and establish a further approximation to a real solution that eliminates the terms of order in the errors.
Let us now choose .
In all what follows, we will assure the validity of the following constraint on the parameter f By interpolation, it also holds that To cancel S 1 in (4.1), by the method in [14], the interior correction term can be chosen in the form In the above, the function H 1 is the unique, odd and decaying solution to the problem The assumption (A4) will imply that Combining with (4.4), this will change the first terms in (4.2) and (4.3) into Whence we will get rid of the terms by using the following lemma to add one more boundary correction term, while the terms involving f will be concerned by the standard reduction procedure.
Lemma 4.1. [15] Let us consider the following problem (4.12) The problem has a unique solution φ * ∈ H 2 (S) which is odd in x for each z. Besides, there is a constant C such that for all small , φ * H 2 (S) ≤ C. In addition, there exist constants 0 < ζ < 1 4 , µ > 0, C > 0 such that We define Hence, φ 21 satisfies the following problem (4.13) and also φ 22 satisfies the problem (4.14) Such functions enjoy the following estimates: and We define the boundary correction term as follows: where Note that φ 2 is an odd function in the variable x for each z ∈ (0, 1/ ).
as the local approximate solution to (3.31). The change of coordinates in (1.16), (2.1), (2.13) and (3.19) will give the local relation between (x, z) andỹ. The function v 2 can be extended globally on Ω , which gives the expression of H in the form It is obvious that the function H will satisfies (3.1) and (3.2).
Remark 4.1. Please note that for the solutionũ with profile described in Remark 1.2, we shall define the approximate solution as The new error is (4.24) The first objective of this part is to compute the terms in (4.22).
It is easy to compute that Recalling the expression of φ 2 and using the equation of φ 21 and φ 22 in (4.13)-(4.14), we get 25) where K 12 (x, z) is the high order term. According to the expression of B 2 in (3.20), it is easy to obtain Recalling the definition of B 5 and B 6 in (3.32), (3.33), we get that Moreover, we can decompose N 0 (φ 1 + φ 2 ) as following where K 42 (x, z) is the high order term. So, according to the above rearrangements, we rewrite the expression (4.22) in terms of where S 3 , S 5 , φ 1,zz , K 11 (x, z), K 21 (x, z), K 31 (x, z) and K 41 (x, z) are the odd terms in the variable x. Observe that since φ 1 , φ 2 and f are of size O( ) all terms in E carry 2 in front. Moreover, K 12 (x, z), K 23 (x, z), K 33 (x, z), K 42 (x, z) are high order terms.
Here are the boundary terms in local forms The boundary error term g 0 has the form (4.33) The terms D 1 3 (v 2 ) and g 1 have similar forms. Moreover, we decompose g 0 = g 01 + g 02 , g 1 = g 11 + g 12 , (4.34) where For further reference, it is useful to estimate the L 2 (S) norm of E. We claim that A rather delicate term in E is the one carrying f , since we only assume a uniform bound on f L 2 (0,1) . For example, we have a term It is easy to check that the term without involving f , is exponentially decaying with respect to the variable x, and is bounded with respect to the variable z. Since (4.37) Other terms can be estimated in the similar way. Hence, we obtain (4.35). Similarly, we have the following estimate and for any f 1 and f 2 satisfying (4.5), we have the following estimates It is therefore of crucial importance to carry out computations of the term R EH x dx, and, similarly, some other terms involving φ.

5.1.
Estimates for projections of the error. In this section, we carry out some estimates for the term R EH x dx. We denote b l , l = 1, 2, generic, uniformly bounded continuous functions of the form where additionally b 1 is uniformly Lipschitz in its last two arguments.
Notice that the odd terms in the variable x in E, say S 3 , S 5 , φ 1,zz , K 11 (x, z), K 21 (x, z), K 31 (x, z), K 41 (x, z), do not contribute to the value of the integral since H x is even. Therefore, := I 1 + I 2 + I 3 + I 4 + I 5 + I 6 . (5.5) In the following, we are in a position to compute above integrals term by term. It is easy to get According to the expression of S 4 in (4.1), f = f 0 + f , direct computation leads to where gives that the term I 3 is Recalling the definition of K 22 (x, z), it follows that where By the definition of K 32 (x, z), we obtain where Recalling the expression of K 12 (x, z), K 23 (x, z), K 33 (x, z) and K 42 (x, z), we get In summary, the above estimates will lead to the conclusion of this section where β 0 is given in (5.6) and and At the end of this section, we will verify the following Lipschitz dependence of We only need to prove (5.16) for the term f f , since the other terms can be proved similarly. For any two functions f 1 , f 2 satisfying (4.5), we have Hence (5.16) holds true.

5.2.
Projection of terms involving φ. In this section, we will estimate the terms involving φ in (5.1) integrated against H x , which can be decomposed as It is easy to get Hence, , we single out two less regular terms. The one whose efficient depends on f explicit has the form To estimate Lipschitz dependence of Λ 1 * on f , it is suffice to evaluate Lipschitz dependence of the following term on f . By Proposition 3.1, we get from which it follows The other less regular one from Λ 1 , which containing φ zz , is the following In order to get Lipschitz dependence of Λ 1 * * on f , we can compute (5.20) by the standard computation, we obtain The remainder Λ 1 − Λ 1 * − Λ 1 * * actually defined, for fixed , a compact operator for f in L 2 (0, 1). This is a consequence of the fact that weak convergence in H 2 (S) implies local strong convergence in H 1 (S), and the same as the case for H 2 (0, 1) and C 1 [0, 1]. If {f j } is a weakly convergent sequence in H 2 (0, 1), then clearly the functions φ(f j ) constitute a bounded sequence in H 2 (S). In the above remainder one can integrate by parts if necessary once in x. Averaging against H x which decays exponentially localizes the situation and the desired fact follows.
From the definition of N in (3.45), we obtain Since we easily see that These terms Λ 2 + Λ 3 define compact operators for f similarly as the remainder Λ 1 − Λ 1 * − Λ 1 * * .

5.3.
Projection of the errors on the boundary. Without loss of generality, in this section we only compute the projection of error on the boundary ∂ 0 S. The main errors on the boundary integrated against H x in the variable x can be calculated as the following:

FEIFEI TANG, SUTING WEI AND JUN YANG
Higher order errors can be proceeded as follows: The other terms D 1 0 (φ) and D 0 0 (φ) on the boundary integrated against H x in the variable x is of size O( 3 ).
It is easy to give the proof of Lemma 6.1 due to the coercivity of linear operator L.
From (6.6), 2 A is contraction mappings of their arguments. By Banach Contraction Mapping Theorem and Lemma 6.2, we can prove the nonlinear problem with the boundary conditions defined in (6.4) is uniquely solvable forf provided that d < 3 2 + ρ for some ρ > 0. The desired result for full problem (6.3)-(6.4) then follows directly from Schauder's fixed-point Theorem.