Oscar AGUDELOManuel DEL PINOJuncheng WEI

(Dedicated to Haim Brezis on the occasion of his 70th birthday)

1 Introduction

1.1 Preliminary discussion

In this paper,we study the singularly perturbed boundary value problem

where α >0 is a small parameter,Ω ? RNis a smooth bounded domain and n is the inner unit normal vector to?Ω.

Solutions to(1.1)correspond exactly to the critical points of the Allen-Cahn energy

Equation(1.1)arises for instance in the gradient theory of phase transition when modelling the phase of a material placed in Ω or when studying stationary solutions for bistable reaction kinetics(see[2]).

Observe that u= ±1 are global minimizers of Jα,representing stable phases of two different materials in Ω.

We are interested in solutions u connecting the stable phases±1.As described in[29],solutions of this type are expected to have a narrow transtition layer from?1 to+1 with a nodal set that is asymptotically locally stationary for the perimeter functional.To be more precise,in[29]the author showed that a family of local minimizers{uα}αof Jαwith uniformly bounded energy must converge in L1(Ω),up to a subsequence,to a function u?of the form

where χEis the characteristic function of a set E and Λ ? Ω minimizes perimeter.In this case,as α → 0

where w(t)=tanh()is the solution of

The above assertion means that for any c∈(?1,1),the level sets{uα=c}converge to?Λ as α → 0.This result provided the intuition that ultimately led to important developments in the theory of Γ-convergence and put into light a deep connection between the Allen-Cahn equation and the theory of minimal surfaces.We refer reader to[4–5,23,31,33]for related results and stronger notions of convergence.

The connection between the Allen-Cahn equation and the theory of minimal surfaces has been explored in order to produce nontrivial solutions of equation(1.1),but the general understanding of solutions to this equation is far from being complete.In this regard,it is natural to ask for existence and asymptotic behavior of solutions to(1.1)in general smooth domains.For the case of minimizers,we refer the reader to[3,18,31,35]and references therein.We also refer the reader to[6,28],where it is established that the only local minimizers in convex domains are the constants±1.

The authors in[25],used a measure theoretical approach and the aforementioned intuition in dimension N=2,to construct local minimizers uαto(1.1)with interfaces collapsing onto afixed minimizing segment Γ0inside Ω that cuts ?Ω perpendicularly.

In[26],the author considers a situation similar to that one described in[25],but where Γ0is a non-degenerate segment,instead of a stable one.Non-degeneracy of Γ0respect to Ωis stated as

where K0,K1are the curvatures of?Ω at the points where Γ0cuts ?Ω orthogonally and|Γ0|corresponds to the length of the segment.

This geometric condition is equivalent to the fact that the eigenvalue problem

does not have λ=0 as an eigenvalue.The author also provides information about the Morse index of these solutions,which is either one or two,depending on the sign of K0and K1.

The previous construction was generalized in[9]under the same geometrical setting described in[26].Solutions in[9]have multiple interfaces that in the limit collapse onto the segment Γ0.Also,at main order,the transition layers interact exponentially respect to their mutual distances giving rise to the Toda system of ODEs.

In dimension N=3,Sakamoto[34]constructed solutions to(1.1)having a narrow transition through a planar disk orthogonal to the boundary of the domain and being non-degenerate in a suitable sense.The author also provides a characterization for this non-degeneracy in terms of the spectrum of the Dirichlet to Neumann map of the planar disk.As for higher dimensions in the setting of manifolds,Pacard and Ritoré[30]constructed solutions having one transition along a codimension one non-degenerate minimal submanifold.

In the spirit of the results mentioned above,we also want to refer the reader to[11–14]dealing with similar results for the inhomogeneous Allen-Cahn equation and[19,36–37]for semilinear elliptic problems,where resonance phenomena are present.

The underlying geometric problem when constructing solutions to(1.1)with narrow interfaces is the existence of minimal surfaces inside the domain Ωintersecting the boundary ?Ω orthogonally.This problem,in a general three dimensional compact Riemannian manifold,has been completely settled in a recent paper by Li[27].For earlier results in this direction,we refer to[15–16].One instance is the critical catenoid in a ball whose uniqueness was established by Fraser and Schoen[17].

1.2 Main result

Our goal in this paper is to generalize the results in[26,34]by taking N=3 and a more general class of minimal surfaces for limiting nodal set.

Let M be a complete embedded minimal surface of finite total curvature in R3.For over a century,there were known only two examples of such surfaces,namely the plane and the catenoid.In[7–8],Costa gave the first nontrivial example of such a surface with genus one,being properly embedded and having two catenoidal connected components outside a large ball sharing an axis of symmetry and another planar component perpendicular to this axis.Later this construction was generalized in[21–22]to surfaces having the same look as the Costa’s surface far away but with arbitrary genus.We refer the interested reader to[24,32]and references therein,for related results and further generalizations.

For y=(y1,y2,y3)∈R3,let us denote

It is known that for some large but fixed R0>0,outside the cylinder{y∈R3:r(y)>R0},M decomposes into finite connected components,say M1,···,Mm,which from now on we will refer to as the ends of M.

For every k=1,···,m,there exists a smooth function Fk=Fk(y?)with

and there exist constants ak,bk,biksatisfying

such that

and relation(1.4)can be differentiated.

It is also known that M is orientable and R3?M has exactly two connected components namely S+and S?(see[20]).Let S+be the connected component of R3?M containing the axis x3which corresponds to the axis of symmetry of the ends M1,···,Mk.

Let ν :M →S2be the unit normal vector to M pointing towards S+and consider Fermi coordinates near M

where η,δ>0 are small but fixed.Observe that z corresponds to the signed distance to M,i.e.,

for every y∈M and z small.

Next,consider a smooth bounded domain Ω,such that

(i)Ω contains a portion of the surface M,denoted by M,i.e.,M:=Ω∩M is non-empty.

(ii)Ω?M has two connected components which,abusing the notation,we denote by S+,S?with the same convention as above.

(iii)For every k=1,···,m,Ck:=Mk∩ ?Ω is a smooth simple closed curve and

Observe that?Ω∩M consists of m non-intersecting closed curves.

Following[9,25,26],in order to produce solutions to(1.1),M must be critical and nondegenerate in a suitable sense respect to?Ω.To make this concepts precise,let us introduce ΔMand|AM|the Laplace-Beltrami operator and the norm of the second fundamental form of M,respectively.The fact that M is a minimal surface is equivalent to saying that its mean curvature HM=0.This implies that|AM|2=?2KM,where KMis the Gaussian curvature of M.

Recall that n is the inner unit normal vector to?Ω and consider the eigenvalue problem

where τ represents the inward unit normal direction to ?M respect to M and I(y)is given by

Our crucial assumptions on M are the following:

(I)M cuts orthogonally ?Ω along the curve Ckfor every k=1,···,m.

(II)λ=0 is not an eigenvalue for the problem(1.5)in H1(M).

As stated in[19],assumption(I)implies that τ and n must coincide along every curve Ckand consequently these curves are geodesics in ?Ω in the direction of ν since their normal vectors in?Ω are parallel to n.Therefore,the quantity I(y)corresponds to the geodesic curvature of?Ω in the direction ν(y)for y ∈ Ck.

Our main result is the following.

Theorem 1.1Assume conditions(i)–(iii)and(I)–(II).Then for every α >0 small enough,there exists a solution uαto(1.1),such that for every x ∈ {dist(·,M)< η} ∩ Ω,

where w(t)is determined by(1.3)and at main order h solves the boundary value problem

Even more,in the set Ω ? {dist(·,M)< η},as α → 0,

Theorem 1.1 provides us with solutions having limiting nodal with multiple catenoidal ends intersecting?Ω orthogonally.We also remark that in the case that the surface M and the domain Ω enjoy axial symmetry,our developments can be carry out in this setting and condition(II)for problem(1.5)is required only in the space in

The paper is organized as follows.In Section 2,we present the invertibility theory of the operator described in(1.6)with Robin boundary conditions and discuss some examples,where our result applies.In Section 3,we present the geometric framework,which we will use to set up the proof of Theorem 1.1.In Section 4,we construct an accurate approximation of the solution to problem(1.1)and Section 5 presents the proof of our main result.The final sections are devoted to provide detailed proofs of lemmas and propositions used in Section 5.

2 Jacobi Operator with Robin Boundary Conditions and Examples

In this part,we consider the equation

for given functions f ∈ Lp(M)and g∈ Lp(?M).

Let us assume hypothesis(II)from the introduction.In the case g=0,using the Fourier decomposition method,it is straight-forward to verify that for any f∈L2(M),there exists a unique solution h∈W2,2(M)satisfying

By standard regularity theory,still assuming that g=0,if p>2 and f∈Lp(M),then problem(2.1)has a unique solution h∈W2,p(M)satisfying

From the previous discussion,it follows that for any p>2,there exists C>0 such that given arbitrary functions f∈ Lp(M),g∈ Lp(?M),problem(2.1)has a unique solution h∈satisfying

2.1 Comments about conditions(I)–(II)

We comment first on conditions(I)–(II)in a particular case.Let M be the catenoid in R3parameterized by the mapping

which provides coordinates on M in terms of the signed arch-length of the profile curve and the rotation around the x3-axis,which in our setting corresponds to the axis of symmetry of M.

Since S+is the connected component of R3?M containing the x3-axis,the unit normal vector to M pointing towards S+is given by

Consider the Fermi coordinates

which define a change of variables for instance on the neighborhood of M

for some fixed and small η>0.

Let Ω be an axially symmetric domain and recall that M=Ω∩M.Since M has two ends and?Ω∩M is axially symmetric,?Ω∩M=C1∪C2,where C1,C2are parallel,non-intersecting circles.C1,C2are parameterized respectively by

for some fixed y1

To describe ?Ω close to the circles Yi(θ),we assume the existence of two smooth functions

and also that the systems of coordinates

describe the set

Normal deformations of M within Ω can be described by

where?h?C2(M)<η and

for y1

Take any arbitrarywithDenote

and let ghbe its respective induced metric,with the convention that g0is the induced metric of M.

The area functional of Mhis computed as

This area functional is of class C2and its first variation around M is given by

where we recall that HMis the mean curvature of M.

From(2.7),we conclude that M is critical for the area functional(2.6)if and only if

Since M is a minimal surface,automatically HM=0.Therefore,(2.8)states that condition(I)is equivalent to the fact that M is critical for the functional(2.6)respect to normal perturbations of M.

Assuming condition(I),the second variation of the area functional around M is given by the quadratic form

and stability properties of M respect to Ω are analyzed through the linear eigenvalue problem

A similar analysis,but with more careful computations,can be carry out for a more general geometric setting,leading to the same interpretation.

2.2 Examples

In the entire catenoid,the linear equation,

has two axially symmetric entire solutions,namely

corresponding respectively to the invariances of M under translations along the vertical axis and dilations.We refer the reader to Section 4 in[1,10]for full details.

In the coordinates y=Y(y,θ)∈ M,we have that

from where we observe that Z1is odd and Z2is even.

Observe also that in(0,∞),Z1is positive and Z1(0)=0,while Z2changes sign only at one point y=y0>0 and?1

Since

we notice that?yZi>0 in(0,∞)for i=1,2.Therefore,Z1,Z2are strictly increasing in(0,∞).

Observe that assumption(I)implies that along the circles Ci,n= τi,where τiis the inward unit tangent vector of the profile curve of the catenoid M along the circles Ci,i=1,2.

Also Ki:=(?1)i+1?zzGi(0)corresponds to the curvature of the integral curve of?Ω in the direction of ν along the circle Ci.

The axially symmetric framework and computations similar to those carried out in[1],yield that in the coordinates y=Y(y,θ),

Hence,non-degeneracy of M respect to?Ω is equivalent to saying that the only solution to

is the trivial one.

Basic theory of ODEs and the developments from Section 4 in[1]imply that λ=0 is not an eigenvalue of(2.10)if and only if

Observe that condition(2.11)is clearly invariant under dilations.

2.2.1 Example 1

If in addition to the axial symmetry of Ω,we assume that?Ω is almost flat along the circles C1,C2in the direction of the normal ν,i.e.,K1=K2=0,then M is non-degenerate respect to Ω.

To verify this claim,we notice that in this case,condition(2.11)is equivalent to

which holds true since the functionis strictly increasing and y1

2.2.2 Example 2

Assume now that the catenoidal portion M is even respect to the vertical axis,i.e.,?y1=y2=:>0.Let

be an ellipsoid of revolution,where a,b>0.

Using the coordinates Xi(θ,z)from(2.4)with G=G2(z)= ?G1(z)such that G(0)=y,?Ω near M is described by the implicit relation

Implicit function theorem yields that

so that M is critical respect to the ellipsoid Ω if a,b satisfy

and since min{Z2(y):y∈R}=?1,this imposes a restriction on a and b that

The monotonicity of Z2(y)allows the following interpretation of the criticality of M:Once the ellipsoid has been fixed satisfying(2.13),there is exactly one catenoid that cuts the boundary of the ellipsoid perpendicularly.

Next,assume that R=a=b,so that Ω=BR(0).In this case,M corresponds to the so-called critical catenoid.This situation was treated in[16],where M is the solution of a maximization problem for the first Steklov eigenvalue of the Dirichlet to Neumann mapping in bounded domains.

Using the same notation as above,one can verify that for the critical catenoid M,K1=From the determinant in(2.11)and relation(2.12),the non-degeneracy of the critical catenoid respect to the sphere?BR(0)is equivalent to the expression

which holds true since all the quantities involved are positive.

2.2.3 Example 3

Concerning stability issues let us consider the quadratic form in

We first establish conditions on M to be minimizer of the area functional.

Proposition 2.1Assume that Z is a smooth positive solution to the linear equation(2.9)in an open set of the catenoid M,containing the portion M=Ω∩M.For every smooth function ? in M it holds that

where τ is the inner normal vector to ?M.Consequently,if

then M is minimizer for the area functional.

ProofThe proof follows directly testing equation(2.9)against ψ =and integrating by parts.

Assume that K1,K2<0,so that Ω is a non-convex domain.If in addition M is an even catenoidal portion with small area,then M is a local minimizer for the area functional(2.6).To see this,it suffices to consider M an even piece of catenoid contained in an open set of M where Z=?Z2>0,where we also have that

The claim follows by a direct application of the previous proposition.

On the other hand,if the area of M is large enough,M resembles the entire catenoid M,which has Morse index 1.Cutting o ffan eigenfunction of ΔM+|AM|2associated to its positive eigenvalue,in a way that the boundary condition plays no role at infinity,we obtain a direction in H1(M)where the second variation of surface area is negative and therefore a catenoidal portion M with large area would be unstable.

The former situation ocurrs also in a general complete embedded minimal surface with finite total curvature for which the Morse index is also finite.

3 Geometric Computations

Let M be a complete embedded minimal surface with finite total curvature.In this part,we compute the Euclidean Laplacian in an open neighborhood of M inside Ω and the normal derivatein ?Ω near?M in suitable systems of coordinates.

First we compute the Euclidean Laplacian well inside the set Ω close to M.Following the developments from[10],denote by M0the part of M inside the cylinder{y:r(y)

where g?1=(gij)2×2is the inverse of the metric g,and the coefficients,are smooth.

Next,consider the set

For k=1,···,m,we parameterize Mk,the k-th end of M with the mapping

Notice that the unit normal vector to M at a point y∈Mkhas the expression in coordinates

so that?iν =O(r?2)and|AM|2=O(r?4)as r → ∞.

In coordinates Yk(y)on Mk,the metric g:=(gij)2×2satistisfies

and this relations can be differentiated.

We compute the Laplace-Beltrami operator on Mk,using again formula(3.1)to find that

The surface M is parameterized completely by the m+1 local coordinates described above and we observe that expression(3.1)holds in the entire M,where for k=1,···,m the coefficients on Mksatisfy

Fermi coordinates given by the mapping X(y,z):=y+zν(y)provide a change of variables in the neighborhood of M

In the set N,the formula

is valid,where ΔMis computed in(3.1)for N ∩M0and(3.2)for N ∩Mkand

On the ends of M,the smooth functions aij(y,z),bi(y,z)satisfy

as r→∞,uniformly on z in the neighborhood N of M(see[10,Lemma 2.1]).

Let us comment further on expression(3.3).For fixed and small z,the mean curvature of the normally translated surface

is given by

where k1,k2are the principal curvatures of M.Since M is a minimal surface,HM=k1+k2=0.It follows that+=0 and so

From the asymptotics of?Mν,we have that ki=O(r?2),and thus we obtain the expansion forin(3.4)follows.

In what follows,we consider a large dilation of M,denoted by Mα:= α?1M for α >0 small.Let us denote the dilated ends of M by Mk,α:= α?1Mkfor k=1,···,m.

For a smooth function h defined in M,we consider dilated and translated Fermi coordinates

for y∈Mαand|t+h(αy)|<.

Scaling and translating expression(3.3),we obtain

where

Expression(3.5)holds true in the region

and we will use it to handle equation(1.1)well inside the region α?1(Ω ∩ N).

The previous geometric considerations above do not take into account the effect of?Ω∩N.

To handle boundary computations,we use that the surface M is orthogonal to?Ω.Let usfix k=1,···,m.From assumption(iii)in the introduction,we may assume that the closed simple curve Ck:=Mk∩Ω is parameterized by

The mapping γkhas a smooth orthogonal extension to an open neighborhood of Ckin Mk.Abusing the notation,we write this extension as

which satisfies

and γk([0,δ)× (0,lk)) ? Mk∩ Ω.The coordinates(ρ,υ)can be thought as polar coordinates in Mknear Ck.

Using formula(3.1)and coordinates γk(ρ,υ)and omitting the depedence on k,the Laplace-Beltrami operator of M close to Ck,takes the form

where

Associated to the coordinate system y= γk(ρ,υ),we consider Fermi coordinates

in the neighborhood of Mk

To described Ω ∩ Nknear Ck,we assume the existence of a smooth function Gk=Gk(υ,z)with Gk(υ,0)=0 and such that

A translation along the integral lines of Mkassociated to the parameterization γk(ρ,υ),is given by

taking δ>0 smaller if necessary.

Modified Fermi coordinates

describe also the set

Observe that

and

so that,assumption(I)is equivalent to saying that?z(0,υ,0)and ?s(0,υ,0)are orthogonal vectors,and hence ?zGk(υ,0)=0.

Summarizing,the function Gk(υ,z)satisfies

From(3.10),the asymptotic expansion in powers of z of(s,υ,z)reads as

where q1,q2⊥ν with expressions given by

Taking derivatives in expression(3.11)and omitting the dependence on k,we compute the induced metric on Ω∩Nk,which takes the form

where

where all the entries of the matrices above are evaluated at(s,υ).

The inverse of the metric=()3×3,has the asymptotic expression

Following[19],we denote

Consider again the function h ∈ C2(M),written in the coordinates(3.8)as h=h(ρ,υ).Taking dilated and translated modified Fermi coordinates

for

and after a series of lengthy,but necessary computations,we arrive to the expressions

and in the coordinatesin the set α?1(Ω ∩ Nk),

The asymptotic expressions for D0,read as follows:

and

where the functionsare smooth with bounded derivatives,andis a differential operator having C1dependence on h and its derivatives.

Next,we turn our attention to the boundary condition.Since at?Ω∩Nk,the vectorsspan the tangent space of?Ω and

along the curve Ck,we can write

It can be check directly from(3.12)–(3.13)that the boundary condition reads as

so that,after dilating and translating with the coordinates(s,θ,t),we find that the boundary condition becomes

where

andhas C1dependence on its variables.

We remark that in the case that M is the catenoid and Ω is an axially symmetric domain containing a catenoidal portion,following the scheme in[9,26],one can parameterize with only one set of coordinates and the calculations reduce considerably.

4 Approximation of the Solution

The proof of our main result relies on a Lyapunov-Schmidt procedure near an almost solution of equation(1.1).This section is devoted to find an accurate global approximation to perform this reduction.

For this,denote f(u):=u(1?u2)and we consider

the solution of the ordinary differential equation(ODE for short)

which has the asymptotics

Set Ωα:= α?1Ω,?Ωα= α?1?Ω and recall that Mα= α?1M.After a rescaling,we are led to consider the problem

where nαstands for the inward unit normal vector to?Ωα.

In what follows for a function U=U(x),we denote

4.1 The inner approximation

Let p>2 and take h∈W2,p(M)satisfying the a priori estimate

where the constant K is going to be chosen large but independent of α>0.

Using the coordinates Xα,hin the set Nα,hdescribed in(3.7),we set as first local approximation

When computing the error created by u0using(3.5)–(3.6),in α?1(Ω ∩ N),we find that

where|AM|,h,?ih,?ijh are evaluated at αy.

Observe that if we take h=0,the size and behavior of the error in expression(4.4)is given by

As in[10],due to the presence of the O(α2)term,we need to improve this approximation.Hence we consider the function ψ1(t)solving the ODE

Using variations of parameters formula and the fact that

we obtain that ψ1(t)given by the formula

from where it follows that for any

So,we consider as a second approximation in the region α?1(Ω ∩ N)the function

where in the coordinates Xα,h

Computing the inner error of this new approximation,we find that

where

and the differential operator R1,αhas C1dependence on all of its variables with

From the error(4.7),we see that in the open neighborhood of M,α?1(Ω ∩N)

4.2 Boundary correction

It is clear that our approximation u1can be defined in the set α?1(Ω ∩ N),but u1does not satisfy in general the boundary condition.In this regard,we need to make a further improvement of the approximation u1by adding boundary correction terms.

Let us consider a smooth cut-o fffunction β such that

where δ>0 is the constant in(3.8).

For the k-th end of α?1M,Mk,α,we consider a cut-o fffunction βα= βk,α(x)= β(αs)for

Near the boundary,we consider an approximation of the form

where φ2,k, φ3,k(x)are defined in α?1(Ω ∩ Nk)and will be chosen of order O(α)and O(α2)respectively.

We first compute the error of the boundary condition created by the approximation u2using the coordinatesand expression(3.18)for a fixed end Mk.We again omit the explicit dependence of k,but noticing that the developments in this part hold true regardless of the end we are working with,since the supports of the cut-o fffunctions βα,kwithin the region Nα,hclose to every end Mk,αare far away from each other.

From(4.3),we know that h=OW2,p(M)(α).Splitting the boundary error in powers of α,we find from(3.18),

where the term?B0,αsatisfies that

Our goal is to get a boundary error of order O(α3e?σ|t|)by choosing h,φ2,φ3satisfying

We do this step by step.First,let us choose φ2solving the equation

from where we obtain that φ2(·,t)is odd in the variable t,and from Proposition 6.1 it follows that in the norms described in(5.2)–(5.3)and for p>3 andthere exists C>0 such that

To choose φ3,we make the decomposition

and we write the second line in(4.9)as

Hence,let φ3satisfy the boundary condition on ?Mα×R written in coordinates(s,θ,t)

Next we compute the error of the approximation u2near the boundary.Denote

Using our choice for φ2and expression(3.15),in coordinates(s,θ,t)this error is written as

where=(αs,αθ,t,h,?Mh,)=O(α4)and

Notice that

and

Hence in order to improve the approximation,we need to get rid of the terms in the first line of(4.12)and the term in(4.13).Let φ3solve the linear problem

From Proposition 6.1,φ3satisfies

From expression(4.12),we directly check that

where Ei,α=Ei,α(αs,αθ,t,h,?Mh,h)=O(α3+i)and

From(4.9)and(4.11),the boundary error takes the form

where c1,c2are described in(4.10),?c2?L∞(?Mα)≤ Cα and

Finally,to get the right size of the boundary error in(4.9),we impose on h the boundary condition

and pulling back the rescaling in α>0,

4.3 Global approximation

Observe that so far,our approximation is defined only in the open set α?1(Ω ∩ N).

The idea to get a global approximation is to interpolate the approximation u2well inside Nα,h,with the function

outside Nα,h.

Let us take a non-negative function∈C∞(R)such that

and consider the following cut-o fffunction in Nα,hgiven by

With the aid of this,we set up as approximation in Ωαthe function

We compute the new error created by this approximation as follows:

where

Using z=|t+h(αy)|,we see that the derivatives of βηdo not depend on the derivatives of h.On the other hand,due to the choice of βηand the explicit form of E,the error created only takes into account the values of βηin the set

so we get the following estimate for the error E:

5 The Proof of Theorem 1.1

The proof of Theorem 1.1 is fairly technical.To keep the presentation as clear as possible,we sketch the steps of the proof,and in the next sections,we give the detailed proofs of the lemmas and propositions mentioned here.

We introduce suitable norms to set up an appropriate functional analytic scheme for the proof of Theorem 1.1.For α >0,1

We also consider for functions g=g(y,t),φ= φ(y,t),defined in the whole Mα×R,the norms

where dVα:=dygMαdt.In the case p=+∞,we have that L∞(B1(y,t),dVα)=L∞(B1(y,t)).

For a function G defined in?Mα×R,we set

and we recall the norm for the parameter function h

We look for a solution to equation(1.1)of the form

where U(x)is the global approximation defined in(4.17)and ? is going to be chosen small in some appropriate sense.Thus,we need to solve the problem

or equivalently

where

5.1 Gluing procedure

To solve problem(5.5),we consider again the cut-o fffunction?β from Subsection 4.3,to define for every n∈N

We look for a solution ?(x)to(5.5)of the form

where φ(y,t)is defined for every(y,t) ∈ Mα× R and ψ(x)is defined in the whole Ωα.So,wefind from equation(5.5)that

Hence,we will have constructed a solution to equation(5.5),if solve the system

As for the boundary conditions,we compute

Therefore,the boundary condition is reduced to the boundary system

We solve first(5.8)–(5.10),using the fact that the potential 2?(1?ζ2)[f?(U)+2]is uniformly positive,so that the linear operator behaves like Δ?2.A solution ψ = Ψ(φ)is then found from the contraction mapping principle.We collect this discussion in the following lemma,that will be proven in detail in Subsection 7.1.

Proposition 5.1Let 30 be sufficiently small.For every h satisfying(4.3)and every φ such that?φ?2,p,σ≤ 1,problem(5.8)–(5.10)has a unique solution ψ = Ψ(φ)and the operator Ψ(φ)is Lipschitz in φ.More precisely, Ψ(φ)satisfies that

and the constant C>0 depends only on p.

Next,we extend(5.7)–(5.9)to a qualitatively similar equation in Mα×R.Let us set

Observe that R(φ)is understood to be zero for|t+h(αy)|>+2,and so we consider the equation

where from expression(4.12)and omitting the depedence on k,we have on the k-th end Mk,αthat

and from expression(4.7),we write

Observe thatandcoincide with S(u1),S(u2)but the parts that are not defined for all t∈ R are cut-o ffoutside the support of ζ4.

As for the boundary condition,we proceed in the same fashion by writing

It suffices to consider φ satisfying

where τα=s is the normal inward direction to ?Mα,and in expression(4.9),we cut-o ffthe parts that are not defined for every t.We write also for further purposes

Observing that again,we have omitted the depedence on the end Mk,αfor notational convenience.

Next,using Proposition 5.1,we solve equation(5.13)with ψ = Ψ(φ).Let us set

So,we only need to solve

To solve problem(5.17)–(5.19),we solve a nonlinear problem in φ,that basically eliminates the parts of the error,that do not contribute to the projections.

The linear theory we develop to solve problem(5.17)–(5.19),considers right-hand sides and boundary data with a behavior similar to that of the errorand,that as we have seen,are basically of the form O(α3e?σ|t|).

Using the fact that N(φ)is Lipschitz with small Lipschitz constant and contraction mapping principle in a ball of radius O(α3)in the norm? ·?2,p,σ,we solve problem(5.17)–(5.19).This solution φ,defines a Lipschitz operator φ = Φ(h).This information is collected in the following proposition.

Proposition 5.2Assume that 3

0 is small.For every α >0 small,problem(5.17)–(5.19)has a unique solution φ = Φ(h),satisfying

and

where the constant C>0 depends only on p.

5.2 Adjusting h,to make the projection equal zero

We denoteTo conclude the proof of Theorem 1.1,we adjust h so that

Integrating(5.17)against w?(t)and using that the function βαdefined in Subsection 4.2 does not depend on the variable t,we compute

From(5.15),we compute

On the other hand,from(5.14),the reduced error near the boundary reads as

From assumption(4.3),and the estimates in Section 7 for the nonlocal terms,we have

Therefore,

where

As for the boundary condition,directly from(5.16)we ask h to satisfy

where we recall that

and the right-hand side in(5.21)does not depend on h.

We solve then

with the boundary condition(5.21)by a direct application of the theory developed in Section 2 and a fixed point argument for h in a ball or order O(α)in the topology induced by the norm?·??described in(4.3).This completes the proof of our theorem.

6 Projected Linear Problem

In this part,we provide the linear theory for the problem

which relies strongly on the fact that solutions to

are the scalar multiples of w?(t).The proof follows the same lines of Lemma 5.1 in[10].We simply remark that when decomposing the solution φ as

from maximum principle one obtains that|φ⊥(y,t)|≤ Ce?σ|t|for some 0< σ

it follows that for certain positive constant λ

where ταis the inward unit normal to ?Mαin Mα.Clearly it follows that ψ =0 and

Consequently,c(y)is a constant function.

Proceeding as in[9,Section 3],it suffices to solve the case G=0 and"Rg(·,t)·w?(t)dt=0.To prove existence,we set

and we consider the space H of function φ∈H1(Mα×R),such that

Sinceas|t|→ ∞,the equation can be put into the setting

where K:H → H is a compact operator.From Fredholm alternative,we obtain a solution φ,such that

As for the a priori estimates,we can proceed using a blow up argument following the same lines as in the local elliptic regularity developed in[10].In our case,we also need to consider two limiting blow up situations:The case of R2×R when taking limit well inside Mα×R and the case of the half space R+×R2when taking the limit in coordinates close to?Mα×R inside one of the sets Mk,α×R.The former case is reduced to the case of R2×R as limiting situation by using an odd reflection respect to the boundary of R+×R2.

Thus we have proven the following proposition.

Proposition 6.1For every p>3 and for every α>0 small enough and given functions g defined in Mα×R and G defined in?Mα×R such that

there exists a unique pair(φ,c)solving problem(6.1)–(6.2)satisfying the a priori estimate

where the constant C dependens only on p.

7 Gluing Reduction and Solution to the Projected Problem

In this section,we prove Lemma 5.1 and then we solve the nonlocal projected problem(5.17)–(5.19).The notations we use in this section have been set up in Sections 4–5.

7.1 Solving the gluing system

Given a fixed φsuch that?φ?2,p,σ≤ 1,we solve problem(5.8)with boundary condition(5.10).To begin with,we observe that there exist constants a

where Qα(x)=2?(1?ζ2)[f?(U)+2].Using this remark,we study the problem

for givenConcerning solvability of this linear problem,we have the following lemma.

Lemma 7.1Assume 30 is small.For any given?g,?G with

equation(7.1)has a unique solution ψ = ψ(g),satisfying the a priori estimate

The proof of this lemma is standard,and we refer the reader to[9,Section 2]for details.

Now we prove Proposition 5.1.Denote by X,the space of functions ψ ∈ W2,p(Ωα)such that?ψ?X< ∞ and let us denote by Γ(,)= ψ the solution to the equation(7.1),from the previous lemma.We see that the bilinear map Γ is continuous,i.e.,

Thus,(5.8)is restated as a fixed point problem

Using the norms described in(5.3)and(5.4),let us take φ and h satisfying

We next estimate the size of the right-hand side in(7.2).Recall thatso that

This means that

and so

As for the second term in the right-hand side of(7.2),the following holds true:

This implies that

Proceeding in the same fashion,we obtain the estimate for the boundary condition

Finally,we check the Lipschitz character on ψ of the term(1?ζ2)N[ζ2φ+ψ].Take ψ1,ψ2∈ X and notice that

from where it follows that

and in particular we see that

Consider:X→X,=(ψ)the operator given by the right-hand side of(7.2).From the previous remarks,we have thatis a contraction provided that α is small enough,and so we have found ψ=(ψ)the solution to(5.8).

We can check directly that Ψ(φ)= ψis Lipschitz in φ,i.e.,

Hence,for α small,we conclude

7.2 Solving the projected problem

Now we solve problem(5.17)–(5.19)using the linear theory developed in Section 6,together with a fixed point argument.From the discussion in Subsection 7.1,we have a nonlocal operator ψ = Ψ(φ).

Recall that

Let us denote

We need to investigate the Lipschitz character of Ni,i=1,2,3.We see that

Using the norm described in(5.2),we find that

As for the term N1(φ),we just have to pay attention to the term R(φ).Notice that R(φ)is linear on φ and

Hence,from the assumptions made on h,we have

Observe also that under the assumption made on h we have

Hence,for?φ?2,p,σ≤ Aα2,we have that?N(φ)?p,σ≤ Cα4.

As for the boundary condition,we check directly from expressions(3.18),(5.16)and(5.18)that on every end Mk,αthe following estimates hold:

with B(φ)linear in φ.

Setting T(g,G)=φ the bilinear operator given from Proposition 6.1,we recast problem(5.17)–(5.19)as the fixed point problem

in the ball

where X is the space of function φ ∈(Mα× R)with the norm?φ?2,p,σ.Observe that

On the other hand,because C and A do not depend on α>0,we take A large enough,so that

Hence,the mapping T is a contraction from the ballonto itself.From the contraction mapping principle,we get a unique solution φas required.We denote the solution to(5.17)–(5.19)for h fixed.

As for the Lipschitz character of Φ(h),it comes from a lengthy by direct computation.We left to the reader to check on the details of the proof of the following estimate:

and this completes the proof of Proposition 5.2.

[1]Agudelo,O.,del Pino,M.and Wei,J.C.,Solutions with multiple catenoidal ends to the Allen-Cahn equation in R3,J.Math.Pures Appl.(9),103(1),2015,142–218.

[2]Allen,S.and Cahn,J.W.,A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening,Acta Metall.,27,1979,1084–1095.

[3]Bardi,M.and Perthame,B.,Exponential decay to stable states in phase transitions via a double logtransformation,Comm.Partial Differential Equations,15(12),1990,1649–1669.

[4]Caffarelli,L.and Cordoba,A.,Uniform convergence of a singular perturbation problem,Comm.Pure and Appl.Math.,XLVI,1995,1–12.

[5]Caffarelli,L.and Cordoba,A.,Phase transitions:Uniform regularity of the intermediate layers,J.Reine Angew.Math.,593,2006,209–235.

[6]Casten,R.and Holland,C.J.,Instability results for reaction diffusion equations with Neumann boundary conditions,J.Differential Equations,27(2),1978,266–273.

[7]Costa,C.J.,Imersoes minimas en R3de genero un e curvatura total finita,PhD thesis,IMPA,Rio de Janeiro,Brasil,1982.

[8]Costa,C.J.,Example of a complete minimal immersions in R3of genus one and three embedded ends,Bol.Soc.Bras.Mat.,15(1–2),1984,47–54.

[9]Del Pino,M.,Kowalczyk,M.and Wei,J.C.,The Toda system and clustering interfaces in the Allen-Cahn equation,Arch.Ration.Mech.Anal.,190(1),2008,141–187.

[10]Del Pino,M.,Kowalczyk,M.and Wei,J.C.,Entire solutions of the Allen-Cahn equation and complete embedded minimal surfaces of finite total curvature,J.Di ff.Geometry,93,2013,67–131.

[11]Del Pino,M.,Kowalczyk,M.,Wei,J.C.and Yang,J.,Interface foliations of a positively curved manifold near a closed geodesic,Geom.Funct.Anal.,20(4),2010,918–957.

[12]Du,Z.and Gui,C.,Interior layers for an inhomogeneous Allen-Cahn equation,J.Differential Equations,249,2010,215–239.

[13]Du,Z.and Lai,B.,Transition layers for an inhomogeneus Allen-Cahn equation in Riemannian Manifolds,preprint.

[14]Du,Z.and Wang,L.,Interface foliation for an inhomogeneous Allen-Cahn equation in Riemmanian Manifolds,Calc.Var,2012.DOI:10.1007,March s00526-012-0521-4

[15]Fraser,A.and Li,M.,Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex boundary,J.Differential Geom.,96(2),2014,183–200.

[16]Fraser,A.and Schoen,R.,The first Steklov eigenvalue,conformal geometry,and minimal surfaces,Adv.Math.,226(5),2011,4011–4030.

[17]Fraser,A.and Schoen,R.,Sharp eigenvalue bounds and minimal surfaces in the ball,2012.arXiv:1209.3789

[18]Garza-Hume,C.E.and Padilla,P.,Closed geodesics on oval surfaces and pattern formation,Comm.Anal.Geom.,11(2),2003,223–233.

[19]Guo,Y.and Yang,J.,Concentration on surfaces for a singularly perturbed Neumann problem in threedimensional domains,J.Differential Equations,255(8),2013,2220–2266.

[20]Hoffman,D.and Karcher,H.,Complete embedded minimal surfaces of finite total curvature,Geometry V,Encyclopaedia Math.Sci.Vol.90,pp.5–93,262–272,Springer-Verlag,Berlin,1997.

[21]Hoffman,D.and Meeks,W.H.,III,A complete embedded minimal surface in R3with genus one and three ends,J.Di ff.Geom.,21,1985,109–127.

[22]Hoffman,D.and Meeks,W.H.,III,Embedded minimal surfaces of finite topology,Ann.Math.,131,1990,1–34.

[23]Hutchinson,J.E.and Tonegawa,Y.,Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory,Calc.Var.,10,2000,49–84.

[24]Kapouleas,N.,Complete embedded minimal surfaces of finite total curvature,J.Differential Geom.,45,1997,95–169.

[25]Kohn,R.V.and Sternberg,P.,Local minimizers and singular perturbations,Proc.Royal Soc.Edinburgh,111A,1989,69–84.

[26]Kowalczyk,M.,On the existence and Morse index of solutions to the Allen-Cahn equation in two dimensions,Ann.Mat.Pura Appl.,1844(1),2005,17–52.

[27]Li,M.M.-C.,A general existence theorem for embedded minimal surfaces with free boundary,Comm.Pure Appl.Math.,68(2),2015,286–331.

[28]Matano,H.,Asymptotic behavior and stability of solutions of semilinear diffusion equations,Publ.Res.Inst.Math.Sci.,15(2),1979,401–454.

[29]Modica,L.,Convergence to minimal surfaces problem and global solutions of Δu=2(u3?u),Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis(Rome,1978),223–244,Pitagora,Bologna,1979.

[30]Pacard,F.and Ritoré,M.,From the constant mean curvature hypersurfaces to the gradient theory of phase transitions,J.Differential Geom.,64(3),2003,359–423.

[31]Padilla,P.and Tonegawa,Y.,On the convergence of stable phase transitions,Comm.Pure Appl.Math.,51(6),1998,551–579.

[32]Pérez,J.and Ros,A.,The space of properly embedded minimal surfaces with finite total curvature,Indiana Univ.Math.J.,45(1),1996,177–204.

[33]Tonegawa,Y.,Phase field model with a variable chemical potential,Proc.Roy.Soc.Edinburgh Sect.,A132(4),2002,993–1019.

[34]Sakamoto,K.,Existence and stability of three-dimensional boundary-interior layers for the Allen-Cahn equation,Taiwanese J.Math.,9(3),2005,331–358.

[35]Sternberg,P.and Zumbrun,K.,Connectivity of phase boundaries in strictly convex domains,Arch.Rational Mech.Anal.,141(4),1998,375–400.

[36]Wei,J.C.and Yang,J.,Concentration on lines for a singularly perturbed Neumann problem in twodimensional domains,Indiana Univ.Math.J.,56(6),2007,3025–3073.

[37]Wei,J.C.and Yang,J.,Toda system and interior clustering line concentration for a singularly perturbed Neumann problem in two dimensional domain,Discrete Contin.Dyn.Syst.,A22(3),2008,465–508.