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Advanced Nonlinear Studies

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Volume 16, Issue 1

Issues

On the Profile of Globally and Locally Minimizing Solutions of the Spatially Inhomogeneous Allen–Cahn and Fisher–KPP Equations

Christos Sourdis
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  • Department of Mathematics “G. Peano”, University of Turin, Via Carlo Alberto 10, 10123 Turin, Italy
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Published Online: 2016-01-21 | DOI: https://doi.org/10.1515/ans-2015-5016

Abstract

We show that the spatially inhomogeneous Allen–Cahn equation -ε2Δu=u(u-a(x))(1-u) in a smooth bounded domain ΩN, u=0 on Ω, with 0<a()<1 continuous and ε>0 a small parameter, cannot have globally minimizing solutions with transition layers in a smooth subdomain of Ω whereon a-12 does not change sign and a-120 on that subdomain’s boundary. Under the assumption of radial symmetry, this property was shown by Dancer and Yan in [5]. Our approach may also be used to simplify some parts of the latter and related references. In particular, for this model, we can give a streamlined new proof of the existence of locally minimizing transition layered solutions with nonsmooth interfaces, considered originally by del Pino in [6] using different techniques. Besides of its simplicity, the main advantage of our proof is that it allows one to deal with more degenerate situations. We also establish analogous results for a class of problems that includes the spatially inhomogeneous Fisher–KPP equation -ε2Δu=ρ(x)u(1-u) with ρ sign-changing.

Keywords: Singular Perturbations; Variational Methods; Elliptic Equation; Transition Layer; Spatial Inhomogeneity; Allen–Cahn Equation; Fife–Greenlee Problem; Fisher Equation

MSC 2010: 35J60; 35B25

1 Introduction and Main Results

Consider the well-studied elliptic problem

{-ε2Δu=u(u-a(x))(1-u)in Ω,u=0on Ω,(1.1)

where a() is a continuous function satisfying 0<a(x)<1 for xΩ¯, Ω is a bounded domain in N, N1, with smooth boundary, and ε>0 is a small number. In [15], this problem was referred to as the spatially inhomogeneous Allen–Cahn equation, while in [7] as the Fife–Greenlee problem.

For the physical motivation behind this problem as well as for the extensive mathematical studies that have been carried out over the last decades, we refer the interested reader to the recent articles [7, 15] and the references therein.

The functional corresponding to (1.1) is

Iε(u)=ε22Ω|Du|2𝑑x-ΩF(x,u)𝑑x,uH01(Ω),

where

F(x,t)=0ts(s-a(x))(1-s)𝑑s.(1.2)

In this paper, we will study the behavior of global and local minimizers of the above functional as ε0. Using the same techniques, we will also study the globally minimizing solutions of the spatially inhomogeneous Fisher–KPP type equation. In the appendixes, we state two variational lemmas that we will use throughout this paper.

1.1 Global Minimizers of the Spatially Inhomogeneous Allen–Cahn Equation

It is easy to see that the minimization problem

inf{Iε(u):uH01(Ω)}

has a minimizer. Minimizers furnish classical solutions of (1.1) (at least when a is Hölder continuous) with values in [0,1] and, more precisely, in (0,1), provided that ε is sufficiently small (see [5, Lemma 2.2]). Let

A={x:xΩ,a(x)<12}andB={x:xΩ,a(x)>12}.

In [5, Theorem 1.1], Dancer and Yan show that any global minimizer uε of Iε in H01(Ω) satisfies

uε{1uniformly on any compact subset of A,0uniformly on any compact subset of B,(1.3)

as ε0. However, this result provides no information about the global minimizers near the set S={xΩ: a(x)=12}. Their proof uses a comparison argument (see Lemma B.1 below) together with a result from [3] (see also Lemma A.1 herein) that the minimizer of the problem

inf{ε22Bτ(x0)|Du|2𝑑x-Bτ(x0)Fb(u)𝑑x:u-φH01(Bτ(x0))}(1.4)

with Fb(t)=0ts(s-b)(1-s)𝑑s tends to 1 (or 0) uniformly on Bτ2(x0) if b<12 (or b>12), as ε0, for any φ with 0φ1; here, Bτ(x0)={x:xN,|x-x0|<τ}. There is no similar result for the case b=12. Actually, in the latter case, the minimizer may have an interior transition layer for some φ with 0φ1 (see [2] and the references therein). On the other hand, if Ω is a ball centered at the origin and a() is radially symmetric, then so is every global minimizer uε of Iε in H01(Ω) (see [5, Proposition 2.6]). Moreover, [5, Theorem 1.3 (i)–(ii)] tells us that for any 0<r1<r2r3<r4 with a(ri)=12, i=1,2,3,4, such that a(r)<12 (or >12) for r(r1,r2)(r3,r4) and a(r)=12 for r[r2,r3], we have that uε1 (or 0) uniformly on any compact subset of (r1,r4), as ε0. The proof of this result relies heavily on the radial symmetry of uε making use of a blow-up argument together with results stemming from the proof of De Giorgi’s conjecture in low dimensions and an energy comparison argument (using the same approach, with a few modifications, a more general radially symmetric problem was treated in [16]). As was pointed out in [5], the nonsymmetric case is far from understood. Nevertheless, in the current paper, we are able to verify the validity of the corresponding nonradial version of the above result as follows.

Assume that a(x)12 (or 12) in a smooth domain A1 (or B1) such that A¯1Ω (or B¯1Ω) and a(x)<12 (or >12) on A1 (or B1). Then, any global minimizer uε of Iε in H01(Ω) satisfies uε1 (or uε0) uniformly on A¯1 (or B¯1), as ε0.

Proof.

We will only consider the case A, since the case B is identical. Let η>0 be any number such that

2η<minxΩ¯(1-a(x)).(1.5)

For small δ>0, we have a(x)<12 if dist(x,A1)δ. Therefore, by (1.3), we deduce that uε1 uniformly on the compact subset of A that is described by {xΩ:dist(x,A1)δ2}, as ε0. Consider the subset of Ω that is defined by A2=A1{xΩ:dist(x,A1)<δ2}. We fix a small δ such that A2A1 is smooth and A¯2Ω. Since any global minimizer satisfies 0<uε<1 if ε is small, we have that

1-uε(x)η,xA2.(1.6)

We claim that 1-uε(x)η, xA¯2, which clearly implies the validity of the assertion of the theorem. Suppose that the claim is false. Then, for some sequence of small ε’s, there exists an xεA2 such that

1-uε(xε)=maxxA¯2(1-uε(x))>η.(1.7)

We will first exclude the possibility that

1-uε(x)2η,xA¯2.(1.8)

To this end, we will argue by contradiction. Let

u~ε(x)={max{uε(x),2-2η-uε(x)},xA2,uε(x),xΩA2.

Since max{uε,2-2η-uε} is the composition of a Lipschitz function with an H1(A2) function, it follows from [8] that u~εH1(A2). Furthermore, from (1.6) and the Lipschitz regularity of A2 we obtain that u~εH01(Ω), see again [8]. Note that u~εC(Ω¯). On the other hand, (1.8) implies that

1-2ηuε(x)u~ε(x)1,xA¯2.

In turn, recalling (1.2) and (1.5), this implies that

F(x,uε(x))F(x,u~ε(x)),xA¯2.(1.9)

To see this, observe that

for each xΩ¯ the function F(x,t) is increasing with respect to t[1-2η,1],(1.10)

since Ft(x,t)=t(t-a(x))(1-t). (Note that tF(t,x) changes monotonicity in (0,1) only at t=a(x)). From (1.7), which implies that uε(xε)<u~ε(xε), it follows that F(x,uε(x))<F(x,u~ε(x)) on an open subset of A2 containing xε. Hence,

ΩF(x,uε(x))𝑑x<ΩF(x,u~ε(x))𝑑x.(1.11)

Moreover, it holds that

Ω|Du~ε|2𝑑xΩ|Duε|2𝑑x,(1.12)

see [14, p. 93]. The above two relations yield that Iε(u~ε)<Iε(uε), contradicting the fact that uε is a global minimizer of Iε in H01(Ω). Consequently, we have that

0<1-uε(xε)<1-2η.(1.13)

Now, let

u^ε(x)={min{1,max{uε(x),2-2η-uε(x)}},xA2,uε(x),xΩA2,

see also [11]. As before, it is easy to see that u^εH01(Ω). Since a(x)12, xA¯2, it follows readily that

F(x,t)<F(x,1)for all t(0,1),xA¯2.

Hence, as before, making use of (1.10), (1.13), and the above relation, we get (1.11), (1.12) with u^ε in place of u~ε, which again contradict the minimality of uε. ∎

In the radially symmetric case, if 0<r1<r2r3<r4 satisfy a(ri)=12, i=1,2,3,4, and a(r)<12 (or >12) for r(r1,r2), a(r)>12 (or <12) for r(r3,r4), and a(r)=12 for r[r2,r3], incorporating our approach into the proof of [5, Theorem 1.3 (iii)–(iv)] can lead to a simpler proof of the fact that global minimizers have only one transition layer in (r1,r4), see also [1], which for N2 takes place near r2 (or r2).

1.2 Local Minimizers of the Spatially Inhomogeneous Allen–Cahn Equation

In the case where there exists a smooth (n-1)-dimensional submanifold Γ of Ω that divides Ω in an interior and an exterior subdomain, which we denote by Ω- and Ω+, respectively, such that a=12 and aν>0 (or <0) on Γ, where ν denotes the outer normal to Γ, it was shown in the pioneering work of Fife and Greenlee [9] that (1.1) has a solution 0<wε<1 such that

wε{1(or 0),uniformly on any compact subset of Ω-,0(or 1),uniformly on any compact subset of Ω+,(1.14)

as ε0. Their approach was based on matched asymptotics and on bifurcation arguments. Such a solution is said to have a transition layer along the interface wε=0, which collapses in a smooth manner to Γ, as ε0. In fact, they considered more general equations of the form ε2Δu=f(x,u) and their proof carries over to the case of finitely many such interfaces. This result was extended by del Pino in [6], via degree-theoretic arguments, for general (even nonsmooth) interfaces. In the following theorem, we present a truly simple proof of the result in [6] for (1.1), which also allows for transition layers between degenerate stable roots of the equation f(x,)=0 (see also [1, Hypothesis (h)]). In fact, with a little more work in the proof and using some ideas from [21], even more degenerate situations can be allowed.

Assume the existence of a closed set ΓΩ and of open disjoint subsets Ω+ and Ω- of Ω such that

Ω=Ω+ΓΩ-.

Assume also the existence of an open neighborhood 𝒩 of Γ such that

a(x)<12(or >12)for x𝒩Ω-,a(x)>12(or <12)for x𝒩Ω+.

Then, there exists a solution 0<wε<1 of (1.1) that satisfies (1.14). Moreover, wε is a local minimizer of Iε in H01(Ω).

Proof.

We will only consider the first scenario, since the one depicted in parentheses can be handled identically. Let η,δ be any positive numbers such that

4η<minxΩ¯a(x)+minxΩ¯(1-a(x))and{x:dist(x,Γ)δ}𝒩.

For convenience purposes, we will assume that Ω is a part of Ω+ (otherwise, the solution would also have a boundary layer along Ω). Let

Ω±δ={xΩ±:dist(x,Γ)>δ}

and

C={uH01(Ω):u2η a.e. on Ω¯+δ,1-u2η a.e. on Ω¯-δ}.

It is easy to verify that the constrained minimization problem

inf{Iε(u):uC}

has a minimizer wεC such that 0wε1 (see the related paper [11]). Our goal is to show that wε does not realize (touch) the constraints if ε>0 is sufficiently small. Naturally, this will imply that wε is a local minimizer of Iε(u) in H01(Ω) and thus a classical solution of (1.1) satisfying the desired assertions of the theorem. The minimizer wε of the constrained problem is a classical solution of the equation (1.1) in {x:dist(x,Γ)<δ}, and in fact a global minimizer in the sense that Iε(wε)Iε(wε+ϕ) for every ϕ that is compactly supported in this region. Furthermore, by the strong maximum principle (see, for example, [12, Lemma 3.4]), we deduce that 0<wε<1 in the same region. As in [5], making use of Lemma B.1 in Appendix B, we can bound wε from below by the minimizer of (1.4), with b=max{a(x),x𝒩Ω-¯}<12 and φ0, over every ball that is contained in Ω-{x:dist(x,Γ)<δ}. From the result of [3] which we mentioned in the introduction (see also Lemma A.1 herein), we obtain that wε1, uniformly on Ω-{x:dist(x,Γ)[δ4,δ2]}, as ε0. In particular, for small ε>0, we have

0<1-wε(x)ηif xΩ- such that dist(x,Γ)=δ2.

As in the part of the proof of Theorem 1.1 that is below (1.6), it follows that the above relation holds for all xΩ- such that dist(x,Γ)δ2. We point out that here the function wε may not be continuous in the vicinity of the constraints, but it is as long as it does not touch them, since there it is a classical solution of (1.1), which suffices for our purposes. Analogous relations hold in Ω+. Consequently, wε stays away from the constraints for small ε>0 and is therefore a local minimizer of Iε in H01(Ω) with the desired asymptotic behavior (1.14), since η,δ>0 can be chosen arbitrarily small. ∎

1.3 Global Minimizers of the Spatially Inhomogeneous Fisher–KPP Equation

Using the same approach, we can treat the elliptic problem

{-ε2Δu=ρ(x)g(u)in Ω,u=0on Ω,(1.15)

where Ω is as before, gC1 such that

g(0)=g(1)=0,g(t)>0 for t(0,1),g(t)<0 for t(0,1),

ρC(Ω¯), and ε>0 is a small number. Note that this includes the important Fisher–KPP equation, where g(t)=t(1-t), arising in population genetics (see [10]).

The functional corresponding to (1.15) is

Jε(u)=ε22Ω|Du|2𝑑x-Ωρ(x)G(u)𝑑x,uH01(Ω),

where

G(t)=0tg(s)𝑑s.(1.16)

It is easy to see that the minimization problem

inf{Jε(u):uH01(Ω)}

has a minimizer. Minimizers furnish classical solutions of (1.15) (at least when ρ is Hölder continuous) with values in [0,1] and, more precisely, in (0,1), provided that ε is sufficiently small. Let

A={x:xΩ,ρ(x)>0}andB={x:xΩ,ρ(x)<0}.

Similarly to [5, Theorem 1.1], using Lemmas A.1 and B.1 below, we can show that any global minimizer uε of Jε(u) satisfies (1.3) (related results can be found in [4] and in [13, Chapter 10]).

In the nondegenerate case, where Γ is a finite union of smooth (n-1)-dimensional submanifolds of Ω such that ρ=0 and ρν0 on Γ, where ν denotes the outer normal to Γ, it can be shown that the width of the transition region of wε is of order ε23 (see [18]). On the other side, in the corresponding nondegenerate case of (1.1) considered in [9], the width of the transition region is of order ε. This difference can be traced back to the fact that the one-dimensional version of (1.1) falls in the framework of standard geometric singular perturbation theory, see [20] (u=0, u=1 are asymptotically stable roots of f(x,u)=0, with respect to the dynamics of u˙=f(x,u), for all xΩ¯), whereas the corresponding version of (1.15) is not (here, the roots u=0, u=1 of g(u)=0 exchange stability as x crosses Γ) and one has to use a blow-up transformation (see [17]).

A Minimizers of a Homogeneous Problem over Balls

The following lemma can be found in [19] and generalizes the result of [3] that we mentioned in relation to (1.4).

Suppose that WC2 satisfies 0=W(μ)<W(t), t[0,μ), W(t)0, t, W(-t)W(t), t[0,μ], or W(t)<0, t<0, for some μ>0. Let x0N, τ>0, η(0,μ), and D>D, where D is determined from the relation 𝐔(D)=μ-η, where in turn 𝐔 is the only function in C2[0,) that satisfies

𝑼′′=W(𝑼) for s>0,𝑼(0)=0,lims𝑼(s)=μ,

(keep in mind that 𝐔>0). There exists a positive constant ε0, depending only on τ,η,D,W, and n, such that there exists a global minimizer uε of the energy functional

E(v)=ε22Bτ(x0)|Dv|2𝑑x+Bτ(x0)W(v)𝑑x,vH01(Bτ(x0)),

which satisfies 0<uε(x)<μ, xBτ(x0), and

μ-ηuε(x),xB¯(τ-Dε)(x0),

provided that ε<ε0.

B A Comparison Lemma from [5]

The following result is [5, Lemma 2.3].

Let 𝒟 be a bounded domain in N with smooth boundary. Let g1(x,t),g2(x,t) be locally Lipschitz functions with respect to t, measurable functions with respect to x, and for any bounded interval I, there exists a constant C such that supx𝒟,tI|gi(x,t)|C, i=1,2, holds. Let

Gi(x,t)=0tgi(x,s)𝑑s,i=1,2.

For φiW1,2(𝒟)=H1(𝒟), i=1,2, consider the minimization problem

inf{Ji(u;𝒟):u-φiW01,2(𝒟)=H01(𝒟)},

where

Ji(u;𝒟)=𝒟{12|u|2-Gi(x,u)}𝑑x.

Let uiW1,2(𝒟), i=1,2, be minimizers to the minimization problems above. Assume that there exist constants m<M such that

  • mui(x)M a.e. for i=1,2, x𝒟,

  • g1(x,t)g2(x,t) a.e. for x𝒟, t[m,M],

  • Mφ1(x)φ2(x)m a.e. for x𝒟.

Suppose further that φiW2,p(𝒟) for any p>1 and that they are not identically equal on 𝒟. Then, we have

u1(x)u2(x),x𝒟.

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About the article

Received: 2015-01-20

Accepted: 2015-04-25

Published Online: 2016-01-21

Published in Print: 2016-02-01


Funding Source: University of Crete

Award identifier / Grant number: DIKICOMA

This project has received funding from the European Union’s Seventh Framework programme for research and innovation under the Marie Skłodowska-Curie grant agreement No. 609402 – 2020 researchers: Train to Move (T2M). At its first stages, it was funded by the DIKICOMA project of the University of Crete.


Citation Information: Advanced Nonlinear Studies, Volume 16, Issue 1, Pages 67–73, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365, DOI: https://doi.org/10.1515/ans-2015-5016.

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