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

Assume that a⁢(x)≤12 (or ≥12) in a smooth domain A1 (or B1) such that A¯1⊂Ω (or B¯1⊂Ω) and a⁢(x)<12 (or >12) on ∂⁡A1 (or ∂⁡B1). Then, any global minimizer uε of Iε in H01⁢(Ω) satisfies uε→1 (or uε→0) uniformly on A¯1 (or B¯1), as ε→0.

Proof.

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

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

We claim that 1-uε⁢(x)≤η, x∈A¯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 xε∈A2 such that

We will first exclude the possibility that

To this end, we will argue by contradiction. Let

u ~ ε ⁢ ( x ) = { max ⁡ { u ε ⁢ ( x ) , 2 - 2 ⁢ η - u ε ⁢ ( x ) } , x ∈ A 2 , u ε ⁢ ( x ) , x ∈ Ω ∖ A 2 .

Since max⁡{uε,2-2⁢η-uε} is the composition of a Lipschitz function with an H1⁢(A2) function, it follows from [8] that u~ε∈H1⁢(A2). Furthermore, from (1.6) and the Lipschitz regularity of A2 we obtain that u~ε∈H01⁢(Ω), see again [8]. Note that u~ε∈C⁢(Ω¯). On the other hand, (1.8) implies that

1 - 2 ⁢ η ≤ u ε ⁢ ( x ) ≤ u ~ ε ⁢ ( x ) ≤ 1 , x ∈ A ¯ 2 .

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

To see this, observe that

since Ft⁢(x,t)=t⁢(t-a⁢(x))⁢(1-t). (Note that t↦F⁢(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,

Moreover, it holds that

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

Now, let

u ^ ε ⁢ ( x ) = { min ⁡ { 1 , max ⁡ { u ε ⁢ ( x ) , 2 - 2 ⁢ η - u ε ⁢ ( x ) } } , x ∈ A 2 , u ε ⁢ ( x ) , x ∈ Ω ∖ A 2 ,

see also [11]. As before, it is easy to see that u^ε∈H01⁢(Ω). Since a⁢(x)≤12, x∈A¯2, it follows readily that

F ⁢ ( x , t ) < F ⁢ ( x , 1 )   for all ⁢ t ∈ ( 0 , 1 ) , x ∈ A ¯ 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<r2≤r3<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 N≥2 takes place near r2 (or r2).

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

Ω = Ω + ∪ Γ ∪ Ω - .

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

a ( x ) < 1 2 ( or > 1 2 )   for x ∈ 𝒩 ∩ Ω - , a ( x ) > 1 2 ( or < 1 2 )   for x ∈ 𝒩 ∩ Ω + .

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

4 ⁢ η < min x ∈ Ω ¯ ⁡ a ⁢ ( x ) + min x ∈ Ω ¯ ⁡ ( 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 = { x ∈ Ω ± : dist ⁡ ( x , Γ ) > δ }

and

C = { u ∈ H 0 1 ⁢ ( Ω ) : u ≤ 2 ⁢ η ⁢ a.e. on ⁢ Ω ¯ + δ , 1 - u ≤ 2 ⁢ η ⁢ a.e. on ⁢ Ω ¯ - δ } .

It is easy to verify that the constrained minimization problem

inf ⁡ { I ε ⁢ ( u ) : u ∈ C }

has a minimizer wε∈C such that 0≤wε≤1 (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),x∈𝒩∩Ω-¯}<12 and φ≡0, 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 wε→1, uniformly on Ω-∩{x:dist⁡(x,Γ)∈[δ4,δ2]}, as ε→0. In particular, for small ε>0, we have

0 < 1 - w ε ⁢ ( x ) ≤ η   if ⁢ x ∈ Ω - ⁢ such that ⁢ dist ⁡ ( x , Γ ) = δ 2 .

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 x∈Ω- 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. ∎