1 Introduction and Main Results
Consider the well-studied elliptic problem
where is a continuous function satisfying for , Ω is a bounded domain in , , with smooth boundary, and is a small number. In , this problem was referred to as the spatially inhomogeneous Allen–Cahn equation, while in  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
In this paper, we will study the behavior of global and local minimizers of the above functional as . 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
has a minimizer. Minimizers furnish classical solutions of (1.1) (at least when a is Hölder continuous) with values in and, more precisely, in , provided that ε is sufficiently small (see [5, Lemma 2.2]). Let
In [5, Theorem 1.1], Dancer and Yan show that any global minimizer of in satisfies
as . However, this result provides no information about the global minimizers near the set . Their proof uses a comparison argument (see Lemma B.1 below) together with a result from  (see also Lemma A.1 herein) that the minimizer of the problem
with tends to 1 (or ) uniformly on if (or ), as , for any φ with ; here, . There is no similar result for the case . Actually, in the latter case, the minimizer may have an interior transition layer for some φ with (see  and the references therein). On the other hand, if Ω is a ball centered at the origin and is radially symmetric, then so is every global minimizer of in (see [5, Proposition 2.6]). Moreover, [5, Theorem 1.3 (i)–(ii)] tells us that for any with , , such that (or ) for and for , we have that (or ) uniformly on any compact subset of , as . The proof of this result relies heavily on the radial symmetry of 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 ). As was pointed out in , 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 (or ) in a smooth domain (or ) such that (or ) and (or ) on (or ). Then, any global minimizer of in satisfies (or ) uniformly on (or ), as .
We will only consider the case A, since the case B is identical. Let be any number such that
For small , we have if . Therefore, by (1.3), we deduce that uniformly on the compact subset of A that is described by , as . Consider the subset of Ω that is defined by . We fix a small δ such that is smooth and . Since any global minimizer satisfies if ε is small, we have that
We claim that , , 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 such that
We will first exclude the possibility that
To this end, we will argue by contradiction. Let
Since is the composition of a Lipschitz function with an function, it follows from  that . Furthermore, from (1.6) and the Lipschitz regularity of we obtain that , see again . Note that . On the other hand, (1.8) implies that
To see this, observe that
since . (Note that changes monotonicity in only at ). From (1.7), which implies that , it follows that on an open subset of containing . Hence,
Moreover, it holds that
see [14, p. 93]. The above two relations yield that , contradicting the fact that is a global minimizer of in . Consequently, we have that
see also . As before, it is easy to see that . Since , , it follows readily that
In the radially symmetric case, if satisfy , , and (or ) for , (or ) for , and for , 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 , see also , which for takes place near (or ).
1.2 Local Minimizers of the Spatially Inhomogeneous Allen–Cahn Equation
In the case where there exists a smooth -dimensional submanifold Γ of Ω that divides Ω in an interior and an exterior subdomain, which we denote by and , respectively, such that and (or ) on Γ, where ν denotes the outer normal to Γ, it was shown in the pioneering work of Fife and Greenlee  that (1.1) has a solution such that
as . Their approach was based on matched asymptotics and on bifurcation arguments. Such a solution is said to have a transition layer along the interface , which collapses in a smooth manner to Γ, as . In fact, they considered more general equations of the form and their proof carries over to the case of finitely many such interfaces. This result was extended by del Pino in , via degree-theoretic arguments, for general (even nonsmooth) interfaces. In the following theorem, we present a truly simple proof of the result in  for (1.1), which also allows for transition layers between degenerate stable roots of the equation (see also [1, Hypothesis (h)]). In fact, with a little more work in the proof and using some ideas from , even more degenerate situations can be allowed.
Assume the existence of a closed set and of open disjoint subsets and of Ω such that
Assume also the existence of an open neighborhood of Γ such that
We will only consider the first scenario, since the one depicted in parentheses can be handled identically. Let be any positive numbers such that
For convenience purposes, we will assume that is a part of (otherwise, the solution would also have a boundary layer along ). Let
It is easy to verify that the constrained minimization problem
has a minimizer such that (see the related paper ). Our goal is to show that does not realize (touch) the constraints if is sufficiently small. Naturally, this will imply that is a local minimizer of in and thus a classical solution of (1.1) satisfying the desired assertions of the theorem. The minimizer of the constrained problem is a classical solution of the equation (1.1) in , and in fact a global minimizer in the sense that 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 in the same region. As in , making use of Lemma B.1 in Appendix B, we can bound from below by the minimizer of (1.4), with and , over every ball that is contained in . From the result of  which we mentioned in the introduction (see also Lemma A.1 herein), we obtain that , uniformly on , as . In particular, for small , we have
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 such that . We point out that here the function 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 . Consequently, stays away from the constraints for small and is therefore a local minimizer of in with the desired asymptotic behavior (1.14), since 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
where Ω is as before, such that
, and is a small number. Note that this includes the important Fisher–KPP equation, where , arising in population genetics (see ).
The functional corresponding to (1.15) is
It is easy to see that the minimization problem
has a minimizer. Minimizers furnish classical solutions of (1.15) (at least when ρ is Hölder continuous) with values in and, more precisely, in , provided that ε is sufficiently small. Let
In the nondegenerate case, where Γ is a finite union of smooth -dimensional submanifolds of Ω such that and on Γ, where ν denotes the outer normal to Γ, it can be shown that the width of the transition region of is of order (see ). On the other side, in the corresponding nondegenerate case of (1.1) considered in , 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  (, are asymptotically stable roots of , with respect to the dynamics of , for all ), whereas the corresponding version of (1.15) is not (here, the roots , of exchange stability as x crosses Γ) and one has to use a blow-up transformation (see ).
A Minimizers of a Homogeneous Problem over Balls
Suppose that satisfies , , , , , , or , , for some . Let , , , and , where is determined from the relation , where in turn is the only function in that satisfies
(keep in mind that ). There exists a positive constant , depending only on , and n, such that there exists a global minimizer of the energy functional
which satisfies , , and
provided that .
B A Comparison Lemma from 
The following result is [5, Lemma 2.3].
Let be a bounded domain in with smooth boundary. Let 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 , , holds. Let
For , , consider the minimization problem
Let , , be minimizers to the minimization problems above. Assume that there exist constants such that
a.e. for , ,
a.e. for , ,
a.e. for .
Suppose further that for any and that they are not identically equal on . Then, we have
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About the article
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.