Double layered solutions to the extended Fisher–Kolmogorov P.D.E.

We construct double layered solutions to the extended Fisher–Kolmogorov P.D.E., under the assumption that the set of minimal heteroclinics of the corresponding O.D.E. satisfies a separation condition. The aim of our work is to provide for the extended Fisher–Kolmogorov equation, the first examples of two-dimensional minimal solutions, since these solutions play a crucial role in phase transition models, and are closely related to the De Giorgi conjecture.


Introduction and Statements
We consider the extended Fisher-Kolmogorov P.D.E.
(3) d 4 u dx 4 − βu ′′ + u 3 − u = 0, u : R → R, which was proposed in 1988 by Dee and van Saarloos [10] as a higher-order model equation for bistable systems. Equation (3) has been extensively studied by different methods: topological shooting methods, Hamiltonian methods, variational methods, and methods based on the maximum principle (cf. [5], [20], and the references therein). In these monographs, a systematic account is given of the different kinds of orbits obtained for O.D.E. (3), which has a considerably richer structure than second order phase transition models.
The existence of heteroclinic orbits of (3) via variational arguments was investigated for the first time by L. A. Peletier, W. C. Troy and R. C. A. M. VanderVorst [21], and W. D. Kalies, R. C. A. M. VanderVorst [15]. In the vector case m ≥ 1, we established [24] the existence of minimal heteroclinics for a large class of fourth order systems, including the O.D.E.: (4) d 4 u dx 4 (x) − βu ′′ (x) + ∇W (u(x)), u : R → R m (m ≥ 1), β > 0, x ∈ R, with a double well potential W as in (2). By definition, a heteroclinic orbit is a solution e ∈ C 4 (R; R m ) of (4) such that lim x→±∞ (e(x), e ′ (x), e ′′ (x), e ′′′ (x)) = (a ± , 0, 0, 0) in the phase-space. A heteroclinic orbit is called minimal if it is a minimizer of the Action functional associated to (4): in the class A := {u ∈ H 2 loc (R; R m ) : lim x→±∞ u(x) = a ± }, i.e. if J R (e) = min u∈A J R (u) =: J min . Assuming (2), we know that there exists at least one minimal heteroclinic orbit e (cf. [24]). In addition, since the minima a ± are nondegenerate, the convergence to the minima a ± is exponential for every minimal heteroclinic e, i.e.
where the constants k, K > 0 depend on e (cf. [24,Proposition 3.4.]). Clearly, if x → e(x) is a heteroclinic orbit, then the maps (7) x explicit examples of potentials having at least two minimal heteroclinics can be given. More precisely, Lemma 2.5 provides the existence of potentials W for which the set F of minimal heteroclinics of (4) satisfies the separation condition where d stands for the distance in L 2 (R; R m ), and d(F − , F + ) := inf{ e − − e + L 2 (R;R m ) : e − ∈ F − , e + ∈ F + }. Under this structural assumption, we are going to construct heteroclinic double layers for (1), that is, solutions The existence of double layered solutions for the system ∆u − ∇W (u) = 0, goes back to the work of Alama, Bronsard and Gui [1]. Subsequently, Schatzman [23] managed to remove the symmetry assumption on W considered in [1]. We also mention the work of Alessio [2], where the separation condition (8) has first been introduced, and the new developments on these results presented in [14,19]. Our construction of heteroclinic double layers for (1) is inspired in the approach from Functional Analysis used in the classical theory of evolution equations. This method has recently been applied in the elliptic context (cf. [25]) to give an alternative proof of Schatzman's result [23]. The idea is to view a solution x)] taking its values in an appropriate space of functions, and reduce the initial P.D.E. to an O.D.E. problem for U . Indeed, the uniform in t boundary conditions (9b) suggest to set a map and work in the affine subspace H := e 0 +L 2 (R; R m ) = {u = e 0 +h : h ∈ L 2 (R; R m )} which has the structure of a Hilbert space with the inner product We also denote by · H the norm in H, and by d(u, v) := u − v L 2 (R;R m ) the corresponding distance. In view of (6), it is clear that e ∈ H, and e ′ , e ′′ ∈ L 2 (R; R m ), for every minimal heteroclinic e ∈ F . Next, we reduce system (1) together with the boundary conditions (9), to a variational problem for the orbit U : We shall proceed in several steps. The idea is to split between the variables x and t, the terms appearing in the energy functional (10) E Ω (u) := associated to (1). By gathering the derivatives of u with respect to x, and the potential term, we first define in H, the effective potential W : H → [0, +∞] by u ∈ A, and thus J R (u) ≥ J min . It is also obvious that W only vanishes on the set F of minimal heteroclinics. Subsequently, we define the constrained class 1 in H, and the numbers t − V < t + V depend on V . Finally, we define the Action functional in H by (12) J R (V ) := where we have set for h ∈ L 2 (R; R m ): One can see that the definitions of W and J are relevant, since on a strip [t 1 , t 2 ] × R, the energy functional E is equal to J up to constant. More precisely, if In the proof of Theorem 1.1 below, we show the existence of a minimizer U of J in the constrained class A. This result follows from an argument first introduced in [24, Lemma 2.4.], and from the nice properties of the effective potential W and the set F established in section 2. Let us just mention that W : H → [0, +∞] is sequentially weakly lower semicontinuous (cf. Lemma 2.3), and that the sets F ± intersected with closed balls are compact in H. Next, from the orbit U : R → H, we recover a solution u of (1). On the one hand, the constraint imposed in the class A, forces U to behave asymptotically as in (9a). On the other hand, the second boundary condition (9b) follows from the definition of the space H. In addition, since U is a minimizer, the double layered solution u obtained is minimal, in the sense that (14) E supp φ (u) ≤ E supp φ (u + φ), ∀φ ∈ C 2 0 (R 2 ; R m ). 1 The method of constrained minimization to construct minimal heteroclinics for the system u ′′ − ∇W (u) = 0, goes back to [3]. We refer to [18], [16], [9] and [8], for the general theory of Sobolev spaces of vector-valued functions.
This notion of minimality is standard for many problems in which the energy of a localized solution is actually infinite due to non compactness of the domain. Thus, Theorem 1.1 provides the existence of two-dimensional minimal solutions for (1), whenever the potential W satisfies the separation condition (8).
In contrast with second order phase transition models, the development of the theory of fourth order phase transition models in the P.D.E. context is very recent. The second order Allen-Cahn equation ∆u = u 3 − u, u : R n → R, has been the subject of a tremendous amount of publications in the past 30 years, certainly motivated from several challenging conjectures raised by De Giorgi (cf. [11] and [12]). As far as fourth order P.D.E. of phase transition type are concerned, only a few aspects of this theory have been investigated. Let us mention: the Γ-convergence results obtained in [13,17], the saddle solution constructed in [6], and the one-dimensional symmetry results established in [7], where an analog of the De Giorgi conjecture is stated, and a Gibbon's type conjecture is proved.
The aim of our present work is to provide for equation (1), the first examples of two-dimensional minimal solutions, since these solutions play a crucial role in phase transition models, and are closely related to the De Giorgi conjecture (cf. [22] in the case of second order phase transition models). After these explanations, we give the complete statement of Theorem 1.1: (2) and (8). Then, there exists a minimal solution u ∈ C 4 (R 2 ; R m ) of (1) such that such that For the sake of simplicity we only focused in this paper on the extended Fisher-Kolmogorov equation, since it is a well-known fourth order phase transition model. However, the proof of Theorem 1.1 can easily be adjusted to provide the existence of heteroclinic double layers for a larger class of P.D.E. than (1). It trivially extends to operators such as a 11 u tttt + a 12 We also point out that by dropping the term σ(V ′ (t)) appearing in the definition of J , we still obtain a minimizer U in the class A, and thus a weak solution u of x) ∈ R 2 , satisfying (15a). Finally, instead of the space H = e 0 +L 2 (R; R m ), other Hilbert spaces may be considered in the applications of Theorem 1.1. Indeed, since the properties of the effective potential W and the sets F ± (established in section 2) hold for the H 2 norm, we may construct an heteroclinic orbitŨ connecting F − and F + , either in e 0 +H 1 (R; R m ) or e 0 +H 2 (R; R m ). Then, we recover from each of these orbits, a weak solution of a sixth (resp. eighth) order P.D.E. satisfying (15a). We refer to [25, section 5] for a similar construction in the space e 0 +H 1 (R; R m ), and for the adjustments to make in the proof of these results.

2.
Properties of the effective potential W and the sets F ± , F ± Assuming that (2) holds, we establish in Lemma 2.3 below some properties of the functions W and σ defined in the previous section. We first recall two lemmas from [24].
Then, the maps u ∈ A b as well as their first derivatives are uniformly bounded and equicontinuous.
As a consequence, d(u, F ) → 0, as W(u) → 0, and for every c 1 > 0, there exists c 2 > 0 such that , and let us assume that l = lim inf k→∞ W(u k ) < ∞ (since otherwise the statement is trivial). By extracting a subsequence we may assume that lim k→∞ W(u k ) = l. Next, in view of Lemma 2.1, we can apply to the sequence {u k } the theorem of Ascoli, to deduce that u k → u in C 1 loc (R; R m ), as k → ∞ (up to subsequence). On the other hand, since u ′′ . In addition, one can easily see that u ∈ H 2 loc (R; R m ), and u ′ = v 1 as well as u ′′ = v 2 . Finally, by the weakly lower semicontinuity of the L 2 (R; R m ) norm we obtain u ′′ 2 Gathering the previous results, we conclude that W(u) ≤ l i.e. W(u) ≤ lim inf k→∞ W(u k ). To show the sequentially weakly lower semicontinuity of σ, we proceed in a similar way.
(ii) We first establish that given u ∈ H such that u ′ , u ′′ ∈ L 2 (R; R m ), and e ∈ F , we have In view of (6), it is clear that e ′ , e ′′ , e ′′′ , e ′′′′ as well as ∇W (e) belong to L 2 (R; R m ). As a consequence, we can see that Finally, by substracting (18) (17) follows. Now, we consider a sequence {u k } ⊂ H such that lim k→∞ W(u k ) = 0. According to Lemma 2.2, there exist a sequence {x k } ⊂ R, and e ∈ F , such that (up to subsequence) the mapsū k (x) := u k (x − x k ) satisfy (19) lim Our claim is that According to hypothesis (2b) we have Let µ > 0 be such that , and define the comparison map An easy computation shows that Clearly, by reproducing the same argument in the interval [λ + , ∞), we can find a comparison map z ∈ As a consequence, we obtain (24) inf{J since otherwise we can construct a map in A whose action is less than J min . On the other hand we have Indeed, for such a map v, there exists an interval [x 1 , , thus we can check that In the sequel, we fix an ǫ ∈ (0, r/2) such that [2(4 + β + µ) + 1]ǫ 2 < √ βc(r/2) 2 , and choose an interval According to (19), we have for k ≥ N large enough: Then, combining (24) with (26d), one can see that Therefore, in view of (25) and (26a), it follows that |ū k (x) − a − | ≤ r, ∀x ≤ λ − (resp. |ū k (x) − a + | ≤ r, ∀x ≥ λ + ). Furthermore, as a consequence of (21b) we get To conclude, we apply formula (17) toū k , and combine (26d) with (26b) and (28), to obtain Finally, in view of (26c), we have ū k − e L 2 (R;R m ) < 1 + 2 √ c )ǫ. This establishes our claim (20), from which the statement (ii) of Lemma 2.3 is straightforward.
From the arguments in the proof of Lemma 2.3, we deduce some useful properties of the sets F ± and F ± (defined in the previous section).

Lemma 2.4.
(i) Let {e k } ⊂ F be bounded in H, then there exists e ∈ F , such that up to subsequence lim k→∞ e k − e H 2 (R;R m ) = 0. (ii) There exists a constant γ > 0, such that for every e ∈ F , we can find T ∈ R such that setting The sets F ± are sequentially weakly closed in H, and strongly closed in H. Furthermore, we have Proof. (i) Since {e k } ⊂ F is bounded in H, we have up to subsequence e k ⇀ e in H, as k → ∞, for some e ∈ H. Proceeding as in the proof of Lemma 2.3 (i), we first obtain that (up to subsequence) e k → e in C 1 loc (R; R m ), as k → ∞, with e ∈ F . Next, we reproduce the arguments after (20), with e k instead ofū k . (ii) Assume by contradiction the existence of a sequence N ∋ k → e k ∈ F , such that e T k −e 0 H 2 (R;R m ) ≥ k, ∀T ∈ R. Then, by Lemma 2.3 (ii), there exists a sequence {x k } ⊂ R, and e ∈ F , such that (up to subsequence) the maps e x k k satisfy lim k→∞ e x k k − e H 2 (R;R m ) = 0. Clearly, this is a contradiction.
Then, in view of (i) we have (up to subsequence) e ± k → e ± in H, as k → ∞, with e ± ∈ F ± . As a consequence d(v, F ± ) = v − e ± L 2 (R;R m ) . (iv) We are going to check that F − is sequentially weakly closed (the proof is similar for F + ). Let . Since the sequences {e ± k } are bounded, it follows from (i), that (up to subsequences) e ± k → e ± holds in H, for some e ± ∈ F ± . Our claim is that d(v, e ± ) = d(v, F ± ). Indeed, given f ± ∈ F ± , we have and as k → ∞, we get which proves our claim. To show that d(v, F − ) ≤ d(v, F + ), we proceed as previously. By assumption, we have This establishes that F − is sequentially weakly closed, and thus also strongly closed. In view of the inequality , since otherwise v would be an interior point of F − (resp. F + ).
Finally, in Lemma 2.5 below, we give explicit examples of potentials for which the separation condition (8) holds 2 .

Proof of Theorem 1.1
Existence of the minimizer U . We first establish that inf V ∈A J R < ∞. Indeed, given e ± 0 ∈ F ± , let for t ≥ 1.
To show the existence of the minimizer U , we shall consider appropriate translations of the sequence v k (t, , with respect to both variables x and t. Then, we shall establish the convergence of the translated maps to the minimizer U . Given T ∈ R, and V ∈ H = e 0 +L 2 (R; R m ), we denote by At this stage, we infer that the sequence d(V k (t 2i0−1 (k)), F − ) is bounded. Indeed, when k is large enough, W0 . We also claim that the sets F ± are invariant by the translations L T . To check this, let us pick u ∈ F − (the proof is similar for F + ). By definition, Next, in view of Lemma 2.4 (ii), for every k, we can find T k ∈ R and e k ∈ F − such that e k H ≤ γ and On the one hand, since V k (0) H ≤ η + γ holds for every k, we have that (up to subsequence)V k (0) ⇀ u 0 in H, as k → ∞, for some u 0 ∈ H. On the other hand, sinceV ′ k as well asV ′′ k are uniformly bounded in L 2 (R; L 2 (R; R m )), it follows that up to subsequence Finally, we writeV k (t) =V k (0) + t 0V ′ k (s)ds, and claim that U (t) := u 0 + t 0 U 1 (s)ds is a minimizer of J in A. Indeed, it is clear that U ∈ H 2 loc (R; H), and since t 0V ′ k (s)ds ⇀ t 0 U 1 (s)ds holds in H for every t ∈ R, we also haveV k (t) ⇀ U (t) for every t ∈ R. Similarly, since V ′ k (0) L 2 (R;R m ) is uniformly bounded (cf. Lemma 2.5), it follows that (up to subsequence)V ′ k (0) ⇀ u 1 in L 2 (R; R m ). Thus, for every t ∈ R, we havē , that we are going to determine. On the one hand, in view of the bound V ′ k (t) L 2 (R;R m ) ≤ M ′ , ∀k, ∀t ∈ R, we obtain by dominated convergence that lim k→∞ On the other hand, using the weak convergenceV ′ k ⇀ U 1 in L 2 (R; L 2 (R; R m )), we deduce that lim k→∞ Thus h = 0, and we have established thatV ′ k (t) ⇀ U 1 (t) holds for every t ∈ R. Now, the sequentially weakly lower semicontinuity of W and σ (cf. Lemma 2.3 (i)), implies that ] for every t ∈ R, thus by Fatou's Lemma we obtain To conclude it remains to show that U ∈ A. In view of the above property (b) it follows that U (t) ∈ F − , for every t ≤ 0. Similarly, in view of (a) and (c), we have U (t) ∈ F + , for t ≥ T > 0 large enough.
By elliptic regularity, it follows that u ∈ C 4 (R 2 ; R m ) is a classical solution of (1). On the other hand, it is clear in view of Lemma 3.1 that (15a) holds. Thus to complete the proof of Theorem 1.1, it remains to show (15b). Let us first establish the uniform continuity of u in the strips [t 1 , t 2 ] × R (with [t 1 , t 2 ] ⊂ R). To see this, we shall consider an arbitrary disc D of radius 1 included in the strip [t 1 , t 2 ] × R, and check that for such discs, u H 2 (D;R m ) is uniformly bounded. Indeed, we notice that u L 2 (D;R m ) is uniformly bounded (independently of D), since the function R ∋ t → u(t, ·) − e 0 (·) L 2 (R;R m ) is continuous. Next, in view of the L 2 bounds obtained in Lemma 3.2 for the first and second derivatives of u, we deduce our claim. To prove (15b), assume by contradiction the existence of a sequence (s k , x k ) such that lim k→∞ x k = ∞, s k ∈ [t 1 , t 2 ], and |u(s k , x k ) − a + | > ǫ > 0. As a consequence of the uniform continuity of u, we can construct a sequence of disjoint discs of fixed radius, centered at (s k , x k ), over which W (u) is bounded uniformly away from zero. This clearly violates the finiteness of E [t1,t2]×R (u) = J [t1,t2] (U ) + J min (t 2 − t 1 ).