Heteroclinic connections and Dirichlet problems for a nonlocal functional of oscillation type

We consider an energy functional combining the square of the local oscillation of a one--dimensional function with a double well potential. We establish the existence of minimal heteroclinic solutions connecting the two wells of the potential. This existence result cannot be accomplished by standard methods, due to the lack of compactness properties. In addition, we investigate the main properties of these heteroclinic connections. We show that these minimizers are monotone, and therefore they satisfy a suitable Euler-Lagrange equation. We also prove that, differently from the classical cases arising in ordinary differential equations, in this context the heteroclinic connections are not necessarily smooth, and not even continuous (in fact, they can be piecewise constant). Also, we show that heteroclinics are not necessarily unique up to a translation, which is also in contrast with the classical setting. Furthermore, we investigate the associated Dirichlet problem, studying existence, uniqueness and partial regularity properties, providing explicit solutions in terms of the external data and of the forcing source, and exhibiting an example of discontinuous solution.


Introduction
One of the most classical problems in ordinary differential equations consists in the study of second order equations coming from mechanical systems having a Hamiltonian structure. For instance, one can consider the simple one-dimensional case in which the Hamiltonian has the form H(p, q) = p 2 2 − W (q) , p, q ∈ R, giving rise to the system of equations Noticing thatq =ṗ, this system of ordinary differential equations reduces to the single second order equation An interesting analogue of (1.1) in partial differential equations arises in the study of phase coexistence models, and in particular in the analysis of the Allen-Cahn equation (1.5) ∆u + u − u 3 = 0.
A well established topic for the dynamical systems as in (1.4) is the search for heteroclinic orbits, that are orbits which connect two (Lyapunov unstable) equilibria. These solutions have the special feature of separating the phase space into regions in which solutions exhibit different topological behaviors (e.g. oscillations versus librations), and, in higher dimensions, they provide the essential building block to chaos.
The analogue of such heteroclinic connections for the phase coexistence problems in (1.5) provides a transition layer connecting two (variationally stable) pure phases of the system. In higher dimensions, these solutions constitute the cornerstone to describe at a large scale the phase separation, as well as the phase parameter in dependence to the distance from the interface.
In this article we explore a brand new line of investigation focused on a nonlocal analogue of (1.1), in which the second derivative is replaced by a finite difference. More concretely, we will consider a functional similar to that in (1.2), but in which the derivative is replaced by an oscillation term. We recall that other nonlocal analogues of (1.1) have been considered in the literature, mainly replacing the second derivative with a fractional second derivative, see [4,6,14,18,19]. Other lines of investigation took into account the case in which the second derivative is replaced by a quadratic interaction with an integrable kernel, see [1] and the references therein.
The interest in this problem combines perspectives in pure and in applied mathematics. Indeed, from the theoretical point of view, nonlocal functionals typically exhibit a number of novel features that are worth exploring and provide several conceptual difficulties that are completely new with respect to the classical cases. On the other hand, in terms of applications, nonlocal functionals can capture original and interesting phenomena that cannot be described by the classical models.
In our framework, in particular, we take into account a nonlocal interaction which is not scale invariant. This type of nonlocal structures is closely related to several geometric motions that have been recently studied both for their analytic interest in the calculus of variations and for their concrete applicability in situations in which detecting different scales allows the preservation of details and irregularities in the process of removing white noises (e.g. in the digitalization of fingerprints, in which one wants to improve the quality of the image without losing relevant features at small scales). We refer in particular to [5,[7][8][9][10][11][12][13]16] for several recent contributions in the theoretical and applied analysis of nonlocal problems without scale invariance.
While the previous literature mostly focuses on geometric evolution equations, viscosity solutions, perimeter type problems and questions arising in the calculus of variations, in this paper we aim at investigating the existence and basic properties of heteroclinic minimizers for nonlocal problems with lack of scale invariance.
Since this topic of research is completely new, we will need to introduce the necessary methodology from scratch. In particular, one cannot rely on standard methods, since: • the problems taken into account do not possess standard compactness properties, • the functional to minimize cannot be easily differentiated, • the solutions found need not to be (and in general are not) regular, • the solutions found need not to be (and in general are not) unique.
In the rest of the introduction, we give formal statements concerning the mathematical setting in which we work and we present our main results, regarding the existence of the heteroclinic connections, their monotonicity properties, the Euler-Lagrange equation that they satisfy, and their lack of regularity and uniqueness. Then, we take into account the Dirichlet problem, obtaining explicit solutions and optimal oscillation bounds. Given r > 0, a < b, with b − a > 2r, and W ∈ C(R), we consider the energy functional As customary, we say that u ∈ L ∞ loc (R) is a local minimizer of E if, for any a < b and any v ∈ L ∞ loc (R) such that u = v outside [a + r, b − r], we have that Notice that, due to the nonlocal character of the oscillation functional, we require the competitor v to coincide with the minimizer u outside [a + r, b − r] instead of [a, b] (see [8] for a discussion of this issue).
We shall assume the following structural conditions on the potential W : W (−1) = W (1) = 0 < W (t) for all t ∈ R \ {−1, 1}, W is strictly decreasing in (−∞, −1) and strictly increasing in (1, +∞), W is an even function in [−1, 1] and has a unique local maximum at t = 0, and we denote (1.10) The last assumption in (1.9) can be slightly generalized by simply assuming that W has a unique local maximum in [−1, 1], with some minor technical adaptations of our arguments. We point out that condition (1.9) is satisfied by the standard "double well" potentials, e.g. the ones in (1.3) and (1.6).
In the forthcoming Subsections 1.2, 1.3 and 1.4 we give precise statements of our main results concerning the existence, possible uniqueness, and geometric properties of the minimizers of the functional in (1.8), specifically focused on heteroclinic connections, that, in our setting, are critical points of the functional which connect the two equilibria −1 and 1. The Dirichlet problem associated to (1.8) (when restricted to monotone functions) will be described in detail in Subsection 1.5.

1.2.
Existence of minimal heteroclinic connections. Now we discuss the construction of local minimizers to (1.8) which connect the two stable equilibria −1 and 1. Our main result on this topic goes as follows: Theorem 1.1 (Existence of minimal heteroclinic connections). Assume that (1.9) holds true. Then, there exists a local minimizer u ∈ L ∞ (R) to (1.8) such that u is monotone nondecreasing and satisfies Moreover, We stress that the existence of heteroclinic connections in nonlocal problems is usually a rather difficult task in itself, which cannot be achieved by standard ordinary differential equations methods and cannot rely directly on conservation of energy formulas. For problems modeled on fractional equations a careful investigation of heteroclinic solutions and of their basic properties has been recently performed in [2,4,6,14,15,[17][18][19].
The case that we treat in this paper is very different from the existing literature, due to the lack of scale invariance. In particular, the proof of the existence result in Theorem 1.1 is more involved than the standard argument based on direct methods, due to the lack of appropriate compactness results for the oscillation functional. In order to gain compactness in our case, we will need to prove that it is possible to restrict the space of competitors for the Dirichlet problem to monotone functions, and then we utilize suitable approximation arguments in compact intervals.
The technical arguments utilized in the proofs are specifically tailored to our case, since we do not have any a priori information on the regularity of the competitors involved in the minimization, and solutions may be discontinuous. Therefore all the methods based on pointwise analysis and geometric considerations are not available at once in our setting, and they need to be replaced by ad-hoc arguments.
See also [7] for further discussions about compactness issues for oscillatory functionals, and [8] for existence and rigidity results for minimizers when W ≡ 0. The case W ≡ 0 that we consider in this paper cannot be reduced to the existing literature on the subject, since it is the presence of a nontrivial potential that defines the notion of equilibria and makes the construction of heteroclinic orbits meaningful.
1.3. Geometric properties of minimal heteroclinic connections. Now we describe the main characteristics of the minimal heteroclinic connections given by Theorem 1.1. In particular we will show that: • they are monotone, • they satisfy an appropriate "finite difference" Euler-Lagrange equation, • and they are not necessarily continuous, since there exists at least one minimal heteroclinic connection which is piecewise constant on intervals of length 2r. In our setting, the simplest competitor for heteroclinic connections is the piecewise constant function defined (up to translations) as It is easy to check that A natural question is whether this is also a minimal heteroclinic connection. This would be the case if one considers discontinuous double well potentials of the form W (t) := χ (−1,1) (t). On the other hand, we can rule out the possibility that u 0 is a minimizer when either r is sufficiently small or W is sufficiently regular, as stated precisely in the next result: Assume that (1.9) holds true. Then, the function u 0 is not a minimal heteroclinic connection if We also show that all the heteroclinic connections are necessarily monotone: Theorem 1.3 (Monotonicity of the heteroclinics). Assume that (1.9) holds true. Let u ∈ L ∞ loc (R) be a local minimizer to (1.8) which satisfies (1.11). Then, u is monotone nondecreasing.
An interesting consequence of Theorem 1.3 is that every minimal heteroclinic connection satisfies an appropriate finite difference equation, which can be seen as an Euler-Lagrange equation associated to the energy functional in (1.8). The precise result that we have goes as follows: Theorem 1.4 (Euler-Lagrange equation). Assume that (1.9) holds true and that W restricted to [−1, 1] is a C 1 function. Let u ∈ L ∞ loc (R) be a a local minimizer for the functional in (1.8) which satisfies (1.11). Then u satisfies the Euler-Lagrange equation We stress that it is not evident to obtain pointwise equations as in (1.15) directly from the minimization of oscillation functionals as in (1.8) since, roughly speaking, it is not easy to carry the derivatives inside the oscillation terms (for instance, while in the classical case one can obtain equation (1.1) by simply taking derivatives of the functional in (1.2), this approach does not lead to equation (1.15) by direct differentiation of the functional in (1.8)).
On the other hand, it is always desirable to find necessary conditions for minimization, and, in our case, the identity in (1.15) plays an important role since it allows us to reconstruct certain values of the minimizers by a partial knowledge of the values nearby. In this sense, the operator on the left hand side of (1.15) is a discretization of the second derivative, and (1.15) can be seen as a discrete version of the classical pendulum and Allen-Cahn equations (compare with (1.1), (1.4) and (1.5)).
It is also interesting to observe that, as r ց 0, the heteroclinic orbits found in Theorem 1.1 recover the classical heteroclinics. This result makes use of the Euler-Lagrange equation given by Theorem 1.4, and its statement goes as follows: Proposition 1.5 (Limit behavior as r ց 0). Assume that (1.9) holds true and that W restricted to [−1, 1] is a C 1 function. For every r > 0, let u r ∈ L ∞ (R) be a local minimizer of (1.8) which is monotone nondecreasing and satisfies (1.11), as given in Theorem 1.1.
Then, up to a translation, we have that u r converges a.e. to the classical heteroclinic u, namely the unique solution to for all x ∈ R, (1.16) u(0) = 0, (1.17) and lim x→±∞ u(x) = ±1. (1.18) In the next result, we show that minimal heteroclinic connections are not necessarily regular, and this is a fundamental difference with respect to the classical case of ordinary differential equations. To this end, we establish the existence of a piecewise constant (and, in particular, discontinuous) minimal heteroclinic connection. Theorem 1.6 (Lack of regularity and discontinuity of heteroclinc connections). Assume that (1.9) holds true and moreover that W restricted to [−1, 1] is a C 1 function. Then, there exists at least one minimal heteroclinic connection u which is piecewise constant, on intervals of length 2r.
In particular, there exists a sequence (u n ) n∈Z such that u(x) ≡ u n for all x ∈ [2nr, 2(n + 1)r).
The sequence u n is monotone nondecreasing, that is u n u n+1 , it satisfies lim n→±∞ u n = ±1, and the recurrence relation  (1.11). This is based on the construction of two different heteroclinic sequences satisfying the recurrence relation in (1.19), as given in the following result: Proposition 1.7. Assume that (1.9) holds true and that W restricted to [−1, 1] is a C 1 function. Then, there exist two different sequences (w n ) n∈Z and (z n ) n∈Z which satisfy the following properties: • (w n ) n∈Z and (z n ) n∈Z are monotone nondecreasing in n, • (w n ) n∈Z and (z n ) n∈Z satisfy the recurrence relation (1.19), • (w n ) n∈Z and (z n ) n∈Z satisfy the limit property We recall that existence of heteroclinic solutions to discrete recurrence relations such as (1.19) has been also considered in the literature, see e.g. [20,21]. A consequence of Proposition 1.7 is the following result. 1.5. The Dirichlet problem. We now observe that the lack of regularity that we pointed out for solutions to the Euler-Lagrange equation (1.15) is a general phenomenon in equations involving the discrete difference operator In particular, we consider the Dirichlet problem associated to this operator with source term f ∈ L ∞ loc (R) and boundary data α, β ∈ L ∞ loc (R): In this setting, we provide basic existence, uniqueness and regularity properties of the solutions of (1.23).
We start by showing that there exists a unique solution to (1.23), according to the following result: Theorem 1.9 (Dirichlet problem for D r ). The system in (1.23) has a unique solution u (up to sets of zero measure), which is given by the function A useful tool towards the proof of the uniqueness result in Theorem 1.9 consists in a suitable Maximum Principle for the operator D r (which will be presented in Lemma 8.1).
We analyze now the regularity of the solution to (1.23). In particular, we obtain a uniform bound on the solution in terms of the external data α and β, and of the source function f . Then, we bound the possible jumps of the solution by a quantity that depends on r, α, β and f (and which becomes small as r ց 0). Then .
Moreover, at any points of J, the function u jumps by at most for some C > 0 depending on a, b, r.
It is interesting to observe that the jump bound in (1.27) improves as r ց 0 and in fact it recovers continuity in the limit (and this fact can be also compared with the asymptotic result of Proposition 1.5). As a counterpart of this observation, we stress that the Dirichlet problem run by the operator D r does exhibit, in general, discontinuous solutions: Corollary 1.11 (Lack of regularity and discontinuity of the solutions of the Dirichlet problem). Fix n ∈ N, with n > 1. The solution of the Dirichlet problem (1.23) with r := 1/n, α := 0, β := 1, f := 0, a := 0 and b := 1 is discontinuous.
The discontinuous example in Corollary 1.11 can be seen as a natural counterpart in the setting of the Dirichlet problem (1.23) of the phenomenon discussed in Theorem 1.6 in the case of global heteroclinics.
Plan of the paper. The rest of this paper is organized as follows. Section 2 contains the construction of a local minimizer to (1.8) which connects monotonically −1 and 1, that is the proof of Theorem 1.1. This construction is obtained by approximation, by solving suitable Dirichlet problems. In Section 3 we prove that every local minimizer to (1.8) which connects the two variationally stable equilibria is monotone, namely Theorem 1.3. Section 4 is devoted to the proof of Theorem 1.4. This is obtained by introducing a new functional F, which coincides with E on monotone functions.
The asymptotics as r ց 0 is discussed in Section 5, which contains the proof of Proposition 1.5. Then, Section 6 contains the proofs of Theorem 1.6, Proposition 1.7, and Corollary 1.8, so in particular it contains the analysis of the discrete version of the Euler-Lagrange equation (1.15), and the nonuniqueness issues described in Proposition 1.7 and Corollary 1.8 are discussed in Section 7.
Finally, in Section 8 we consider the Dirichlet problem for D r , and we present the proofs of Theorem 1.9, and of Corollaries 1.10 and 1.11.
Notation. In the sup, inf, lim sup and lim inf notation, we mean the "essential supremum and infimum" of the function (i.e., sets of null measure are neglected) and the essential superior and inferior limit of a function at a point. Moreover, we shall identify a set E ⊆ R n with its points of density one, and ∂E with the topological boundary of the set of points of density one.
For x ∈ R, we will denote with ⌈x⌉ (resp. ⌊x⌋ ) the smallest integer z such that x z (the biggest integer z such that x z) that is Finally for any u : I ⊂ R → R monotone function, we will always identify u with its right continuous representative.

2.
Existence of minimal heteroclinic connections, and proofs of Theorem 1.1 and Proposition 1.2 The construction of the local minimizer given by Theorem 1.1 will be obtained by approximations, using solutions to appropriate Dirichlet problems.
To this end, we fix R > 2r and consider the minimization problem To prove that (2.1) admits a minimum, we will use standard direct method in the calculus of variations. First of all, though, we need to restrict the space of competitors to gain some more compactness. Namely, we prove that we can consider monotone nondecreasing competitors, as stated in the following result: Then, there exists a monotone nondecreasing functionṽ such thatṽ Proof. We first prove that it is enough to consider competitors v taking values in [−1, 1]. For this, we claim that To prove (2.2), we observe that, by definition, for every c ∈ R and x ∈ R, Then, given v as in the statement of Lemma 2.1, by (2.3) we get that for all x ∈ R Moreover, by the hypothesis on W in (1.9), we have that Hence, (2.2) follows from (1.8), (2.4), and (2.5). Now, if v is monotone nondecreasing, the proof of Lemma 2.1 is completed by takingṽ := v.
Hence, we suppose that v is not monotone nondecreasing, and we provide a method to modify v in [−R + r, R − r] in order to get a monotone nondecreasing functionṽ with lower energy, as desired.
Since v is not monotone nondecreasing, there exist a, b ∈ R such that The idea is to consider all possible quadruples as in (2.6), by substituting v with a functionṽ which coincides with v outside [−R + r, R − r] and has lower energy than v, and this will imply the thesis of Lemma 2.1. The precise details go as follows.
Using (2.3) and the first assumption on W in (1.9), we conclude that otherwise, and we get that . As a consequence, from now on we assume that −1 < B < A < 1. We define Then up to extracting a subsequence, we get that η j → b 0 , for some b 0 ∈ R. In this case, due to (2.6), we have that b 0 > b > a.
We also notice that Hence, we suppose that A + B 0 < 0, and we define Observe that A 0 A, and v(x) A 0 for all x b 0 .
Then up to extracting a subsequence, we get that η j → a 0 , for some a 0 < b 0 .
We observe that if A 0 = 1, we argue as before, definingṽ = 1 as in (2.7) with a 0 in place of a, and we conclude that Now we iterate this procedure, setting So, we are left with the case B 1 < B 0 . The possibility that B 1 = −1 can be dealt with as before. Hence, Observe that necessarily As above, we separate two cases, namely we consider the case in which A 0 + B 1 0 and the one in which A 0 + B 1 > 0. In the first case, we defineṽ as in (2.8) with b 1 in place of b, while in the second case we setṽ as in (2.7) with a 0 in place of a. In both cases, we obtain that These observations took into account all possible cases given by (2.6), and so the proof of Lemma 2.1 is complete.
With the aid of Lemma 2.1, we can prove existence of a solution to the minimization problem in (2.1).
In addition, where c W is as in (1.10).
Proof. In light of Lemma 2.1, the minimization problem in (2.1) is equivalent to the following minimization problem where We start proving (2.9). We consider the function v ±1 ∈ M R with v ±1 := ±1 in (−R + r, R − r). Then, in view of the properties of W given in (1.9), and this implies that Furthermore, we letṽ ∈ M R such that We note that 2 − R + r < R − r, since R > 1 + r. By (1.9) and (1.10), we get that Since r 1 (and so in particular −R + 2r −R + 2) and R > r + 1, we get As a consequence of this and (2.14), From this and (2.12), we obtain (2.9).
We show now that a minimizer u R does exist. To this aim, we consider a minimizing sequence u n ∈ M R , and we have that u n is uniformly bounded. Moreover, the sequence u n has uniformly bounded variation (since it consists of equibounded monotone functions). By compact embeddings of BV (−R + r, R − r) in L p (−R + r, R − r) for every p 1 (see [3,Corollary 3.49]), we obtain that, up to a subsequence, u n → u R pointwise and in L 1 (−R + r, R − r), as n → +∞, for some u R ∈ M R . Now, by the lower semicontinuity of the oscillation functional with respect to L 1 convergence (see [7]), and the continuity of the potential term of the energy with respect to pointwise convergence, we conclude that u R is a solution to (2.10), and therefore to (2.1). Now we are in the position of completing the proof of Theorem 1.1.
Proof of Theorem 1.1. For any R > 2r + 1, we consider the solution u R of (2.1) constructed in Proposition 2.2. We recall the notation in (2.11), and we observe that u R ∈ M R ′ , for any R ′ > R. Hence, we obtain that Therefore, recalling also the uniform bound in (2.9), we conclude that Now, up to a translation, we can assume that, for all R > 2r + 1, u R (x) < 0 for any x < 0 and u R (x) 0 for any x 0. Moreover, we observe that the sequence u R is equibounded, and has equibounded total variation, since the functions u R are all monotone. Thus, by compactness theorem (see [3,Corollary 3.49]) we get that, up to extracting a subsequence, u R → u pointwise and locally in L p for every p 1, as R → +∞. We point out that (2.18) the limit function u ∈ L ∞ (R), with |u| 1, it is monotone nondecreasing and satisfies (1.11).
Now, we set we recall (2.17) and we claim that For this, we fix M > 0. Then, by the lower semicontinuity of the oscillation part of the functional with respect to L 1 convergence and the continuity of the potential part with respect to the pointwise convergence, we get where we used the notation in (2.16).
Now, for any v ∈ L ∞ (R) such that v is monotone nondecreasing and and for any M > 0, we define the function We claim that For this, we fix ε > 0 and we take M sufficiently large such that for any x ∈ [M − 3r, +∞) and for any y ∈ (−∞, −M + 3r], in light of (2.21). This gives that, for any Therefore, using (2.24), As a consequence, which implies the desired result in (2.23) sending ε → 0 and M → +∞. From (2.23), we have that for any ε > 0 there exists M (ε, v) such that We also observe that v M ∈ M M , and therefore, by the minimality of u M and using (2.25), (2.16) and (2.17), we obtain that By the arbitrariness of ε, we conclude that To this end, we argue towards a contradiction, assuming that there exist M 0 > 0, a function v ∈ L ∞ (R) such that and ε > 0 such that Also, recalling (2.17) and (2.19), and using ( Consequently, recalling also (2.30) and the notation in (2.16), we conclude that Furthermore, the functionṽ M belongs to M M (recall the definition of this space in (2.11)). As a consequence of this observation and of (2.31), we find that which is in contradiction with the minimality of u M . This conclude the proof of Theorem 1.1.
We conclude the section proving Proposition 1.2.
Proof of Proposition 1.2. In light of (1.14), to prove the statement, it is sufficient to construct a function v : R → R, which is monotone nondecreasing, satisfies (1.11) and such that To this end, we observe that, if r < 4 4+c W , then and therefore the first case in the statement is a consequence of (1.12).
Hence we now focus on the case in which W is differentiable in ±1. For any ε > 0, we consider the function Since W is differentiable in 1, recalling that W (1) = 0 = W ′ (1) (thanks to (1.9)), we get that and therefore there exists ε 0 = ε 0 (r) such that for all 0 < ε < ε 0 we get that This completes the proof of Proposition 1.2.
3. Rigidity results for minimal heteroclinic connections, and proof of Theorem 1.3 We divide the proof of Theorem 1.3 in several steps. From now on, we assume that u is as in the statement of Theorem 1.3.
In a similar manner, one shows that u −1, and then the claim follows, as desired.
Step 2: u has finite global energy. Namely, we show here that For this, we fix R 2(r + 1) and let for any x ∈ (−R + 1 + r, R − 1 − r), we have that osc On the other hand, by Step 1 and the definition of u R , we have that |u R | 1, and therefore osc (x−r,x+r) u R 2.
Combining this and (3.7), we deduce that Similarly, since u R = 1 in (−R + 1, R − 1), Then, we plug this information and (3.8) into (3.6) and we conclude that By taking R as large as we wish, one deduces (3.5), as desired.
Step 3: monotonicity of u. We suppose, by contradiction, that u is not monotone, and so in particular there exist a, b ∈ R Lebesgue points for u such that By contradiction, if it were not the case, we let (a, b, A, B) any quadruple such that (3.9) holds, with In particular, since (3.10) does not hold, necessarily (3.12) u(x) 0 for almost every x ∈ [a, +∞).
We also notice that, in light of (3.11), and recalling the assumptions in (1.9), we get that W (t) > W (A) for every t ∈ [0, A). Now, we define the function Since A < 1 and (1.11) holds true, we see that there exists c > a such that v(x) = u(x) on [c, +∞). Also, due to (2.3), we get that osc u for all x. Moreover, thanks to (3.12), we have that 0 u(x) v(x) = A < 1 for any x ∈ (a, c). Hence, by (1.9), we obtain that W (v(x)) W (u(x)) for any x ∈ (a, c), with strict inequality on the set {x ∈ (a, c) s.t. u(x) < A} which has positive measure.
Collecting all these pieces of information, we conclude that and v is a competitor for u in (a − r, c + r), since v = u on (−∞, a] ∪ [c, +∞). This is in contradiction with the minimality of u, and therefore (3.10) is established. As a consequence, we fix now a quadruple (a, b 0 , A, B 0 ) as in (3.10), with −1 < B 0 < 0, and we define (3.14) .
We observe that, since A 0 > −1 and (1.11) holds true, the sequence η j is uniformly bounded, otherwise, passing to a subsequence, we would have η j → −∞ and u(η j ) → −1 = A 0 . So, passing to a subsequence, we define We claim that Not to interrupt this calculation, we postpone the proof of this claim, which is quite long, to Step 4. We prove now that Assume on the contrary that A 0 + B 0 0. If this is true, since −1 < B 0 0 and B 0 < A 0 < 1, due to assumption (1.9), we get that W (t) > W (B 0 ) for all t ∈ (B 0 , A 0 ). We define the function We observe that, since B 0 > −1 and (1.11) holds true, there exists c 0 < 0 such that v(x) = u(x) on (−∞, c 0 ]. Moreover, by the definition of A 0 in (3.14), we get that u(x) A 0 for almost every x b 0 . Therefore, as shown before, we get that W (v(x)) W (u(x)) for any x ∈ (c 0 , b 0 ), with strict inequality on the set which has positive measure. Therefore v = u on (−∞, c 0 ] ∪ [b 0 , +∞) and has strictly less potential energy in (c 0 − r, b 0 + r). These observations contradict the minimality of u, and so (3.17) holds true.
We define now We show that Indeed, if this were not the case, then, in light of (3.19), we have that B 1 = B 0 and u(x) B 0 for almost every x a 0 . We note that since A 0 + B 0 > 0 by (3.17), and B 0 < 0, then W (t) > W (A 0 ) for every t ∈ [B 0 , A 0 ). We define the function v as in (3.13) with A 0 in place of A and a 0 in place of a. Again, since We observe that, due to the fact that (1.11) holds and B 1 < 1, the sequence ζ j is uniformly bounded, and passing to a subsequence, we define b 1 := lim j ζ j > a 0 .
Following the same arguments as for the proof of (3.16), we can prove that (3.21) B 1 > −1.

See
Step 5 for a brief sketch of this. Next we observe that Indeed, if this were not true, we can argue as in the the proof of claim (3.20), define the function v as in (3.13) with A 0 in place of A and a 0 in place of a, and show that u = v outside a compact interval and moreover that v has strictly less energy of u, since u(x) B 1 for almost every x a 0 , in contradiction with the minimality of u. Then, we claim that Indeed, if this were not the case, then a 0 < b 1 < b 0 , and in particular u(x) A 0 for every x b 1 , and A 0 + B 1 < 0, by (3.22). Hence, we can proceed as in the proof of claim (3.17). That is, briefly, we define the function v as in (3.18) with B 1 in place of B 0 and b 1 in place of b 0 , we show that u = v outside a compact interval and finally we prove that v has strictly less energy of u, since u(x) A 0 for almost every x b 1 , in contradiction with the minimality of u. Now we define (3.24) Then A 1 A 0 > B 1 . Also, we observe that A 1 > A 0 , otherwise we could repeat exactly the same proof of claim (3.23) and obtain a contradiction to the minimality of u. Moreover, we see that A 1 < 1 by using the same argument as for (3.16), see Step 4.
Up to passing to a subsequence we define a 1 := lim j η j .
Since A 1 > A 0 and u(x) A 0 for almost every x b 0 , necessarily a 0 < b 0 < a 1 < b 1 . Moreover u(x) B 1 for almost every x a 1 and u(x) A 1 for almost every x b 1 .
We observe that two possibilities may arise: either A 1 + B 1 < 0 or A 1 + B 1 0. We will show that both of them are in contradiction with the minimality of u and then this implies that quadruples as in (3.9) cannot exist, and then finally that u is monotone.
If A 1 + B 1 0, we argue as in the the proof of claim (3.20), namely we define the function v as in (3.13) with A 1 in place of A and a 1 in place of a, and we show that u = v outside a compact interval and moreover that v has strictly less energy of u, since u(x) B 1 for almost every x a 1 , in contradiction to the minimality of u.
If instead A 1 + B 1 < 0, we can proceed as in the proof of claim (3.17): we define the function v as in (3.18) with B 1 in place of B 0 and b 1 in place of b 0 , we show that u = v outside a compact interval and finally we prove that v has strictly less energy of u, since u(x) A 1 for almost every x b 1 , in contradiction to the minimality of u.
These observations imply that u is monotone, and thus the claim in Step 3 is established.
Now, we observe that, by definition, u ρ = u outside [a 0 , ρ + 3r], so, by the minimality of u, we get that Indeed, if it were not the case, we would have that u(x) = 1 for almost every x a 0 . But this would be in contradiction with the fact that a 0 < b 0 and lim inf x→b 0 u(x) = B 0 < 0. Hence, (3.34) is established. As a consequence of (3.34), for large ρ, we can write Then, we insert this information into (3.33) and we find that Now we observe that, thanks to (3.26), for any x ∈ [ρ, ρ + 3r], Using this, as long as µ > 0 is sufficiently small (possibly in dependence on r), we get from (3.36) that We also observe that if ρ ρ µ + 2r and x ρ − r, then x − r ρ µ > a 0 , and therefore, by the definition of u ρ , and recalling (3.26), we get Then, using this observation and recalling (3.29), we conclude that for any x ρ − r with ρ ρ µ + 2r. From (3.38), for any x ρ − r and ρ ρ µ + 2r, we have that Therefore, we conclude that, if ρ ρ µ + 2r, where E(u) is the energy defined in (3.5). Plugging this information into (3.37), and recalling the claim (3.5) in Step 2, we conclude thatĉ 2 5µ 2 r + 5µr + µE(u) 2r 2 , which leads to a contradiction by sending µ ց 0, and this concludes the proof of (3.16).
Step 5: proof of claim (3.21). For the proof of (3.21), the argument is the same as for the proof of (3.16) in Step 4, with obvious modifications. We sketch it briefly for the reader's convenience.

As done in
Step 4, it is easy to check that for any x ∈ (b 1 − r, b 1 + r) As a consequence of these observations and of the minimality of u,

As in
Step 4, we see that, for λ << −1, for someĉ > 0, independent of µ, λ, otherwise we would get u(x) = −1 for almost every x b 1 in contradiction with the definition of A 0 .
Thus, using the fact that u λ = −1 in [λ, b 1 ), we conclude that Therefore, for λ << −1, we get that Moreover, recalling the definition of u λ , it is easy to check that, for any x λ + r and λ < λ µ − 2r, there holds osc Now the conclusion follows plugging this information in (3.41) and sending µ → 0, obtaining a contradiction as in Step 4.

4.
The difference equation satisfied by minimal heteroclinic connections, and proof of Theorem 1.4 In this section, we provide a proof of Theorem 1.4. In order to get the result, we will need to introduce an auxiliary functional. We observe that, for monotone functions, the oscillation defined in (1.7) reads as osc (x−r,x+r)

Moreover, it is easy to check that for any
We introduce the following auxiliary functional, defined, for any r > 0, a < b, and W as in (1.9), as In this setting, we say that u ∈ L ∞ loc (R) is a local minimizer of F if, for any a < b and any v ∈ L ∞ loc (R) such that u = v outside [a + r, b − r], we have that It is easy to check that if u is a critical point for the operator F (a,b) in (4.1), then Comparing (1.8) and (4.1), one notices that for any v ∈ L ∞ loc (R), and moreover

Combining (4.3) with (4.4) we obtain that
if u is monotone and it is a local minimizer for the functional in (4.1), then it is also a local minimizer for the functional in (1.8). (4.5) In order to prove Theorem 1.4, in light of (4.2), it is sufficient to show that the reverse statement of (4.5) holds true. This will be accomplished in Proposition 4.2 below.
To this aim, we need the following result, which is the analogous of Proposition 2.2 for the functional (4.1). More precisely: Lemma 4.1. Assume that (1.9) holds true and consider R > 2r.
Then there exists a solution v R to the Dirichlet problem (4. 6) inf , v(x) = 1 for a.e. x R − r and v(x) = −1 for a.e. x −R + r .
Moreover v R is monotone nondecreasing and piecewise constant on intervals of length at least 2r.
Finally v R is also a solution to the Dirichlet problem (2.1).
Proof. We notice that for any c ∈ R and for any v ∈ L ∞ loc (R), we get that which implies (4.7) in this case. The case in which v(x − r) c and v(x + r) c can be treated similarly.
which gives (4.7) in this case as well. Therefore, thanks to (4.7) and (1.9), we get that This implies that we can reduce to consider the case in which −1 v(x) 1 for a.e. x. So, we fix v ∈ L ∞ (R) with values in [−1, 1] such that From (4.8), we also see that (4.11) Furthermore, by the change of variable y = x + nr, From this and (4.11), dividing odd and even indexes, we have that Hence, in light of (4.10), if we set K := N 2 , we can write (4.12) Then, we consider the finite minimization problem: where the class of competitors is such that w j = −1 for all j 0, w j = 1 for all j K + 1, and w j ∈ [−1, 1] for all j ∈ Z. Note that, by (4.10) and (4.12), we get that (4.14) With analogous arguments as in the proof of Lemma 2.1, one can see that monotone sequences make the energy functional lower, and therefore, as in Proposition 2.2, one finds that there exists a monotone nondecreasing solution to (4.13), that is a solution with w j w j+1 . Let us denote with (w R j ) this solution. We define now a function v R : [−R − r, R + r] → [−1, 1] as follows: Recalling (4.14), we see that v R attains the minimal possible value, and this concludes the proof of existence of a monotone nondecreasing, piecewise constant solution to (4.6).
Finally, v R is also a solution to the Dirichlet problem (2.1), due to (4.5).
As a consequence of Lemma 4.1, we obtain the counterpart of (4.5).
Proposition 4.2. Let u be a local minimizer for the functional in (1.8) which satisfies (1.11). Then, u is also a local minimizer for the functional in (4.1).
Proof. By Theorem 1.3, we know that u is monotone. Now, we argue by contradiction, assuming that there exist M 0 > 0, a function v ∈ L ∞ (R) such that v = u outside [−M 0 + r, M 0 − r], and ε > 0, such that where (4.4) is used in the last equality.
Since u is a local minimizer for the functional (1.8), and the two functionals (1.8) and (4.1) coincide on monotone functions, we get that v is not monotone in (M 0 + r, M 0 − r). Now we take M > M 0 + r and we consider v M and u M as given in (2.22). Then, we get from (4.16) that Consequently, recalling the notation of e M and e, as given in (2.16) and (2.17), and exploiting (2.19) and (2.25), we see that From (4.17) and (4.19) we get that which gives a contradiction.
From Proposition 4.2 we obtain Theorem 1.4 by arguing as follows: Proof of Theorem 1.4. By Proposition 4.2, we get that u is a local minimizer of (4.1). Therefore, in view of (4.2), it is a solution to (1.15).

5.
Asymptotics as r ց 0, and proof of Proposition 1.5 In this section, we show that the heteroclinic connections constructed in this paper approach, for small r, the classical heteroclinics arising in ordinary differential equations, as stated in Proposition 1.5.
Proof of Proposition 1.5. Let u r be as in Theorem 1.1, for a given r > 0. Up to a translation, we will suppose that (5.1) u r (x) < 0 for all x < 0 and u r (x) 0 for all x > 0.
Since u r is monotone and bounded uniformly in r, and therefore bounded in BV uniformly in r, up to a subsequence we may suppose that u r converges to some u a.e. in R.
Consequently, from (1.15), for every φ ∈ C ∞ 0 (R), This gives that 4u ′′ = W ′ (u), in the distributional sense, and hence in the smooth sense as well, and this proves (1.16). Also, passing to the limit in (5.1), u(x) 0 for all x < 0 and u(x) 0 for all x > 0, which gives that 0 lim and hence (1.17) is established. Now, in light of (1.12), we can write that Accordingly, using Fatou's Lemma, Furthermore, we know that u is monotone nondecreasing and with values in [−1, 1], since so is u r , hence we can define We claim that (5.4) ℓ + = 1.
Indeed, suppose not, say ℓ + < ℓ, for some ℓ < 1. From (1.17) and the monotonicity of u, we know that ℓ + u(0) = 0. Therefore, there exists X > 0 such that for all x X we have that u(x) ∈ − 1 2 , ℓ . This and (1.9) give that, for all x X, In this section, we prove Theorem 1.6. The proof is based on the existence of piecewise constant solutions to the Dirichlet problem (2.1) given in Lemma 4.1.
Proof of Theorem 1.6. By Lemma 4.1, we have that for all R > 2r there exists a monotone solution v R to the Dirichlet problem (2.1), which is piecewise constant.
Thus, up to a translation, we can assume that v R (x) < 0 for any x < 0 and v R (x) 0 for any x > 0.
Recalling the construction in (4.15), we get that there exists n(R) → +∞ as R → +∞ such that (6.1) v R is constant in intervals of the form [2nr, 2(n + 1)r) for all −n(R) n n(R).
Since v R is equibounded and has equibounded total variation, by compactness (see [3,Corollary 3.49]) up to extracting a subsequence, we can define where the limit holds pointwise and locally in L p for every p 1. From this (see (2.27)), we also conclude that v is a local minimizer of (1.8), which satisfies (1.11). Finally, by (6.1) and (6.2), we obtain that v is constant on every interval [2nr, 2(n + 1)r), for n ∈ Z. Proof of Proposition 1.7. Fixed K ∈ N, we consider two finite minimization problems. The first one is the following where the class of competitors (w j ) is such that w 0 = 0, w j = −w −j and w j = 1 for all j K. We note that, for all K > 1, . Indeed we choose the competitor w 0 := 0 and w j := 1 for all j > 0, and then (7.2) plainly follows from (7.1).
Moreover, by (7.1), we see that m The second finite minimization problem is the following: where the class of competitors (z j ) is such that z j = −z −j−1 and z j = 1 for all j K − 1.
Choosing the competitor w j := 1 for all j 0, we get that m (2) K 2 r 2 for all K > 1. Moreover, we see that m K for all K > 1. Arguing as in the proof of Lemma 2.1, one can show that monotone sequences lower the energy, and therefore, as in Proposition 2.2, one obtains that, for all K > 2, there exist monotone nondecreasing solutions (w K j ) and (z K j ) respectively to the minimization problem (7.1) and (7.3). Let us now fix (φ j ) such that φ 0 := 0, φ j = −φ −j and φ j := 0 for all j K − 1. So, the sequence w K j + δφ j is an admissible competitor for the minimization problem (7.1) for all δ ∈ R. Therefore, by the minimality of w K j , we conclude that Now, fixed j * ∈ (0, K − 1), we choose φ j such that φ j * := 1 (and so by assumption φ −j * = −1) and 0 elsewhere. Substituting in (7.4), we thereby get that, for all j ∈ (0, K − 1), From this, using the fact that w K j = −w K −j and that W ′ is an odd function on [−1, 1] by assumption (1.9), we conclude that A similar argument gives that also z K j satisfies (7.5), namely (7.6) Now, using the monotonicity of (w K j ) and (z K j ), we can also take the limit as K → +∞ for the sequences (w K j ) and (z K j ). In this way, we obtain two sequences that we denote by (w j ) and (z j ), respectively. We notice that (w i ) and (z i ) are monotone nondecreasing. Also, they satisfy (1.20) and are odd, in the sense thatw 0 = 0 andw n = −w −n for all n > 0, whereasz n = −z −n−1 . Moreover, using (7.5) and (7.6), we get that they are solutions to (1.19). This conclude the proof of Proposition 1.7.
Remark 7.1. Concerning the proof of Proposition 1.7, we also observe that, arguing as in the proof of Theorem 1.1, we get that (w j ) is a solution to the minimization problem among all sequences such that w j ∈ [−1, 1] for all j ∈ Z, w 0 = 0, w j = −w −j and lim j→+∞ w j = 1, whereas (z j ) is a solution to the minimization problem among all sequences such that z j ∈ [−1, 1] for all j ∈ Z, z i = −z −i−1 for all i 0 and lim j→+∞ z j = 1. Now, we prove Corollary 1.8, as a consequence of Proposition 1.7.
Proof of Corollary 1.8. Let (w n ) n and (z n ) n be as in Proposition 1.7. We define u(x) :=w n and v(x) :=z n for all x ∈ [2nr, 2nr + 2r).
Then, u and v are monotone nondecreasing, satisfy (1.11) and, in view of (1.19), they are solutions to (1.15), as desired.
We stress that u and v are geometrically different (namely, they are not equal up to a translation). Indeed, if we define I n := [2nr, 2nr + 2r), we see that (7.9) u = 0 on an odd number of intervals I n , and v = 0 on an even number of intervals I n .
To check this, we first observe that u =w 0 = 0 in I 0 . By monotonicity, we can taken to be the greatest n for which u = 0 in I n , and then u = 0 in I 0 ∪ . . . In, with u > 0 in In +1 . Then, recalling (1.21), it follows that u = 0 in I −1 ∪ . . . I −n , with u < 0 in I −n−1 . This says that u = 0 in I 0 ∪ I ±1 . . . I ±n and u = 0 elsewhere. Similarly, suppose that v = 0 in some interval I j . From (1.21), we conclude that v = 0 also in I −j−1 . Since j cannot be equal to −j − 1, this argument always provides a couple of intervals on which v = 0. The proof of (7.9) is thereby complete.
8. The Dirichlet problem for D r , and proofs of Theorem 1.9, and of Corollaries 1.10 and 1.11 In this section we provide the proofs of Theorem 1.9, of Corollary 1.10 and of Corollary 1.11. First of all, we observe that uniqueness of solutions to (1.23) is a direct consequence of the following Maximum Principle for the operator D r . Lemma 8.1 (Maximum Principle for D r ). Let r > 0, a < b, and u ∈ L ∞ loc (R) be such that Then u 0 almost everywhere in [a − r, b + r].
Proof. By contradiction, let us assume that σ := sup such that x k →x, x k are density points for u, and u(x k ) → σ as k → +∞ .
We claim that, in fact, this supremum is attained, and To check this, fix m ∈ N, to be taken as large as we wish. By (8.3), we know that there existsx m ∈ S such that |x ⋆ −x m | 1/m. Then, by (8.2), we know that there exists a sequence x k,m ∈ [a − r, b + r] such that Let now x ⋆ m := x km,m . By construction, This and (8.2) imply that x ⋆ ∈ S, which, combined with (8.3), gives (8.4), as desired.
To check this, suppose, by contradiction, that x ⋆ ∈ [a − r, a] (the case x ⋆ ∈ [b, b + r] is similar). Then it must be that Indeed, if x ⋆ ∈ [a − r, a) and x k → x ⋆ as k → +∞, we have that x k ∈ [a − r, a) for large k, and thus which is a contradiction. Then, in view of (8.6), we can take a sequence y k > a such that lim k→+∞ y k = a = x ⋆ and lim k→+∞ u(y k ) = σ.
So we can exploit (8.1) at the point y k , for k sufficiently large such that y k − r < a, and write that 0 Then, passing to the limit as k → +∞, we obtain that 0 σ − 2σ = −σ, which is a contradiction. The proof of (8.5) is thereby complete.
Proof of Theorem 1.9. The uniqueness statement plainly follows from Lemma 8.1, hence we focus on the proof of the fact that the function in (1.24) is a solution to (1.23), which is a direct, albeit tricky, computation.
This proves (1.23) in this case. So, it only remains to check (8.13). To this end, which completes the proof of (1.23) in this case as well. The proof of Theorem 1.9 is thereby finished.
Finally, under the assumption that the functions α, β, f are continuous, we study the regularity of the solution to (1.23) given by Theorem 1.9.