On the loss of compactness in the vectorial heteroclinic connection problem
We give an alternative proof of the theorem of Alikakos-Fusco [AF] concerning existence of heteroclinic solutions to the system
Here are local minima of a potential with . (1) arises in the theory of phase transitions. Our method is variational but differs from the original artificial constraint method of [AF] and establishes existence by analysing the loss of compactness in minimising sequences of the action in the appropriate functional space. Our assumptions are slightly different from those considered previously and also imply a priori estimates for the solution.
Key words and phrases:Heteroclinic connection problem, loss of compactness, phase transitions, Hamiltonian system.
2010 Mathematics Subject Classification:Primary 34C37, 46B50, 82B26; Secondary 37K05
In this paper we consider the problem of existence of heteroclinic solutions to the Hamiltonian ODE system
where is a potential and are local minima of it with . A typical for is shown in Figures 1,2. Solutions to (1) are known as “heteroclinic connections”, being standing waves of the gradient diffusion system
(1) arises in the theory of phase transitions. For details we refer to Alikakos-Bates-Chen [ABC] and to Alberti [Al]. From the viewpoint of physics, (1) is the Newtonian law of motion with force induced by the potential and the trajectory of a test particle which connects two maxima of . In the scalar case of , existence is textbook material by phase plane methods. For a variational approach we refer to Alberti [Al]. Even in this simple case the unboundedness of implies that standard compactness and semicontinuity arguments fail when one tries to obtain solutions to variationally as minimisers of the Action functional
However, for rearrangement methods do apply (Kawohl [Kaw]). When , (1) is much more difficult. It has first been considered by Sternberg in [St], as a problem arising in the study of the elliptic system . Noting the compactness problems, he utilises the Jacobi Principle to obtain solutions by studying geodesics in the Riemannian manifold .
Following a different approach, Alikakos-Fusco [AF] subsequently treated (1) utilising the Least Action Principle. They derived their solutions as minimisers of (3). They introduced an artificial constraint in order to restore compactness and apply the Direct Method and obtained solutions to the (1) by eventually removing the constraint. The same approach has subsequently been applied by Alikakos jointly with the author [AK] to the respective travelling wave problem for (2), establishing existence of solution to the system for . (1) has attracted some attention in connection with the study of system and related material appears also in Alama-Bronsard-Gui [ABG], Bronsard-Gui-Schatzman [BGS], Alikakos [A, A2] and Alikakos-Fusco [AF3].
The problem (1) is nontrivial; except for the failure of the Direct Method for (3) due to the loss of compactness, an additional difficulty when is that the Maximum Principle does not apply. In the papers [AF], [AK] were introduced substitutes of the Maximum Principle for minimisers. Inspired by these results, the author in [Ka] developed related ideas which apply to general nonconvex functionals. A further difficulty of (1) is that additional minima of obstruct existence and suitable assumptions on must be imposed (see [AF]).
In the present work, following [AF], we obtain solutions to (1) as minimisers of (3). We bypass their unilateral constraint method which is of independent interest, but requires a rather delicate analysis. We establish existence for (1) by analysing and then restoring by hand the loss of compactness in minimising sequences. Our motivation comes from the theory of Concentration Compactness (see Lions [L1, L2], and also Bates-Xiaofeng [BX] for a related application of this principle). We note however that Lions’ theory merely motivated the ideas utilised herein and we do not know if the well-known condition of “strict inequality” applies in the present context. Our approach is conceptually different: we introduce a functional space tailored for the study of (1) and show that given any minimising sequence of (3), there exist uniformly decaying translates up to which compactness is restored and passage to a minimiser is available (Theorem 2.1). Our main ingredients are certain energy estimates and measure bounds which relate to those of [AF], [AK]. Herein however we utilise a different method: we control the behaviour of the minimising sequence by the sup-level sets and compactify the sequence by suitable translations.
Our basic assumption (A1) is slightly stronger than the respective of [AF], but we still allow for a certain degree of degeneracy. Under this assumption we obtain the a priori quantitative decay estimates (2.1) by means of energy arguments, without linearising the equation. The rest of the assumptions (A2’), (A2”) allow for ’s with several minima and possibly unbounded from below, being similar to those of [AF]. We believe that our proof of the Alikakos-Fusco theorem [AF] provides further insights to the understanding of the problem.
2. Hypotheses, Setup and the Existence-Compactness Result.
Hypotheses. We assume with local minima at zero: . Moreover:
(A1) There exist and such that for all the sublevel sets contain two stricitly convex components , each enclosing respectively such that and
In addition, at least one of the following two properties is satisfied: either
(A2’) we have
(A2”) there exists a convex bounded (localisation) set and a such that are global minima of , while
(A1) allows for flatness at the minima for all (but not flatness as in [AF], [AK]). The assumption (A2’) requires that are the only components of the sublevel sets . We note that there is a crucial local monotonicity assumption hidden inside (A1). this monotonicity is included in the statement that the level sets coincide with the boundaries of the sublevel sets and hence “flatness” is exluded.
Under assumption (A2’), we immediately obtain . The assumption (A2”) allows for ’s which may be unbounded from below, assuming nonnegativity of only within .
Under (A2”) the existence of a local minimiser of (3) with is a certain issue, but (A1) is more crucial. We shall refer to (A2’) as the “coercive” and to (A2”) as the “non-coercive” assumption.
Functional setup. We derive solutions to (1) as minimisers of (3) in an affine Sobolev space which incorporates the boundary condition and excludes the trivial solutions . Let denote the local Sobolev space of vector functions . For consider the affine function
and set . For , the affine -space, is a complete metric space for the distance. The function (4) will serve also as an a priori upper bound on the action (3) of the minimiser. For , we introduce the affine anisotropic Sobolev space
(Existence - Compactness) Assume that satisfies (A1) and either (A2’) or (A2”), with , , , as in (A1), (A2’), (A2”). There exists a minimising sequence of the problem
In addition, any such minimising solution satisfies the decay estimates
as well as the bound , where
(2.1) imply that the solution is nontrivial. In particular, .
Theorem 2.1 asserts that translation invariance of (1) and (3) causes the only possible loss of compactness to minimising sequences. The space plays a special role to this description. The estimates (2.1) are an essential property, satisfied uniformly by the compactified sequence of the translates and may not be satisfied by the initial . In addition they are quantitative, in the sense that the constant depends explicitly on the potential. Moreover, they guarantee that and , both fully, not merely up to subsequences.
3. Proof of the Main Result.
Control on the minimising sequence. Let be any minimising sequence of (3). We will tacitly identify each with its precise representatives. Since
we have the inclusion . By (4), we obtain
and hence the explicit bounds
We immediately get
is necessarily a strict upper bound since all are merely Lipschitz while minimising solutions to (1) must be smooth (this latter fact is a consequence of standard regularity considerations of the solutions to the Euler-Lagrange equations). Further, for large we have
We now derive bounds. They are obtained in two different ways, depending on whether (A2’) of (A2”) is assumed. In the case of (A2’), it is a consequence of the next energy estimate. For and we define the control set
Let denote the Lebesgue measure on and the constant in estimates (2.1).
(Energy Estimate I) Assume satisfies (A2’). Then we have
for all .
This proves (8). Let now be a subinterval of such that the endpoints of lie on different components of . Hence, we have
by using that , we deduce
( bound under (A2’)) If satisfies (A1), (A2’), then
Now we turn to the case of (A2”). We obtain existence of a minimising sequence of (3) localised inside whereon .
( bound under (A2”)) If satisfies (A1), (A2”), there is a minimising sequence for which and . Moreover,
Proof of Lemma 3.3. We show the existence of a deformation of to a new such that on and all the minimising sequences of (3) relative to in can be chosen to be localised inside . By (A2”), inside and on . We define by reflecting with respect to the hyperplane the portions of the graph of which lie in the halfspace , to .
By construction, , for . Suppose for the shake of contradiction that has a minimising sequence such that for some and , . This is the only case that has to be excluded since by the definition of the ”tails” of each approach asymptotically , at least along a sequence (in general of course there may exists countably many such intervals and we apply this argument to each one of them). By replacing by the straight line segment with the same endpoints, i.e. by defining
we obtain by convexity of that . By pointwise comparison,
In addition, minimises the Dirichlet integral since it is a straight line, thus
In the case that (A2”) is assumed, we fix a sequence valued inside . Moreover,
As the notation suggests, the right hand side will henceforth stand for . Now we employ (A1) to show that is connected. For , , we set
We also set
We note that is the distance between the 2 components of the level set .
(Control on the times) Assume satisfies (A1) and either (A2’) or (A2”). Then, for , if is the minimising sequence constructed previously, then the respective sets are intervals and hence
Proof of Lemma 3.4. The claim follows by a direct application of the Replacement Lemma 12 in p. 1381 of [AK] by choosing as the Lebesgue measure on . In order to make the presentation self-contained, we provide also an alternative proof which bypasses this maximum principle type of result of [AK]. We note that the result follows by the replacement lemma of [AF] as well, but this is not entirely direct since herein we use convex level sets and not balls.
We fix a term of the minimising sequence and a respective and we drop the subscript . Since is open, there exist countably many open intervals such that
Since , each image is connected with endpoints on and
For all , the image has endpoints on different components of .
Indeed, supose for the sake of contradiction that for some , both and are on . The deformation of Lemma 3.3 together with the strictness of assumption (A1) contradicts minimality of . The same holds if the endpoints are on . The claim follows.
The set consists of finitely many intervals of odd number.
and hence for each , by (17),
Hence, by Lemma 3.1, we have
which implies that there exists a no greater than the integer part of such that
and equals , exits for the 1st time at and stays inside after (Figure 4). Since
in view of (18) the number of interval has to odd, for otherwise stays inside for infinite time and this contradicts that (at least along a sequence) converges to as .
All subsets , , … , of the image lie inside the interior and can not touch the boundary (Figure 4).
Fix a and assume for the sake of contradiction that there is , such that lies on the boundary . Then, by replacing by the straight line segment with the same endpoints (as in Lemma 3.1), we obtain a contradiction.
Hence, if touches the boundary , this happens at isolated points (and otherwise it is inside ).
Fix such a point and call it . By continuity and by assumption (A1), there exist such that lies outside .
By replacing by the straight line segment with the same endpoints (as in Lemma 3.1), we obtain a contradiction. By arguing for all such points , we see that lies inside , as desired.
, that is has only one connected component and hence .
We argue by contradiction. Suppose that and consider the set
Since lies strictly inside the sublevel set, we have that . We set
Since the components are finitely many, their distance from the minimum of is bounded away frow zero and hence . By definition of , there exists at least one of the components , say for , which touches only the boundary of and does not intersect . Morover, it can not touch the boundary at more than one points. Hence,
and consequently is contained into and only is on , having both the endpoints , on . By arguing as in Lemma 3.1 for , we obtain a contradiction to the minimality of the action of . Hence, .
The following sharpens (8), under the additional information that is connected.
(Energy estimate II) For all and , we have
where we have also used Lemma 3.4. In addition,
The Lemma follows. ∎
(Uniform bounds on ) For ,, we have
Restoration of Compactness. The bounds (21) provide information which allow to control the behaviour of each by “tracking” the ’s. In the terminology of [ABG], translation invariance of (3) and (1) allows us to “fix a centre” for the ’s and align the minimising sequence, preventing the terms from escaping to . For , we set
which is the centre of the control set . We define the translates of the minimising sequence by:
For these translates, their control sets are centred at , being symmetric (Figure 5). The control sets of and of are related by
The translates defined by (22), (23) will be referred to as the compactified sequence relative to the initial . The sequence will turn out to be weakly precompact in , converging to a solution of (1).
Bounds and Decay Estimates for the Compactified Sequence. The bound on the derivatives is immediate by the kinetic energy term of (3). The more interesting uniform bound is a consequence of our assumption (A1) on the nonconvex potential term.