On the existence of connecting orbits for critical values of the energy
Giorgio Fusco Giovanni F. Gronchi Matteo Novaga






On the existence of connecting orbits for critical values of the energy
    Giorgio
    Fusco,[Dipartimento di Matematica, Università dell'Aquila; e-mail:
      ] 
Giovanni F. Gronchi,[Dipartimento di Matematica, Università
  di Pisa; e-mail:
  ] 
Matteo Novaga[Dipartimento di Matematica, Università di Pisa; e-mail:
  ] 
    Received: date / Accepted: date
==========================================================================================================================================================================================================================================================









We consider an open connected set Ω and a smooth potential U
which is positive in Ω and vanishes on ∂Ω. We
study the existence of orbits of the mechanical system

    ü=U_x(u),

that connect different components of ∂Ω and lie on the
zero level of the energy. We allow that ∂Ω contains a
finite number of critical points of U. The case of symmetric
potential is also considered.





§ INTRODUCTION

Let U:^n→ be a
function of class C^2. We assume that Ω⊂^n is a
connected component of the set {x∈^n: U(x)>0} and
that ∂Ω is compact and is the union of N≥ 1
distinct nonempty connected components Γ_1,…,Γ_N. We
consider the following situations


  H N≥ 2 and, if Ω is unbounded, there is r_0>0 and
  a non-negative function σ:[r_0,+∞)→ such that
    ∫_r_0^+∞σ(r)dr=+∞ and
  
    √(U(x))≥σ(| x|),   x∈Ω,  | x|≥ r_0.


  H_s Ω is bounded, the origin 0∈^n belongs to
  Ω and U is invariant under the antipodal map

    U(-x)=U(x),  x∈Ω.




Condition (<ref>) was first introduced in
<cit.>. A sufficient condition for (<ref>)
is that lim inf_|x|→∞ U(x) >0.


We study non constant solutions u:(T_-,T_+)→Ω, of the
equation

    ü=U_x(u),   U_x=(∂ U/∂ x)^T,

that satisfy

    lim_t→ T_±d(u(t),∂Ω)=0,

with d the Euclidean distance, and lie on the energy surface

    1/2|u̇|^2-U(u)=0.

 We allow that the boundary ∂Ω of Ω contains a
 finite set P of critical
 points of U and assume


  H_1 If Γ∈{Γ_1,…,Γ_N} has positive
  diameter and p∈ P∩Γ then p is a hyperbolic critical
  point of U.

If Γ has positive diameter, then hyperbolic critical points
p∈Γ correspond to saddle-center equilibrium points in the
zero energy level of the Hamiltonian system associated to
(<ref>). These points are organizing centers of complex
dynamics, see <cit.>.

Note that 𝐇_1 does not exclude that some of the
Γ_j reduce to a singleton, say {p}, for some p∈ P.  In
this case nothing is required on the behavior of U in a neighborhood
of p aside from being C^2.

A comment on 𝐇 and 𝐇_s is in order. If P is
nonempty u≡ p for p∈ P is a constant solution of
(<ref>) that satisfies (<ref>) and (<ref>). To
avoid trivial solutions of this kind we require N≥
  2 in 𝐇, and look for solutions that connect different
components of ∂Ω. In 𝐇_s we do not exclude
that ∂Ω is connected (N=1) and avoid trivial solutions
by restricting to a symmetric context and to solutions that pass through
0.

We prove the following results.

Assume that 𝐇 and 𝐇_1 hold. Then for each
Γ_-∈{Γ_1,…,Γ_N} there exist
Γ_+∈{Γ_1,…,Γ_N}∖{Γ_-} and a
map u^*:(T_-,T_+)→Ω, with -∞≤ T_-<T_+≤
+∞, that satisfies (<ref>), (<ref>) and

    lim_t→ T_±d(u^*(t),Γ_±)=0.

Moreover, T_->-∞ (resp. T_+<+∞)  if and only if
Γ_- (resp. Γ_+)
has positive diameter. If T_->-∞ it results

    lim_t→ T_-u^*(t)= x_-,
       lim_t→ T_-u̇^*(t)=0,

for some x_-∈Γ_-∖ P. An analogous statement
holds if T_+<+∞.



Assume that 𝐇_s and 𝐇_1 hold. Then there exist
Γ_+∈{Γ_1,…,Γ_N} and a map
u^*:(0,T_+)→Ω, with 0<T_+≤ +∞, that satisfies
(<ref>), (<ref>) and

    lim_t→ T_+d(u^*(t),Γ_+)=0.

Moreover, T_+<+∞ if and only if Γ_+ has positive diameter. If
T_+<+∞ it results

    lim_t→ T_+u^*(t)=x_+,
       lim_t→ T_+u̇^*(t)=0,

for some x_+∈Γ_+∖ P.

We list a few straightforward consequences of Theorems <ref> and
<ref>.


  Theorem <ref> implies that, if ∂Ω=P, given p_-∈
P there is p_+∈ P∖{p_-} and a heteroclinic connection
between p_- and p_+, that is a solution u^*:→^n of
(<ref>) and (<ref>) that satisfies

    lim_t→±∞u^*(t)=p_±.


The problem of the existence of heteroclinic connections between two
isolated zeros p_± of a non-negative potential has been recently
reconsidered by several authors. In <cit.> existence was established
under a mild monotonicity condition on U near p_±. This
condition was removed in <cit.>, see also <cit.>. The most
general results, equivalent to the consequence of Theorem <ref>
discussed in Section <ref>, were recently obtained in <cit.> and in
<cit.>, see also <cit.>.
All these papers establish existence by a variational
approach. In <cit.>, <cit.> and <cit.> by minimizing the
action functional, and in <cit.> and
<cit.> by minimizing the Jacobi functional.


  Theorem <ref> implies that,
    if Γ_-={p} for some p∈ P and the elements of
  {Γ_1,…,Γ_N}∖{Γ_-} have all positive
diameter, there exists a nontrivial orbit homoclinic to p that satisfies (<ref>), (<ref>).


Let v^*:→Ω∪{x_+}  be the extension
defined by

    v^*(T_++t)=u^*(T_+-t),  t∈(0,+∞),  v^*(T_+)=x_+,

of the solution u^*:(-∞,T_+)→Ω given by
Theorem <ref>.

The map v^* so defined is a smooth non-constant solution of
(<ref>) that satisfies

    lim_t→±∞v^*(t)=p.




Theorem <ref> implies that, if all the sets
Γ_1,…,Γ_N have positive diameter, given
Γ_-∈{Γ_1,…,Γ_N}, there exist
Γ_+∈{Γ_1,…,Γ_N}∖{Γ_-} and a
periodic solution v^*:→Ω of (<ref>) and
(<ref>) that oscillates between Γ_- and
Γ_+. This solution has period T=2(T_+-T_-).



The solution v^* is the T-periodic extension of the map
w^*:[T_-,2T_+-T_-]→Ω defined by w^*(t)=u^*(t) for
t∈(T_-,T_+), where u^* is given by Theorem <ref>, and

    w^*(T_±)=x_±,
       w^*(T_++t)=u^*(T_+-t),   t∈ (0,T_+-T_-].



The problem of existence of heteroclinic, homoclinic and periodic
solutions of (<ref>), in a context similar to the one considered
here, was already discussed in <cit.> where ∂Ω is
allowed to include continua of critical points. Our result concerning
periodic solutions extends a corresponding result in <cit.> where
existence was established under the assumption that P=∅.

The following result is a direct consequence of Theorem <ref>.


Theorem <ref> implies that, if all the sets Γ_1,…,Γ_N have positive diameter,
  there exists Γ_+∈{Γ_1,…,Γ_N} and a
  periodic solution v^*:→Ω of (<ref>) and
  (<ref>) that satisfies
  
    v^*(-t) = -v^*(t),    t∈.

  This solution has period
  T=4T_+, with T_+.



The solution v^* is the T-periodic extension of the map
w^*:[-2T_+,2T_+]→Ω defined by w^*(t) = u^*(t) for
t∈(0,T_+), where u^* is given by Theorem <ref>, and by

    w^*(t) = -w^*(-t), 2cm t∈ (-T_+,0),
    
         w^*(0) = 0,    w^*(± T_+) = ± x_+, 
    
         w^*(T_++t) = w^*(T_+-t),  0.8cm t∈(0,T_+],
    
         w^*(-T_++t) = w^*(-T_+-t),  0.3cm t∈[-T_+,0).

In particular the solution oscillates between x_+ and -x_+ and
this is true also when ∂Ω is connected (N=1).





§ PROOF OF THEOREMS <REF> AND <REF>


We recall a classical result.

Let G:^n→  be a smooth bounded and
non-negative potential, I=(a,b) a bounded interval.


Define the Jacobi functional

    𝒥_G(q, I)=√(2)∫_I√(G(q(t)))|q̇(t)| dt

and the action functional

    𝒜_G(q,I) =
    ∫_I(1/2|q̇(t)|^2+G(q(t)))dt.

Then


  * 
    𝒥_G(q, I)
    ≤𝒜_G(q,I),
    
       q∈ W^1,2(I;^n)

 with equality sign if and only if

    1/2|q̇(t)|^2-G(q(t))=0, t∈ I.


  * 
    min_q∈𝒬𝒥_G(q, I)=min_q∈𝒬𝒜_G(q,I),

where

    𝒬={q∈ W^1,2(I;^n):q(a)=q_a, q(b)=q_b}.



When G=U we shall simply write 𝒥, 𝒜 for
𝒥_U, 𝒜_U.

We now start the proof of Theorem <ref>. Choose
Γ_-∈{Γ_1,…,Γ_N} and set

    d=min{| x-y|: x∈Γ_-, y∈∂Ω∖Γ_-}.

For small δ∈(0,d) let O_δ={x∈Ω:
d(x,Γ_-)<δ} and let U_0=1/2min_x∈∂
  O_δ∩ΩU(x). We note that U_0>0 and define the
admissible set

    𝒰={u∈ W^1,2((T_-^u,T_+^u);^n):
            -∞< T_-^u < T_+^u< +∞, 2.2cm
    
            u((T_-^u,T_+^u))⊂Ω, U(u(0))=U_0,        u(T_-^u)∈Γ_-,        u(T_+^u)∈∂Ω∖Γ_-}.

We determine the map u^* in Theorem <ref> as the limit of a
minimizing sequence {u_j}⊂𝒰 of the action
functional

    𝒜(u,(T_-^u,T_+^u)) =
    ∫_T_-^u^T_+^u(1/2|u̇(t)|^2+U(u(t)))dt,



Note that in the definition of 𝒰 the times
T_-^u and T_+^u are not fixed but, in general,
change with u. Note also that the condition
U(u(0))=U_0 in (<ref>) is

a normalization which can always be imposed by a translation of time
and has the scope of eliminating the loss of compactness due to
translation invariance.

Let x̅_-∈Γ_- and
x̅_+∈∂Ω∖Γ_- be such that |x̅_+-x̅_-|=d and set

    ũ(t)=(1-(t+τ))x̅_-+(t+τ)x̅_+,  t∈[-τ,1-τ],

where τ∈(0,1) is chosen so that U(ũ(0))=U_0. Then
ũ∈𝒰, T_-^ũ=-τ,
T_+^ũ=1-τ and

    𝒜(ũ,(-τ,1-τ))=a<+∞.

Next we show that there are constants M>0 and T_0>0 such that
each u∈𝒰 with

    𝒜(u,(T_-^u,T_+^u))≤ a,

satisfies

    u_L^∞((T_-^u,T_+^u);^n)≤ M,
        T_-^u≤-T_0<T_0≤ T_+^u.


The L^∞ bound on u follows from 𝐇 and from
Lemma <ref>, in fact, if Ω is unbounded, |
u(t̅)|=M for some t̅∈(T_-^u,T_+^u) implies

    a≥𝒜(u,(T_-^u,t̅)) ≥∫_T_-^u^t̅√(2U(u(t)))|u̇(t)|
    dt≥√(2)∫_r_0^Mσ(s)ds.

The existence of T_0 follows from

    d_1^2/|T_-^u|≤∫_T_-^u^0|u̇(t)|^2dt≤ 2a,    
    d_1^2/T_+^u≤∫_0^T_+^u|u̇(t)|^2dt≤ 2a,

where d_1=d(∂Ω,{x: U(x)>U_0}).

Let {u_j}⊂𝒰 be a minimizing sequence

    lim_j→+∞𝒜(u_j,(T_-^u_j,T_+^u_j)) =
    inf_u∈𝒰𝒜(u,(T_-^u,T_+^u)) := a_0≤ a.

We can assume that each u_j satisfies (<ref>) and
(<ref>). By considering a subsequence, that we still denote by {u_j},
we can also assume that there exist T_-^∞, T_+^∞ with -∞≤
T_-^∞≤-T_0<T_0≤ T_+^∞≤ +∞ and a continuous
map u^*:(T_-^∞,T_+^∞)→^n such that

    lim_j→+∞T_±^u_j=T_±^∞,
       lim_j→+∞u_j(t)=u^*(t),  t∈(T_-^∞,T_+^∞),

and in the last limit the convergence is uniform on bounded
intervals. This follows from (<ref>) which
implies that the sequence {u_j} is equi-bounded and from (<ref>) which implies

    | u_j(t_1)-u_j(t_2)|≤|∫_t_1^t_2|u̇_j(t)| dt|≤√(a)| t_1-t_2|^1/2,

so that the sequence is also equi-continuous.

By passing to a further subsequence we can also assume
that u_j⇀ u^* in
W^1,2((T_1,T_2);^n) for each T_1, T_2 with
T_-^∞<T_1<T_2<T_+^∞.  This follows from (<ref>),
which implies

    1/2∫_T_-^u_j^T_+^u_j|u̇_j|^2dt ≤𝒜(u_j,(T_-^u_j,T_+^u_j))≤ a,

and from the fact that each map u_j satisfies (<ref>) and
therefore is bounded in L^2((T_-^u_j,T_+^u_j);^n).

We also have

    𝒜(u^*,(T_-^∞,T_+^∞))≤ a_0.


Indeed, from the lower semicontinuity of the norm, for each
T_1, T_2 with T_-^∞<T_1<T_2<T_+^∞ we have

    ∫_T_1^T_2|u̇^*|^2dt ≤lim inf_j→+∞∫_T_1^T_2|u̇_j|^2dt.

This and the fact that u_j converges to u^* uniformly in
[T_1,T_2]
imply

    𝒜(u^*,(T_1,T_2)) ≤lim inf_j→+∞𝒜(u_j,(T_1,T_2)) ≤lim inf_j→+∞𝒜(u_j,(T_-^u_j,T_+^u_j))=a_0.

Since this is valid for each T_-^∞<T_1<T_2<T_+^∞ the claim
(<ref>) follows.



Define
 T_-^∞≤ T_-≤-T_0<T_0≤ T_+≤
T_+^∞ by setting

    T_-=inf{t∈(T_-^∞,0]:u^*((t,0])⊂Ω}
       T_+=sup{t∈(0,T_+^∞):u^*([0,t))⊂Ω}.

Then


  * 
    𝒜(u^*,(T_-,T_+))=a_0.


  * T_+<+∞ implies lim_t→ T_+u^*(t)=x_+ for some x_+∈Γ_+ and
Γ_+∈{Γ_1,…,Γ_N}∖{Γ_-}.

  * T_+=+∞ implies

    lim_t→+∞d(u^*(t),Γ_+)=0,

 for some Γ_+∈{Γ_1,…,Γ_N}∖{Γ_-}.

Corresponding statements apply to T_-.


We first prove (ii), (iii). If T_+<+∞ the existence of 
lim_t→ T_+u^*(t) follows from (<ref>) which
implies that u^* is a C^0,1/2 map. The limit x_+
belongs to ∂Ω and therefore to Γ_+ for some
Γ_+∈{Γ_1,…,Γ_N}.

Indeed,
x_+∉∂Ω would imply the existence of τ>0 such
that, for j large enough,




    d(u_j([T_+,T_++τ]),∂Ω)≥1/2d(x_+,∂Ω),

in contradiction with the definition of T_+.  If T_+=+∞ and
(iii) does not hold there is δ>0 and a diverging sequence
{t_j} such that

    d(u^*(t_j),∂Ω)≥δ.

Set U_m=min_d(x,∂Ω)=δU(x)>0. From the uniform
continuity of U in {| x|≤ M} (M as in (<ref>)) it
follows that there is l>0 such that

    |
    U(x_1)-U(x_2)|≤1/2U_m,   for  |
    x_1-x_2|≤ l,  x_1,x_2∈{| x|≤ M}.

This and u^*∈ C^0,1/2 imply

    U(u^*(t))≥1/2U_m,  t∈ I_j =
    (t_j-l^2/a,t_j+l^2/a),

and, by passing to a subsequence, we can assume that the intervals
I_j are disjoint. Therefore for each T>0 we have

    ∑_t_j≤ Tl^2U_m/a≤∫_0^TU(u^*(t))dt≤ a_0,

which is impossible for T large. This establishes (<ref>) for
some Γ_+∈{Γ_1,…,Γ_N}.  It remains to show
that Γ_+≠Γ_-. This is a consequence of the minimizing
character of {u_j}. Indeed, Γ_+=Γ_- would imply the
existence of a constant c>0 such that
lim_j→∞𝒜(u_j,(T_-^u_j,T_+^u_j))≥ a_0+c.

Now we prove (i).  T_+- T_-<+∞, implies that u^* is an
element of 𝒰 with T_±^u^*= T_±. It follows that
𝒜(u^*,(T_-,T_+))≥ a_0, which together with (<ref>)
imply (<ref>).  Assume now T_+- T_-=+∞. If T_+=+∞,
(<ref>) implies that, given a small number ϵ>0, there
are t_ϵ and x̅_ϵ∈∂Ω such that
| u^*(t_ϵ)-x̅_ϵ|=ϵ and the
segment joining u^*(t_ϵ) to x̅_ϵ belongs
to Ω.  Set

    v_ϵ(t) =
    (1-(t-t_ϵ))u^*(t_ϵ) +
    (t-t_ϵ)x̅_ϵ,
      t∈(t_ϵ,t_ϵ+1].

From the uniform continuity of U there is η_ϵ>0, lim_ϵ→ 0η_ϵ=0, such that U(v_ϵ(t))≤η_ϵ, for t∈[t_ϵ,t_ϵ+1]. Therefore we have

    𝒜(v_ϵ,(t_ϵ,t_ϵ+1))
     ≤1/2ϵ^2+η_ϵ.

If T_->-∞ the map u_ϵ =
1_[T_-,t_ϵ]u^* +
1_(t_ϵ,t_ϵ+1]v_ϵ
  belongs to 𝒰 and it results

    a_0≤𝒜(u_ϵ,(T_-,t_ϵ+1)) =
    𝒜(u^*,(T_-,t_ϵ)) +
    𝒜(v_ϵ,(t_ϵ,t_ϵ+1))
    ≤𝒜(u^*,(T_-,T_+))+1/2ϵ^2+η_ϵ.

Since this is valid for all small ϵ>0 we get

    a_0≤𝒜(u^*,(T_-,T_+)),

that together with (<ref>)
establishes (<ref>) if T_->-∞ and T_+=+∞. The
discussion of the other cases where T_+-T_- =+∞ is similar.


We observe that there are cases with T_+<T_+^∞ and/or
  T_->T_-^∞, see Remark <ref>.



The map u^* satisfies (<ref>) and (<ref>) in
(T_-,T_+).


  1. We first show that
 


for each T_1, T_2 with T_-<T_1<T_2<T_+ we have

    𝒜(u^*,(T_1,T_2))=inf_v∈𝒱𝒜(v,(T_1,T_2)),

where

    𝒱={v∈ W^1,2((T_1,T_2);^n): v(T_i)=u^*(T_i),i=1,2; 
      v([T_1,T_2])⊂Ω}.

Suppose instead that there are η>0 and v∈𝒱 such that

    𝒜(v,(T_1,T_2))=𝒜(u^*,(T_1,T_2))-η.

Set w_j:(T_-^u_j,T_+^u_j)→Ω defined by

    w_j(t)={[                  u_j(t),  t∈(T_-^u_j,T_1]∪[T_2,T_+^u_j),; v(t)+T_2-t/T_2-T_1δ_1j+t-T_1/T_2-T_1δ_2j,
      t∈(T_1,T_2), ].

where δ_ij=u_j(T_i)-u^*(T_i), i=1,2, with u_j as in (<ref>).

Define v_j:[T_-^v_j,T_+^v_j]→^n by

    v_j(t) = w_j(t - τ_j),

where τ_j is such that U(v_j(0))=U_0, as in
(<ref>). Note that

    𝒜(v_j,(T_-^v_j,T_+^v_j))= 𝒜(w_j,(T_-^u_j,T_+^u_j)).

From (<ref>) we have
  lim_j→∞δ_ij= 0, i=1,2, so that

    lim_j→+∞𝒜(w_j,(T_1,T_2))=
    𝒜(v,(T_1,T_2))
    =
    𝒜(u^*,(T_1,T_2))-η≤lim inf_j→+∞𝒜(u_j,(T_1,T_2))-η.

Therefore we have

    lim inf_j→+∞𝒜(w_j,(T_-^u_j,T_+^u_j))=
      lim_j→+∞𝒜(w_j,(T_1,T_2))+
    lim inf_j→+∞𝒜(u_j,(T_+^u_j,T_1)∪(T_2,T_+^u_j))
    
       ≤lim inf_j→+∞𝒜(u_j,(T_1,T_2))-η+
    lim inf_j→+∞𝒜(u_j,(T_+^u_j,T_1)∪(T_2,T_+^u_j))≤
    a_0-η,

that, given (<ref>), is in contradiction with the
minimizing character of the sequence {u_j}.


The fact that u^* satisfies (<ref>) follows from (<ref>)

and regularity theory, see <cit.>.

To show that u^* satisfies (<ref>) we
distinguish the case T_+-T_-<+∞ from the case
T_+-T_-=+∞.


2. T_+-T_-<+∞.
Given t_0, t_1 with T_-<t_0<t_1<T_+, let
ϕ:[t_0,t_1+τ]→[t_0,t_1] be linear, with |τ|
small, and let ψ:[t_0,t_1]→[t_0,t_1+τ] be the
inverse of ϕ. Define u_τ:[T_-,T_++τ]→^n
by setting

    u_τ(t)={[       u^*(t),  t∈[T_-,t_0],;  u^*(ϕ(t)),  t∈[t_0,t_1+τ],; u^*(t-τ),  t∈(t_1+τ,T_++τ)] ].

Note that u_τ∈𝒰 with T_-^u_τ=T_- and
T_+^u_τ=T_++τ.  Since u^* is a
minimizer we have

    d/dτ𝒜(u_τ,(T_-^u_τ,T_+^u_τ))|_τ=0=0.

From (<ref>), using also the change of variables t=ψ(s), it
follows

    𝒜(u_τ,(T_-^u_τ,T_+^u_τ))-𝒜(u^*,(T_-,T_+))
    
        = ∫_t_0^t_1+τ(ϕ̇^2(t)/2|u̇^*(ϕ(t))|^2 +
      U(u^*(ϕ(t))))dt -∫_t_0^t_1(1/2|u̇^*(t)|^2+U(u^*(t)))dt
    
        = ∫_t_0^t_1(1 -
        ψ̇(t)/2ψ̇(t)|u̇^*(t)|^2 +
      (ψ̇(t)-1)U(u^*(t)))dt
    
        = ∫_t_0^t_1(-τ/t_1-t_0/2(1 +
        τ/t_1-t_0)|u̇^*(t)|^2 +
      τ/t_1-t_0U(u^*(t)))dt
    
       = - τ/t_1-t_0∫_t_0^t_1(
      |u̇^*(t)|^2/2(1 + τ/t_1-t_0) -
      U(u^*(t)))dt .

This and (<ref>) imply

    ∫_t_0^t_1(1/2|u̇^*(t)|^2-U(u^*(t)))dt=0.

Since this holds for all t_0,t_1, with T_-<t_0<t_1<T_+, then (<ref>) follows.


3. T_+-T_-=+∞. We only consider the case
T_+=+∞. The discussion of the other cases is similar.

Let T∈(T_-,+∞), let T_-<t_0<t_1<T and let
ϕ:[t_0,T]→[t_0,T] be linear in the intervals
[t_0,t_1+τ], [t_1+τ,T], with |τ| small, and such that
ϕ([t_0,t_1+τ])=[t_0,t_1].

Define u_τ:(T_-,+∞)→^n by setting

    u_τ(t)={[ u^*(t),  t∈(T_-,t_0]∪[T,+∞);      u^*(ϕ(t)),  t∈[t_0,T]. ].

We have

    𝒜(u_τ,(T_-,T))-𝒜(u^*,(T_-,T))
    
        = ∫_t_0^t_1(-τ/t_1-t_0/2(1 +
        τ/t_1-t_0)|u̇^*(t)|^2 +
      τ/t_1-t_0U(u^*(t)))dt +
      ∫_t_1^T(τ/T-t_1/2(1 +
        τ/T-t_1)|u̇^*(t)|^2
      -τ/T-t_1U(u^*(t)))dt.

Since u^* restricted to the interval [t_0,T] is a minimizer of
(<ref>), by differentiating with respect to τ and setting
τ=0 we obtain

    -1/t_1-t_0∫_t_0^t_1(1/2|u̇^*(t)|^2
    -U(u^*(t)))dt
    +1/T-t_1∫_t_1^T(1/2|u̇^*(t)|^2
    -U(u^*(t)))dt=0.

From (<ref>) it follows that the second term in this expression
converges to zero when T→+∞. Therefore, after taking
the limit for T→+∞, we get back to (<ref>) and,
as before, we conclude that (<ref>) holds.



Assume that lim_t→ T_+u^*(t)=p∈ P. Then

    T_+=+∞.



Since U is of class C^2 and p is a critical point of U there are constants c>0 and ρ>0 such that

    U(x)≤ c| x-p|^2,  x∈ B_ρ(p)∩Ω.

Fix t_ρ so that u^*(t)∈ B_ρ(p)∩Ω for t≥
t_ρ. Then T_+=+∞ follows from (<ref>) and

    d/dt| u^*-p|≥ -|u̇^*|=-√(2U(u^*))≥
    -√(2c)| u^*-p|,   t≥ t_ρ.



We now show that if Γ_+ has positive diameter then T_+<+∞. To prove this we first show that T_+=+∞ implies u^*(t)→ p∈ P as t→+∞, then we conclude that this is in contrast with (<ref>).

If T_+=+∞, then
there is p∈ P such that

    lim_t→+∞u^*(t)=p.

An analogous statement applies to T_-.


If Γ_+={p} for some p∈ P,  then
(<ref>) follows by (<ref>). Therefore we assume that
Γ_+ has positive diameter.  The idea of the proof is to show
that if u^*(t) gets too close to ∂Γ_+∖ P it is
forced to end up on Γ_+∖ P in a finite time in
contradiction with T^*=+∞.

If (<ref>) does not hold there is q>0 and a sequence

{τ_j}, with lim_j→∞τ_j=+∞,
such that d(u^*(τ_j),P)≥ q,

for all j∈.
Since, by
(<ref>) u^* is bounded, using also (<ref>), we can
assume that

    lim_j→+∞u^*(τ_j) = x̅,  for
      some  x̅∈Γ_+∖∪_p∈ P B_q(p).

The smoothness of U implies that there


 are positive constants r̅, r, c and C such
  that


  * the orthogonal projection on
  π:B_r̅(x̅)→∂Ω is well defined
  and π(B_r̅(x̅))⊂∂Ω∖ P;

  * we have

    B_r(x_0)⊂ B_r̅(x̅),  
    x_0∈∂Ω∩ B_r̅/2(x̅);


  * if (ξ,s)∈^n-1× are local coordinates with
  respect to a basis {e_1,…,e_n}, e_j=e_j(x_0), with
  e_n(x_0) the unit interior normal to ∂Ω at
  x_0∈∂Ω∩ B_r̅/2(x̅) it results
  
    1/2c s≤ U(x(x_0,(ξ,s)))≤ 2c s,  |ξ|^2+s^2≤ r^2,  s≥ h(x_0,ξ),

where

    x=x(x_0,(ξ,s))=x_0+∑_j=1^n ξ_je_j(x_0) + se_n(x_0),

and h:∂Ω∩ B_r̅/2(x̅) ×{|ξ|≤ r}→, | h(x_0,ξ)|≤
C|ξ|^2, for |ξ|≤ r, is a local
representation of ∂Ω in a neighborhood of x_0,
that is U(x(x_0,(ξ,h(x_0,ξ))))=0 for |ξ|≤ r.

Fix a value j_0 of j and set t_0=τ_j_0.





If j_0 is sufficiently large, setting t_0=τ_j_0 we
have that x_0=π(u^*(t_0)) is well defined. Moreover
x_0∈∂Ω∩ B_r̅/2(x̅) and

    u^*(t_0)=x_0+δ e_n(x_0),  δ=| u^*(t_0)-x_0|.

For k=8/3√(2) let Q_0 be the set

    Q_0={x(x_0,(ξ,s)):|ξ|^2+(s-δ)^2<k^2δ^2, s>δ/2}.

Since δ→ 0 as j_0→+∞ we can assume
that δ>0 is so small (δ < min{1/2Ck^2,
r/1+k} suffices) that Q_0⊂Ω∩
B_r(x_0).

.2cm
Claim 1. u^*(t)
leaves Q_0 through the disc D_0=∂
Q_0∖∂ B_kδ(u^*(t_0)).

.2cm

From (<ref>) we have a_0≤𝒜(v,(T_-,T_+^v)) for
each W^1,2 map v:(T_-,T_+^v]→^n that coincides
  with u^* for t≤ t_0, and satisfies
  v((t_0,T_+^v))⊂Ω, v(T_+^v)∈∂Ω and
  (<ref>). Therefore if we set
  
    w(s)=x_0+s e_n(x_0),

  s∈[0,δ], we have

    a_0≤𝒜(u^*,(T_-,t_0)) + 𝒥(w,(0,δ)).

On the other hand, if u^*(t_0^')∈∂ Q_0(x_0)∩∂
B_kδ(u^*(t_0)), where

    t_0^'=sup{t>t_0:u^*([t_0,t))⊂Q_0∖∂
      B_kδ(u^*(t_0))},

from (<ref>) it follows

    𝒜(u^*,(T_-,t_0))+𝒥(u^*,(t_0,t_0^'))≤ a_0.

Using (<ref>) we obtain

    𝒥(w,(0,δ))≤4/3c^1/2δ^3/2,

and, since

    cδ/4≤
    U(x(x_0,(ξ,s))),   (ξ,s)∈Q_0(x_0),

we also have, with k defined above,

    8/3c^1/2δ^3/2 =
    k/√(2)c^1/2δ^3/2≤c^1/2δ^1/2/√(2)∫_t_0^t_0^'|u̇^*(t)| dt ≤√(2)∫_t_0^t_0^'√(U(u^*(t)))|u̇^*(t)|
    dt.

From (<ref>) and (<ref>) it follows

    𝒥(w,(0,δ)) ≤1/2𝒥(u^*,(t_0,t_0^')),

and therefore (<ref>) and (<ref>) imply the absurd
inequality a_0<a_0. This contradiction proves the claim.

.2cm From Claim 1 it follows that there is t_1∈(t_0,+∞)
with the following properties:

    u^*([t_0,t_1))⊂ Q_0(x_0),
    
       u(t_1)∈ D_0.

Set x_0,1=π(u^*(t_1)) and δ_1=|
u^*(t_1)-x_0,1|.

Since h(x_0,0)=h_ξ(x_0,0)=0 and the radius
ρ_δ=(k^2-1/4)^1/2δ of D_0 is
proportional to δ, we can assume that δ is so small that
the ratio 2δ_1/δ and | x_0,1 -
  x_0|/| u^*(t_1)-x(x_0,(0,δ/2))| are near
1 so that we have

    δ_1≤ρδ,  for some  ρ<1,
       | x_0,1-x_0|≤ kδ.





















We also have

    t_1-t_0≤ k^'δ^1/2,   k^' =
    8k/c^1/2.

This follows from

    (t_1-t_0)c/4δ≤𝒜(u^*,(t_0,t_1))
    =𝒥(u^*,(t_0,t_1)) 
    
       =√(2)∫_t_0^t_1√(U(u^*(t)))|u̇^̇*̇(t)| dt
    ≤ 2√(cδ)| u^*(t_1)-u^*(t_0)|≤ 2 c^1/2 k δ^3/2.

where we used (<ref>) to estimate 𝒥 on the segment
joining u^*(t_0) with u^*(t_1).

We have u^*(t_1)=x_0,1+δ_1e_n(x_0,1) and we can apply Claim
1 to deduce that there exists t_2>t_1 such that

    u^*([t_1,t_2))⊂ Q_1(x_0,1),
       u^*(t_2)∈ D_1,

where Q_1 and D_1 are defined as Q_0 and D_0 with δ_1
and x(x_0,1,(ξ,s)) instead of δ and
x(x_0,(ξ,s)). Therefore an induction argument yields sequences
{t_j}, {x_0,j}, {δ_j} and {Q_j(x_0,j)} such
that

    u^*([t_j,t_j+1))⊂ Q_j(x_0,j),    x_0,j=π(u^*(t_j)),
       δ_j+1≤ρδ_j≤ρ^j+1δ,
       | x_0,j+1-x_0,j|≤ kδ_j≤ kρ^jδ,
       (t_j+1-t_j)≤ k^'δ_j^1/2≤ k^'ρ^j/2δ^1/2,
       u^*(t_j)=x_0,j+δ_je_n(x_0,j) ∈ D_j.

We can also assume that Q_j(x_0,j)⊂Ω∩ B_r(x_0),
 for all j∈.

This follows from | u^*(t_j+1)-u^*(t_j)|≤
kδ_j≤ kρ^jδ.

From (<ref>) we obtain that

there exists T with
t_0<T≤k^'δ^1/2/1-ρ^1/2 such
that

    u^*(T)=lim_t→ Tu^*(t)=lim_j→+∞
      x_0,j∈∂Ω∖ P,
       | u^*(T)-x_0|≤kδ/1-ρ.

This contradicts the existence of the sequence {τ_j}, with lim_j→∞τ_j=+∞, appearing in
(<ref>) and establishes (<ref>).  The proof
 of the lemma is complete.



We continue by showing (<ref>) contradicts (<ref>).






Assume that Γ_+ has positive diameter. Then

    T_+<+∞.

An analogous statement applies to Γ_- and T_-.


From Lemma <ref>, if T_+=+∞ there exists p∈ P such
that lim_t→+∞u^*(t) = p.  We use a local argument to show
that this is impossible if Γ_+ has positive diameter.

By a suitable change of variable we can
assume that p=0 and that, in a neighborhood of 0∈^n, U reads

    U(u)=V(u)+W(u),

where V is the quadratic part of U:

    V(u)=1/2(-∑_i=1^mλ_i^2u_i^2 +
    ∑_i=m+1^nλ_i^2u_i^2),     λ_i>0
 and W satisfies,

    | W(u)|≤ C| u|^3,   | W_x(u)|≤ C|
    u|^2,   | W_xx(u)|≤ C| u|.

Consider the Hamiltonian system with

    H(p,q) = 1/2| p|^2 - U(q),   p∈^n, 
    q∈Ω⊂^n.

For this system the origin of ^2n is an
equilibrium point that corresponds to the critical point p=0 of
U.

Set D = diag(-λ_1^2,
…,-λ_m^2,λ_m+1^2, …,λ_n^2).
The eigenvalues of the symplectic matrix

    ([ 0 D; I 0 ])

are

    -λ_i,  i=m+1,…,n
       -  λ_i,  i=m+1,…,n
       ± iλ_i,  i=1,…,m.

Let

(e_1,0),…,(e_n,0), (0,e_1),…,(0,e_n) be the basis of
^2n defined by e_j=(δ_j1,…,δ_jn), where
δ_ji is Kronecker's delta.

The stable S^s, unstable S^u and center S^c subspaces invariant
under the flow of the linearized Hamiltonian system at 0∈^2n
are

    S^s=span{(-λ_j e_j,e_j)}_j=m+1^n,
       S^u=span{(λ_j e_j,e_j)}_j=m+1^n,
       S^c=span{(e_j,0), (0,e_j)}_j=1^m.

From (<ref>) and (<ref>) we have

    lim_t→+∞(u̇^*(t),u^*(t))=0∈^2n.

Let W^s and W^u  be the local stable and unstable
manifold and let W^c be a local center
manifold at 0∈^2n. From the center manifold theorem
<cit.>,
<cit.>, there is a constant λ_0>0 such that, for each
solution (p(t),q(t)) that remains in a neighborhood of 0∈^2n
for positive time, there is a solution (p^c(t),q^c(t))∈ W^c that
satisfies

    |(p(t),q(t))-(p^c(t),q^c(t))|=O(e^-λ_0t).

Since W^c is tangent to S^c at 0∈^2n, the projection
W_0^c on the configuration space is tangent to
S_0^c=span{e_j}_j=1^m, which is the projection of S^c on the
configuration space. Therefore, if (p^c,q^c) ≢0, given
γ>0, by (<ref>) there is t_γ such that
d(q(t),S_0^c)≤γ| q(t)|, for t≥ t_γ. For
γ small, this implies that q(t)∉Ω for t≥
t_γ. It follows that (p^c,q^c)≡ 0 and from (<ref>)
(p(t),q(t)) converges to zero exponentially. This is possible only
if (p(t),q(t))∈ W^s and, in turn, only if q(t)∈ W_0^s, the
projection of W^s on the configuration space. This argument leads to
the conclusion that the trajectory of u^* in a neighborhood of 0
is of the form

    u^*(t(s))=𝔲^*(s) = sη+z(s),

where

    η=∑_i=m+1^nη_ie_i

is a unit vector[Actually
  η coincides with one of the eigenvectors of U”(0).],
s∈[0,s_0) for some s_0>0, and z(s) satisfies

    z(s)·η=0,  | z(s)|≤ c| s|^2,
      | z^'(s)|≤
    c| s|

for a positive constant c.


We are now in the position of constructing our local perturbation of
u.  We first discuss the case U=V, z(s)=0.  We set

    u̅(s)=sη

and, in some interval

[1,s_1],
construct a
competing map v̅:[1,s_1]→^n,


    v̅ = u̅ + g e_1,
       g:[1,s_1]→,

with the following
properties:

    V(v̅(1))=0, 
          v̅(s_1)=u̅(s_1),
          𝒥_V(v̅,[1,s_1])<𝒥_V(u̅,[0,s_1]).


The basic observation is that, if we move from u̅ in the
direction of one of the eigenvectors e_1,…,e_m corresponding to
negative eigenvalues of the Hessian of V, the potential V
decreases and therefore, for each s_0∈(1,s_1) we can define the
function g in the interval [1,s_0] so that

    𝒥_V(u̅+ge_1, (1,s_0))=𝒥_V(u̅,
              (1,s_0)).

Indeed it suffices to impose that

g:(1,s_0]→
  satisfies the condition

    √(V(u̅(s))) = √(1+g'^2(s))√(V(u̅(s)+g(s)e_1)),  s∈(1,s_0].

According with this condition we take g  as the
solution of the problem

    {[ g' =
     -λ_1g/√(s^2λ_η^2-λ_1^2g^2) =
    -λ_1g/sλ_η/√(1 -
        λ_1^2g^2/s^2λ_η^2);                                                               g(1)=λ_η/λ_1 ].,

where we have used (<ref>) and set


    λ_η=√(∑_i=m+1^nλ_i^2η_i^2).

Note that the initial condition in (<ref>) implies
V(v̅(1))=0. The solution g of (<ref>) is well defined
in spite of the fact that the right hand side tends to
-∞ as s→ 1.

Since g defined by (<ref>) is positive for s∈[1,+∞),

to satisfy the condition v̅(s_1)=u̅(s_1), we give a
suitable definition of g in the interval [s_0,s_1] in order that
g(s_1)=0. Choose a number α∈(0,1) and extend g with
continuity to the interval [s_0,s_1] by imposing that

    √(V(u̅(s))) = α√(1 + g'^2(s))√(V(u̅(s)+g(s)e_1)),  s∈(s_0,s_1].


Therefore, in the interval (s_0,s_1], we define g by

    g' = -1/α√(1 - α^2 +
        α^2λ_1^2g^2/s^2λ_η^2/1 -
        λ_1^2g^2/s^2λ_η^2)≤ -
    √(1-α^2)/α.




Since (<ref>) implies

    𝒥_V(v̅,[s_0,s_1]) =
    1/α𝒥_V(u̅,[s_0,s_1]),

from (<ref>) we see that v̅ satisfies also the
requirement (<ref>) above if we can choose α∈(0,1) and
1<s_0<s_1 in such a way that

    𝒥_V(u̅,(0,1)) >
    1-α/α𝒥_V(u̅,(s_0,s_1)).

Since (<ref>) implies s_1<s_0+α
  g(s_0)/√(1-α^2) a sufficient condition for this is

    𝒥_V(u̅,(0,1)) >
    1-α/α𝒥_V(u̅,
         (s_0,s_0+α g(s_0)/√(1-α^2))),

or equivalently

    1 > 1-α/α((s_0+α
      g(s_0)/√(1-α^2))^2 - s_0^2) =
    2s_0g(s_0)√(1-α/1+α) + α
      g^2(s_0)/1+α.

By a proper choice of s_0 and α the right hand side of (<ref>)
can be made as small as we like. For instance we can
fix s_0 so that g(s_0)≤1/4 and then choose
α in such a way that
1/2s_0√(1-α/1+α)≤1/4 and
conclude that (<ref>) holds.

Next we use the function g

 to define a comparison map v that coincides with
u^* outside an ϵ-neighborhood of 0 and show that the
assumption that the trajectory of u^* ends up in some p∈ P must
be rejected. For small ϵ>0 we define

    v(ϵ s) = ϵ sη+z(ϵ s) + ϵ
    g(s-σ)e_1,  s∈[1+σ,s_1+σ],

where σ=σ(ϵ) is determined by the condition

    U(v(ϵ(1+σ)))=0,

which, using (<ref>), (<ref>), (<ref>) and
g(1)=λ_η/λ_1, after dividing by ϵ^2,
becomes

    1/2λ_η^2((1+σ)^2-1)=ϵ f(σ,ϵ),

where f(σ,ϵ) is a smooth bounded function defined in a
neighborhood of (0,0). For small ϵ>0, there is a unique
solution σ(ϵ)=O(ϵ) of (<ref>). Note
also that (<ref>) implies that

    v(ϵ(s_1+σ))=𝔲^*(ϵ(s_1+σ)).

We now conclude by showing that, for ϵ>0 small, it results

    𝒥_U(𝔲^*(ϵ·),(0,s_1+σ)) >
    𝒥_U(v(ϵ·),(1+σ,s_1+σ)).

From (<ref>) and (<ref>) we have

    lim_ϵ→ 0^+ϵ^-1|d/ds𝔲^*(ϵ s)|
    =1,   lim_ϵ→ 0^+ϵ^-1|d/dsv(ϵ s)|
    = √(1+g'^2(s)),

and, using also (<ref>) and σ=O(ϵ),


    lim_ϵ→ 0^+ϵ^-2U(𝔲^*(ϵ s)) =
      V(u̅(s)),  s∈(0,s_1),
    
       lim_ϵ→ 0^+ϵ^-2U(v(ϵ s)) =
      V(v̅(s)),  s∈(1,s_1)

uniformly in compact intervals.

The limits (<ref>) and (<ref>) imply

    lim_ϵ→
        0^+ϵ^-2𝒥_U(𝔲^*(ϵ·),
      (0,s_1+σ)) = lim_ϵ→
        0^+√(2)∫_0^s_1+σ√(ϵ^-2U(𝔲^*(ϵ s)))ϵ^-1|d/ds𝔲^*(ϵ s)|
      ds,
    
       =√(2)∫_0^s_1√(V(u̅(s)))
      ds = 𝒥_V(u̅,(0,s_1))
    
       lim_ϵ→
        0^+ϵ^-2𝒥_U(v(ϵ·),(1+σ,s_1+σ))
      =lim_ϵ→
        0^+√(2)∫_1+σ^s_1+σ√(ϵ^-2U(v(ϵ s)))ϵ^-1|d/dsv(ϵ s)| ds,
    
       =√(2)∫_1^s_1√(V(v̅(s)))√(1+g'^2(s)) ds =
      𝒥_V(v̅,(1,s_1)).

This and (iii) above imply that, indeed, the inequality (<ref>)
holds for small ϵ>0. The proof is complete.

We can now complete the proof of Theorem <ref>. We show that the
map u^*:(T_-,T_+)→^n possesses all the required
properties. The fact that u^* satisfies (<ref>) and
(<ref>) follows from Lemma <ref>. Lemma
<ref> implies (<ref>) and, if T_->-∞, also
(<ref>). The fact that x_-∈Γ_-∖ P is a consequence
of Lemma <ref> and implies that Γ_- has positive
diameter. Viceversa, if Γ_- has positive diameter, Lemmas
<ref> and <ref> imply that T_->-∞ and that (<ref>)
holds for some x_-∈Γ_-∖ P. The proof of Theorem
<ref> is complete.



From Theorem <ref> it follows that if N is even then there are
at least N/2 distinct orbits connecting different elements of
{Γ_1,…,Γ_N}.  If N is odd there are at least
(N+1)/2.
 
Simple examples show that, given distinct Γ_i, Γ_j∈{Γ_1,…,Γ_N}, an orbit
connecting them does not always exist.

Let

    𝒰_ij = { u∈
      W^1,2((T_-^u,T_+^u);^n):u((T_-^u,T_+^u))⊂Ω,
        u(T_-^u)∈Γ_i, u(T_+^u)∈Γ_j
      }

with i≠ j and

    d_ij = inf_u∈𝒰_ij𝒜(u,(T_-^u,T_+^u)).

An orbit connecting Γ_i and Γ_j exists if

    d_ij < d_ik + d_kj,   ∀ k≠ i,j.










The proof of Theorem <ref> uses, with obvious
modifications, the same arguments as in
the proof of Theorem <ref> to characterize u^* as the limit of
a minimizing sequence {u_j} of the action functional

    𝒜(u,(0,T^u)) = ∫_0^T^u(1/2|u̇(t)|^2
    + U(u(t)))dt.

in the set


    𝒰 = { u∈ W^1,2((0,T^u);^n) :
    0<T_+^u <+∞, u(0) = 0,  u([0,T_+^u))⊂Ω,  u(T_+^u)
      ∈∂Ω}.















  In the symmetric case of Theorem <ref> it is easy to
  construct an example with T_+<T_+^∞.  For U(x)=1-|
  x|^2, x∈^2, the solution u:[0,π/2]→^2
  of (<ref>) determined by (<ref>) and
  u([0,π/2])={(s,0):s∈[0,1]} is a minimizer of
  𝒜 in 𝒰. For  small, let
  t_=arcsin(1-) and define u_ϵ: [0,T^u_ϵ]
  →^2 as the map determined by (<ref>),
  u_ϵ([0,t_ϵ]) = {(s,0): s∈[0,1-)} and
    u_((t_ϵ, T^u_]) =
  {(1-,s):s∈(0,√(2-^2)]}. In this case
T_+=π/2 and T_+^∞=3π/4.






 §.§ On the existence of heteroclinic connections



Corollary <ref> states the existence of heteroclinic
connections under the assumptions of Theorem <ref> and, in
particular, that U∈ C^2.  Actually, by examining the proof of
Theorem <ref> we can establish an existence result under weaker
hypotheses. In the special case ∂Ω=P, #P≥ 2,
given p_-∈ P, the set 𝒰 defined in (<ref>) takes
the form

    𝒰={u∈ W^1,2((T_-^u,T_+^u);^n):
            -∞< T_-^u < T_+^u< +∞, 2.2cm
    
            u((T_-^u,T_+^u))⊂Ω, U(u(0))=U_0,        u(T_-^u)=p_-,        u(T_+^u)∈ P∖{p_-}}.

In this section we slightly enlarge the set 𝒰 by allowing
T_±^u=±∞ and consider the admissible set

    𝒰={u∈ W^1,2_loc((T_-^u,T_+^u);^n):
            -∞≤ T_-^u < T_+^u≤+∞, 2.2cm
    
            u((T_-^u,T_+^u))⊂Ω, U(u(0))=U_0,lim_t→ T_-^uu(t)=p_-,lim_t→ T_+^uu(t)∈ P∖{p_-}}.



Assume that U is a non-negative continuous function, which vanishes in a
finite set P, #P≥ 2, and satisfies

    √(U(x))≥σ(| x|),   x∈Ω,  | x|≥ r_0

for some r_0>0 and
a non-negative function σ:[r_0,+∞)→ such that
∫_r_0^+∞σ(r)dr=+∞.
 
Given p_-∈ P there is p_+∈ P∖{p_-} and a
  Lipschitz-continuous map u^*:(T_-,T_+)→Ω that satisfies
  (<ref>) almost everywhere on (T_-,T_+), 

    lim_t→ T_±u^*(t)=p_±,

and minimizes the action functional 𝒜 on 𝒰̃.



We begin by showing that

    a_0=inf_u∈𝒰𝒜=inf_u∈𝒰̃𝒜=ã_0.

Since 𝒰⊂𝒰̃ we have
a_0≥ã_0. On the other hand arguing as in the proof of
Lemma <ref>, if T_+-T_-=+∞, given a small number
ϵ>0, we can construct a map u_ϵ∈𝒰 that
satisfies

    a_0≤𝒜(u_ϵ,(T_-^u_ϵ,T_+^u_ϵ))≤𝒜(u,(T_-^u,T_+^u))
    +η_ϵ

where η_ϵ→ 0 as ϵ→ 0. This
implies a_0≤ã_0 and establishes (<ref>). It follows
that we can proceed as in the proof of Theorem <ref> and define
u^*∈𝒰̃ as the limit of a minimizing sequence
{u_j}⊂𝒰. The arguments in the proof of Lemma
<ref> show that (<ref>) holds. It remain to show that
u^* is Lipschitz-continuous. Looking at the proof of Lemma
<ref> we see that the continuity of U is sufficient for
establishing that (<ref>) holds almost everywhere on
(T_-,T_+), and the Lipschitz character of u^* follows. The proof
is complete.


Without further information on the behavior of U in a neighborhood
of p_± nothing can be said on T_± being finite or infinite and
it is easy to construct examples to show that all possible
combinations are possible. As shown in Lemma <ref> a sufficient
condition for T_±=±∞ is that, in a neighborhood of
p=p_±, U(x) is bounded by a function of the form c|
x-p|^2, c>0. U of class C^1 is a sufficient condition in
order that u^* is of class C^2 and satisfies (<ref>).














§ EXAMPLES

In this section we show a few simple applications of Theorems
<ref> and <ref>.

Our first application describes a class of potentials with
the property that, in spite of the existence of possibly infinitely
many critical values, (<ref>) has a nontrivial periodic orbit
on any energy level.

Assume that U:^n→ satisfies

    U(-x)=U(x),  x∈^n,
    
       U(0) = 0,  U(x)< 0  x≠ 0,
       lim_|x|→∞U(x) = -∞

Assume moreover that each non zero critical point
  of U is hyperbolic with Morse index i_m≥ 1. Then there is a
  nontrivial periodic orbit of (<ref>) on the energy level
  1/2|u̇|^2-U(u)=α for each α>0.


For each α>0 we set Ũ=U(x)+α and let
Ω⊂{Ũ>0} be the connected component that contains
the origin. Ω is open, nonempty and bounded and, from the
assumptions on the properties of the critical points of U, it
follows that ∂Ω is connected and contains at most a
finite number of critical points. Therefore we are under the
assumptions of Corollary <ref> for the case N=1 and the
existence of the periodic orbit follows.


An example of potential U:^2→ that satisfies the
assumptions in Proposition <ref> is, in polar coordinates
r,θ,

    U(r,θ) = -r^2+1/2tanh^4(r)
    cos^2(r^-1)cos^2k(2θ),

where k>0 is a sufficiently large number.



Next we give another application of Corollary <ref>.  For the
potential U:^2→, with

    U(x)=1/2(1-x_1^2)^2+1/2(1-4x_2^2)^2,

the energy level

 α=-1/2
is critical and corresponds to
four hyperbolic critical points p_1=(1,0), -p_1,
p_2=(0,1/2) and -p_2. The connected component
Ω⊂{Ũ>0}, (Ũ=U(x)-1/2) that
contains the origin is bounded by a simple curve Γ that
contains ± p_1 and ± p_2. In spite of the presence of these
critical points, from Theorem <ref> it follows that there is a
minimizer u∈𝒰, with 𝒰 as in
(<ref>) and u(T^u)∈Γ∖{± p_1,± p_2}, and
Corollary <ref> implies the existence of a periodic solution v^*.

Note that there are also two heteroclinic orbits, solutions of
(<ref>) and (<ref>):

    u_1(t) = (tanh(t),0),      u_2(t) = (0,1/2tanh(2t)).

These orbits connect p_j to -p_j, for j=1,2. By
Theorem <ref> both u_1 and u_2 have action greater than
v^*|_(-T_+,T_+).



Our last example shows
that Theorems <ref> and <ref> can be used to derive
information on the rich dynamics that (<ref>) can exhibit when
U undergoes a small perturbation. We consider a family of potentials
U:^2×[0,1]→. We assume that U(x,0)=x_1^6+x_2^2
which from various points of view is a structurally unstable potential
and, for λ>0 small, we consider the perturbed potential

    U(x,λ)=2λ^4x_1^2+x_2^2-2λ^2x_1x_2-3λ^2x_1^4+x_1^6.


This potential satisfies U(-x,λ)=U(x,λ) and, for
λ>0, has the five critical points p_0, ± p_1
and ± p_2 defined by

    p_0=(0,0),
    
       p_1 = (λ(1-(2/3)^1/2)^1/2,
      λ^3(1-(2/3)^1/2)^1/2),
    
       p_2 = (λ(1+(2/3)^1/2)^1/2,
      λ^3(1+(2/3)^1/2)^1/2),

which are all hyperbolic.


We have U(p_2,λ)<0=U(p_0,λ)<U(p_1,λ) and p_0 is
a local minimum, p_1 a saddle and p_2 a global
minimum. Let α be the energy level. For
-α<U(p_2,λ) or -α≥ U(p_1,λ) no
information can be derived from Theorems <ref> and <ref>
therefore we assume -α∈[U(p_2,λ),U(p_1,λ)). For
  -α=U(p_2,λ) Corollary <ref> or Corollary
  <ref>

 yields the existence of a heteroclinic connection u_2
between -p_2 and p_2. For -α∈(U(p_2,λ),0) Corollary
<ref> implies the existence of a periodic orbit
u_α. This periodic orbit converges uniformly in compact intervals to
u_2 and the period T_α→+∞ as
-α→ U(p_2,λ)^+. For α=0 Corollary
<ref> implies the existence of two orbits u_0 and -u_0
homoclinic to p_0=0. We can assume that u_0 satisfies the
condition u_0(-t)=u_0(t) and that u_α(0)=0. Then we have that
u_α(·±T_α/4) converges uniformly in compact intervals
to ∓ u_0 and T_α→+∞ as -α→
0^-. For -α∈(0,U(p_1,λ)), ∂Ω is the
union of three simple curves all of positive diameter: Γ_0 that
includes the origin and ±Γ_2 which includes ± p_2 and
Corollary <ref> together with the fact that U(·,λ) is
symmetric imply the existence of two periodic solutions
ũ_α and -ũ_α with ũ_α
that oscillates between Γ_0 and Γ_2 in each time
interval equal to T_α/2. Assuming that
ũ_α(0)∈Γ_2 we have that, as -α→
0^+, ũ_α→ u_0 uniformly in compacts and
T_α→+∞. Finally we observe that, in the limit
-α→ U(p_1,λ)^-, ũ_α converges
uniformly in  to the constant solution u≡ p_1.





 §.§ Acknowledgements

The first author is indebted with Peter Bates for fruitful discussions on
the subject of this paper.

plain
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