Prescribed energy connecting orbits for gradient systems

Abstract

We are concerned with conservative systemsq = ∇V(q), q ∈ R N for a general class of potentials V ∈ C 1 (R N ). Assuming that a given sublevel set {V ≤ c} splits in the disjoint union of two closed subsets V c − and V c + , for some c ∈ R, we establish the existence of bounded solutions q c to the above system with energy equal to −c whose trajectories connect V c − and V c + . The solutions are obtained through an energy constrained variational method, whenever mild coerciveness properties are present in the problem. The connecting orbits are classified into brake, heteroclinic or homoclinic type, depending on the behavior of ∇V on ∂V c ± . Next, we illustrate applications of the existence result to double-well potentials V, and for potentials associated to systems of duffing type and of multiple-pendulum type. In each of the above cases we prove some convergence results of the family of solutions (q c ).

Connecting orbits in Hilbert spaces and applications to P.D.E

“…Assuming (2), we know that there exists at least one minimal 1 heteroclinic orbit e (cf. for instance [7], [14], [23] or [4], for a general theorem about the existence of heteroclinic connections). In addition, since the minima a ± are nondegenerate, the convergence to the minima a ± is exponential for every heteroclinic orbit e, i.e.…”

A Variant of the Mountain Pass Theorem and Variational Gluing

“…The approach taken here is based on Byeon, Montecchiari, and Rabinowitz [28], and on [7]. See also Alikakos and Fusco [29] and Alessio, Montecchiari, and Zuniga [30].…”

Periodic motions for multi-wells potentials and layers dynamic for the vector Allen-Cahn equation

“…1 we sketch a situation where A has four elements and N = 6. If A = {a − , a + } with a − = a + the problem is well understood, see for example [9], [2] and under minimal assumption on W , there is a T 0 > 0 such that, for each T ≥ T 0 , there exists a T -periodic solution u T that oscillates twice for period on the same trajectory with extreme at u T (0) ∈ B δ (a − ) and u T (T /2) ∈ B δ (a + ) and lim…”

“…Remark. A = {a − , a + }, a − = a + implies the existence of a solution ū of (1.1) that connects a − to a + [2], [30], [35], [36], [39]. If A contains more than two elements, the existence of a heteroclinic solution ū that connects a − ∈ A to a + ∈ A, a − = a + is not guaranteed.…”