Heteroclinic connections for multiple-well potentials: the anisotropic case

Abstract

Abstract. We investigate the existence of solutions to systems of N differential equations representing connections between minima of potentials with several equal depths in R N . Using variational techniques and in particular a method introduced in [AF] we first prove such existence for N ≥ 2 and two minima. Dealing next with symmetric potentials corresponding to free bulk energies in crystals we establish existence for N ≥ 2 in various cases of more than two minima. Finally we obtain a sufficient condition establishing existence of connections to not necessarily symmetric potentials for arbitrary N and three minima.

Local minimizers and planar interfaces in a phase-transition model with interfacial energy

“…We conclude this remark on condition (4.9) by noting that Alama et al [1] and Alikakos and Fusco [3] (see also [31]) show that a sufficient condition for the existence of a connection between two given zeroes a 1 and a 2 of W in the presence of a third zero a 3 is that…”

“…This is important because, in general, the existence of more than two zeroes of W can result in the non-existence of connections between a given pair of zeroes. See, for example, Alama et al [1], Alikakos et al [2], Alikakos and Fusco [3], Stefanopolous [31], and also Fife and McLeod [18] and Volpert et al [35]. The condition (4.9) holds in particular for the martensite wells in a cubic to tetragonal transition, when k = 3 and…”

“…Symmetry conditions on the function W can ensure that (4.16) is satisfied; see, for example, Stefanopolous [31], and also Bronsard et al [13]. However, in spite of physically realistic ψ having material-symmetry properties in addition to frame-indifference (f), it is not clear that for such ψ, the function W (·) = ψ(A + · ⊗ N ) satisfies sufficient symmetry to ensure that (4.16) is typically satisfied if (4.9) fails, essentially because imposing the ansatz A + u ⊗ N partially breaks the symmetry of ψ.…”

The Heteroclinic Connection Problem for General Double-Well Potentials

“…(for some small δ > 0), where S n−1 stands for the unit sphere, the existence of a heteroclinic connection was proven recently in [6] (see also [7] and [48]). Their novelty was to employ constraints which are subsequently removed.…”

“…We emphasize that some of these applications require a triple-well or four-well potential. Nevertheless, under a reflection symmetry assumption on W (which is frequently inherited from the physical model), the problem can easily be reduced to the double-well case (see [2] or [48]). The heteroclinic connection problem for nonsymmetric multi-well potentials is much more complicated to treat (see [51]).…”