Abstract
We begin by giving a concise proof of the existence of a heteroclinic connection (Theorem 2.1). The experienced reader then can move on to Sect. 2.6. In Sect. 2.4 we develop an alternative approach via constrained minimization. Most readers will find this easier and also good preparation for the polar form and the cut-off lemma in Chap. 4. In Sect. 2.6 we consider the connection problem for an unbalanced double-well potential, and handle it via the constrained method. Finally in Sect. 2.7 we investigate the failure of the existence of a connection when three or more global minima are present.
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- 1.
If u 0 is minimal and bounded, then it satisfies the equipartition relation (cf. Theorem 2.3).
- 2.
If m = 0, it follows that g(z(t)) = g(α), and since g is analytic z(t) = α. Thus, if u(x) is a heteroclinic, it follows that m ≠ 0 and so g(z i) ≠ g(z j).
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Alikakos, N.D., Fusco, G., Smyrnelis, P. (2018). Connections. In: Elliptic Systems of Phase Transition Type. Progress in Nonlinear Differential Equations and Their Applications, vol 91. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-90572-3_2
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