On the Loss of Compactness in the Heteroclinic Connection Problem

Details On the Loss of Compactness in the Heteroclinic Connection Problem On the Loss of Compactness in the Heteroclinic Connection Problem

2008-04-17

arxiv.org/abs/0804.2692v2

Mathematics

Classical Analysis and ODEs

11 pages, 5 figures, totally revised

Scientific paper

We consider the problem of existence of heteroclinic solutions to the Hamiltonian 2nd order system of ODEs U_{xx} = Grad W (U), U : R \larrow R^N, U(- \infty) = a^{-}, U(+ \infty) = a^{+}, where $a^{\pm}$ are local minima of the potential W in $C^2(R^N)$ with $W(a^\pm)= 0$. This problem arises in the theory of phase transitions and has been considered before by Sternberg [St] and Alikakos-Fusco [A-F]. Herein we present a new {efficient} proof of existence under assumptions different from those considered previously and we derive new {a priori decay estimates}, valid even when $a^\pm$ are degenerate. We establish \emph{existence by analyzing the loss of compactness} in a suitable functional space setup: for any minimizing sequence of the Action functional $E (U) = \int_{R} {1/2 | U_x |^2 + W (U)} dx$, there exist uniformly decaying designated translates, up to which, compactness is restored and passage to a solution is available.

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