On the loss of compactness in the vectorial heteroclinic connection problem

Katzourakis, N. (2016) On the loss of compactness in the vectorial heteroclinic connection problem. Proceedings of the Royal Society of Edinburgh A, 146 (3). pp. 595-608. ISSN 1473-7124 doi: 10.1017/S0308210515000700

Abstract/Summary

We give an alternative proof of the theorem of Alikakos and Fusco concerning existence of heteroclinic solutions U : ℝ → ℝ N to the system Here a± are local minima of a potential W ∈ C2(ℝ N ) with W(a±) = 0. This system arises in the theory of phase transitions. Our method is variational but differs from the original artificial constraint method of Alikakos and Fusco and establishes existence by analysing the loss of compactness in minimizing sequences of the action in the appropriate functional space. Our assumptions are slightly different from those considered previously and also imply a priori estimates for the solution.