2024/12/04 by Zhiyuan Geng, Geng, Zhiyuan
Materials Science · #Solidification and crystal growth phenomena
paper · pdf · doi:10.48550/arxiv.2412.02954
For the two dimensional Allen-Cahn system with a triple-well potential, previous results established the existence of a minimizing solution u:ℝ2→ℝ2 with a triple junction structure at infinity. We show that along each of three sharp interfaces, u is asymptotically invariant in the direction of the interface and can be well-approximated by the 1D heteroclinic connections between two phases. Consequently, the diffuse interface is located in an O(1) neighborhood of the sharp interface, and becomes nearly flat at infinity. This generalizes all the results for the triple junction solution with symmetry hypotheses to the non-symmetric case. The proof relies on refined sharp energy lower and upper bounds, alongside a precise estimate of the diffuse interface location.