Uniqueness of the blow-down limit for triple junction problem

Abstract: We prove the uniqueness of $L^1$ blow-down limit at infinity for an entire minimizing solution $u:\mathbb{R}^2\rightarrow\mathbb{R}^2$ of a planar Allen-Cahn system with a triple-well potential. Consequently, $u$ can be approximated by a triple junction map at infinity. The proof exploits a careful analysis of energy upper and lower bounds, ensuring that the diffuse interface remains within a small neighborhood of the approximated triple junction at all scales.