Asymptotic behavior of the interface for entire vector minimizers in phase transitions

• Published in: Journal of functional analysis Vol. 283; no. 6; p. 109565
• Main Authors: Alikakos, Nicholas D., Geng, Zhiyuan, Zarnescu, Arghir
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15-09-2022
• Subjects: Entire solutions; Phase transitions; Subquadratic potential; Vector minimizers; Subquadratic potential; Vector minimizers; Phase transitions; Entire solutions
• ISSN: 0022-1236; 1096-0783
• DOI: 10.1016/j.jfa.2022.109565

We study globally bounded entire minimizers u:Rn→Rm of Allen-Cahn systems for potentials W≥0 with {W=0}={a1,...,aN} and W(u)∼|u−ai|α near u=ai, 0<α<2. Such solutions are, over large regions, identically equal to some zeroes of the potential ai's. We establish the estimatesLn(I0∩Br(x0))≤c1rn−1,Hn−1(∂⁎I0∩Br(x0))≥c2rn−1,r≥r0(x0) for the diffuse interfaceI0:={x∈Rn:min1≤i≤N⁡|u(x)−ai|>0} and the free boundary∂I0. Furthermore, if α=1 we establish the upper boundHn−1(∂⁎I0∩Br(x0))≤c3rn−1,r≥r0(x0).