Calculus of Variations and Geometric Measure Theory

J. H. Andrade - J. Conrado - S. Nardulli - P. Piccione - R. Resende

Multiplicity of solutions to the multiphasic Allen-Cahn-Hilliard system with a small volume constraint on closed parallelizable manifolds

created by nardulli on 30 Mar 2022
modified on 26 Feb 2024

[BibTeX]

Published Paper

Inserted: 30 mar 2022
Last Updated: 26 feb 2024

Journal: Journal of Functional Analysis
Volume: Volume 286
Number: 7
Pages: 48
Year: 2024
Doi: 10.1016/j.jfa.2024.110345

ArXiv: 2203.05034 PDF
Links: Journal of Functional Analysis

Abstract:

We prove the existence of multiple solutions to the Allen--Cahn--Hilliard (ACH) vectorial equation involving a multi-well (multiphasic) potential with a small volume constraint on a closed parallelizable Riemannian manifold. More precisely, we find a lower bound for the number of solutions depending on topological invariants of the underlying manifold. The phase transition potential is considered to have a finite set of global minima, where it also vanishes, and a subcritical growth at infinity. Our strategy is to employ the Lusternik--Schnirelmann and infinite-dimensional Morse theories for the vectorial energy functional. To this end, we exploit that the associated ACH energy $\Gamma$-converges to the vectorial perimeter for clusters, which combined with some deep theorems from isoperimetric theory yields the suitable setup to apply the photography method. Along the way, the lack of a closed analytic expression for the Euclidean multi-isoperimetric function for clusters imposes a delicate issue. Furthermore, using a transversality theorem, we also show the genericity of the set of metrics for which solutions to our geometric system are nondegenerate.