TY - JOUR

T1 - Matrix-valued Allen–Cahn equation and the Keller–Rubinstein–Sternberg problem

AU - Fei, Mingwen

AU - Lin, Fanghua

AU - Wang, Wei

AU - Zhang, Zhifei

N1 - Funding Information:
The authors are indebted to the anonymous referees for their careful reading and helpful comments which considerably helped to improve the quality of the manuscript. M. Fei is supported by NSF of China under Grant No. 11871075 and 11971357. F.H. Lin is supported by an NSF grant DMS1955249. W. Wang is supported by NSF of China under Grant No. 11922118 and 11871424. Z. Zhang is supported by NSF of China under Grant No. 12171010.
Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2023/7

Y1 - 2023/7

N2 - In this paper, we consider the sharp interface limit of a matrix-valued Allen–Cahn equation, which takes the form: ∂tA=ΔA-ε-2(AATA-A)withA:Ω⊂Rm→Rn×n.We show that the sharp interface system is a two-phases flow system: the interface evolves according to the motion by mean curvature; in the two bulk phase regions, the solution obeys the heat flow of harmonic maps with values in O+(n) and O-(n) (represent the sets of n× n orthogonal matrices with determinant + 1 and - 1 respectively); on the interface, the phase matrices on two sides satisfy a novel mixed boundary condition. The above result provides a solution to the Keller–Rubinstein–Sternberg’s problem in the O(n) setting. Our proof relies on two key ingredients. First, in order to construct the approximate solutions by matched asymptotic expansions, as the standard approach does not seem to work, we introduce the notion of quasi-minimal connecting orbits. They satisfy the usual leading order equations up to some small higher order terms. In addition, the linearized systems around these quasi-minimal orbits needs to be solvable up to some good remainders. These flexibilities are needed for the possible “degenerations” and higher dimensional kernels for the linearized operators on matrix-valued functions due to intriguing boundary conditions at the sharp interface. The second key point is to establish a spectral uniform lower bound estimate for the linearized operator around approximate solutions. To this end, we introduce additional decompositions to reduce the problem into the coercive estimates of several linearized operators for scalar functions and some singular product estimates which are accomplished by exploring special cancellation structures between eigenfunctions of these linearized operators.

AB - In this paper, we consider the sharp interface limit of a matrix-valued Allen–Cahn equation, which takes the form: ∂tA=ΔA-ε-2(AATA-A)withA:Ω⊂Rm→Rn×n.We show that the sharp interface system is a two-phases flow system: the interface evolves according to the motion by mean curvature; in the two bulk phase regions, the solution obeys the heat flow of harmonic maps with values in O+(n) and O-(n) (represent the sets of n× n orthogonal matrices with determinant + 1 and - 1 respectively); on the interface, the phase matrices on two sides satisfy a novel mixed boundary condition. The above result provides a solution to the Keller–Rubinstein–Sternberg’s problem in the O(n) setting. Our proof relies on two key ingredients. First, in order to construct the approximate solutions by matched asymptotic expansions, as the standard approach does not seem to work, we introduce the notion of quasi-minimal connecting orbits. They satisfy the usual leading order equations up to some small higher order terms. In addition, the linearized systems around these quasi-minimal orbits needs to be solvable up to some good remainders. These flexibilities are needed for the possible “degenerations” and higher dimensional kernels for the linearized operators on matrix-valued functions due to intriguing boundary conditions at the sharp interface. The second key point is to establish a spectral uniform lower bound estimate for the linearized operator around approximate solutions. To this end, we introduce additional decompositions to reduce the problem into the coercive estimates of several linearized operators for scalar functions and some singular product estimates which are accomplished by exploring special cancellation structures between eigenfunctions of these linearized operators.

UR - http://www.scopus.com/inward/record.url?scp=85149126999&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85149126999&partnerID=8YFLogxK

U2 - 10.1007/s00222-023-01183-8

DO - 10.1007/s00222-023-01183-8

M3 - Article

AN - SCOPUS:85149126999

SN - 0020-9910

VL - 233

SP - 1

EP - 80

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

IS - 1

ER -