The vector-valued Allen-Cahn equation with potentials of high-dimensional double-wells under Robin boundary conditions

2025/05/31 by Xingyu Wang, Wang, Xingyu
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2506.00392

openalex publication_date 2025/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This work investigates the vector-valued Allen-Cahn equation with potentials of high-dimensional double-wells under Robin boundary conditions. We establish local-in-time convergence of solutions to mean curvature flow with a fixed contact angle 0<α≤ 90^∘, for a broad class of boundary energy densities and well-prepared initial data. The limiting sharp-interface system is derived, comprising harmonic heat flows in the bulk and minimal pair conditions at phase boundaries. The analysis combines the relative entropy method with gradient flow calibrations and weak convergence techniques. These results extend prior works on the analysis of the vector-valued case without boundary effects (Comm. Pure Appl. Math., 78:1199-1247, 2025) and the scalar-valued case with boundary contact energy (Calc. Var. Partial Differ. Equ., 61:201, 2022).

Citations

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• Weak-strong uniqueness for the mean curvature flow of double bubbles
• Matrix-valued Allen-Cahn equation and the Keller-Rubinstein-Sternberg problem
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• The local structure of the energy landscape in multiphase mean curvature flow: Weak-strong uniqueness and stability of evolutions
• Convergence rates of the Allen-Cahn equation to mean curvature flow: A\n short proof based on relative entropies
• Harmonic maps in connection of phase transitions with higher dimensional potential wells
• Convergence of the Allen-Cahn Equation to the Mean Curvature Flow with 90^∘-Contact Angle in 2D
• Convergence of the Allen-Cahn Equation to multi-phase mean-curvature\n flow
• On the motion of a curve by its binormal curvature
• The coarea formula for Sobolev mappings

Related

• Convergence rates for the Allen-Cahn equation with boundary contact energy: The non-perturbative regime
• Convergence of the Scalar- and Vector-Valued Allen-Cahn Equation to Mean Curvature Flow with 90°-Contact Angle in Higher Dimensions
• Convergence of the Allen-Cahn equation with a nonlinear Robin Boundary Condition to Mean Curvature Flow with Contact Angle close to 90°
• Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow
• Asymptotic limit of a vector-valued Allen-Cahn equation for phase transition dynamics