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<p>In this work, we investigate the vector-valued Allen-Cahn equation with potentials of high-dimensional double wells under Robin boundary conditions. For a broad class of boundary energy densities and well-prepared initial data, we establish local-in-time convergence of solutions to the mean curvature flow with a fixed contact angle <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(0&lt;\alpha \leqslant 90^\circ \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>&lt;</mo> <mi>α</mi> <mo>⩽</mo> <msup> <mn>90</mn> <mo>∘</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>. We derive the limiting sharp-interface system, which comprises harmonic map heat flows in the bulk and minimal pair conditions on the interface. Our analysis combines the relative entropy method with gradient flow calibrations and employs tools from geometric measure theory to handle the boundary terms. 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).</p>

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

  • 期刊论文

  • 发表日期：

    2026年4月1日

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  • Xingyu Wang

摘要

In this work, we investigate the vector-valued Allen-Cahn equation with potentials of high-dimensional double wells under Robin boundary conditions. For a broad class of boundary energy densities and well-prepared initial data, we establish local-in-time convergence of solutions to the mean curvature flow with a fixed contact angle \(0<\alpha \leqslant 90^\circ \) 0 < α ⩽ 90 ∘ . We derive the limiting sharp-interface system, which comprises harmonic map heat flows in the bulk and minimal pair conditions on the interface. Our analysis combines the relative entropy method with gradient flow calibrations and employs tools from geometric measure theory to handle the boundary terms. 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).