The sharp interface limit of the matrix-valued Allen-Cahn equation
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866908829326770176 |
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| author | Wang, Xingyu |
| author_facet | Wang, Xingyu |
| contents | In this work, we study a matrix-valued Allen-Cahn equation with a Saint Venant-Kirchhoff potential $F(\mathbf{A})=\frac{1}{4}\|\mathbf{A}\mathbf{A}^\top-\mathbf{I}\|^2$. Our approach employs the modulated energy method together with weak convergence methods for nonlinear partial differential equations. This avoids the subtle spectrum analysis of the linearized operator at the so-called quasi-minimal orbits as well as the construction of asymptotic expansion. Moreover, it relaxes the assumption on the admissible initial data, which exhibits a phase transition along an initial interface. As a byproduct, we construct a weak solution to the limiting harmonic heat flow system with both minimal pair and Neumann-type boundary conditions across the interface. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_11485 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The sharp interface limit of the matrix-valued Allen-Cahn equation Wang, Xingyu Analysis of PDEs In this work, we study a matrix-valued Allen-Cahn equation with a Saint Venant-Kirchhoff potential $F(\mathbf{A})=\frac{1}{4}\|\mathbf{A}\mathbf{A}^\top-\mathbf{I}\|^2$. Our approach employs the modulated energy method together with weak convergence methods for nonlinear partial differential equations. This avoids the subtle spectrum analysis of the linearized operator at the so-called quasi-minimal orbits as well as the construction of asymptotic expansion. Moreover, it relaxes the assumption on the admissible initial data, which exhibits a phase transition along an initial interface. As a byproduct, we construct a weak solution to the limiting harmonic heat flow system with both minimal pair and Neumann-type boundary conditions across the interface. |
| title | The sharp interface limit of the matrix-valued Allen-Cahn equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2602.11485 |