Homogenization of a vertical oscillating Neumann condition
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915300324147200 |
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| author | Feldman, William M Huang, Zhonggan |
| author_facet | Feldman, William M Huang, Zhonggan |
| contents | We homogenize the Laplace and heat equations with the Neumann data oscillating in the ``vertical" $u$-variable. These are simplified models for interface motion in heterogeneous media, particularly capillary contact lines. The homogenization limit reveals a pinning effect at zero tangential slope, leading to a novel singularly anisotropic pinned Neumann condition. The singular pinning creates an unconstrained contact set, generalizing the contact set in the classical thin obstacle problem. We establish a comparison principle for the heat equation with this new type of boundary condition. The comparison principle enables a proof of homogenization via the method of half-relaxed limits from viscosity solution theory. Our work also demonstrates, for the first time in a PDE problem in multiple dimensions, the emergence of rate-independent pinning from gradient flows with wiggly energies. Prior limit theorems of this type, in rate-independent contexts, were limited to ODEs and PDEs in one dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_17298 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Homogenization of a vertical oscillating Neumann condition Feldman, William M Huang, Zhonggan Analysis of PDEs We homogenize the Laplace and heat equations with the Neumann data oscillating in the ``vertical" $u$-variable. These are simplified models for interface motion in heterogeneous media, particularly capillary contact lines. The homogenization limit reveals a pinning effect at zero tangential slope, leading to a novel singularly anisotropic pinned Neumann condition. The singular pinning creates an unconstrained contact set, generalizing the contact set in the classical thin obstacle problem. We establish a comparison principle for the heat equation with this new type of boundary condition. The comparison principle enables a proof of homogenization via the method of half-relaxed limits from viscosity solution theory. Our work also demonstrates, for the first time in a PDE problem in multiple dimensions, the emergence of rate-independent pinning from gradient flows with wiggly energies. Prior limit theorems of this type, in rate-independent contexts, were limited to ODEs and PDEs in one dimension. |
| title | Homogenization of a vertical oscillating Neumann condition |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2505.17298 |