Divergence-form equations admitting nowhere $C^1$ Lipschitz weak solutions
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866915806400479232 |
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| author | Liao, Menglan Yan, Baisheng |
| author_facet | Liao, Menglan Yan, Baisheng |
| contents | We study a class of partial differential equations in divergence form that admit highly irregular Lipschitz weak solutions. By reformulating these divergence-form equations as a first-order partial differential relation and adapting the convex integration scheme recently developed in \cite{GKY26} for irregular diffusion equations, we show that the same structural Condition~$O_N$ introduced there also ensures the existence of Lipschitz weak solutions that are nowhere $C^1$ for the corresponding time-independent equations in bounded domains, under suitable boundary data. In particular, for the smooth strongly polyconvex functions on $\mathbb{R}^{2\times n}$ constructed in that paper for all $n \ge 2$, the associated Euler--Lagrange equations admit Lipschitz weak solutions that are nowhere $C^1$ and satisfy zero boundary conditions in any bounded domain of $\mathbb{R}^n$. Our approach relies on new building blocks constructed from the same wave cone and $\mathcal{T}_N$-configurations employed in the analysis of diffusion equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_17012 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Divergence-form equations admitting nowhere $C^1$ Lipschitz weak solutions Liao, Menglan Yan, Baisheng Analysis of PDEs 35A01, 35D30, 35F20, 49A30, 49Q20 We study a class of partial differential equations in divergence form that admit highly irregular Lipschitz weak solutions. By reformulating these divergence-form equations as a first-order partial differential relation and adapting the convex integration scheme recently developed in \cite{GKY26} for irregular diffusion equations, we show that the same structural Condition~$O_N$ introduced there also ensures the existence of Lipschitz weak solutions that are nowhere $C^1$ for the corresponding time-independent equations in bounded domains, under suitable boundary data. In particular, for the smooth strongly polyconvex functions on $\mathbb{R}^{2\times n}$ constructed in that paper for all $n \ge 2$, the associated Euler--Lagrange equations admit Lipschitz weak solutions that are nowhere $C^1$ and satisfy zero boundary conditions in any bounded domain of $\mathbb{R}^n$. Our approach relies on new building blocks constructed from the same wave cone and $\mathcal{T}_N$-configurations employed in the analysis of diffusion equations. |
| title | Divergence-form equations admitting nowhere $C^1$ Lipschitz weak solutions |
| topic | Analysis of PDEs 35A01, 35D30, 35F20, 49A30, 49Q20 |
| url | https://arxiv.org/abs/2602.17012 |