Boundary-driven patterns in elongated convex domains
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866917356106678272 |
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| author | Sonego, Maicon |
| author_facet | Sonego, Maicon |
| contents | We consider the heat equation in a smooth bounded convex domain $Ω\subset \mathbb{R}^2$ with nonlinear Neumann boundary condition $\partial_νu = λ(u - u^3)$. Stable non-constant stationary solutions do not exist when $Ω$ is a ball. We show that this behavior is not a consequence of convexity alone. More precisely, if the inradius of $Ω$ is fixed and its diameter is sufficiently large, then there exists $λ>0$ for which the problem admits such a solution. The result reveals a geometric mechanism for the emergence of stable non-constant stationary solutions in elongated convex domains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_20938 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Boundary-driven patterns in elongated convex domains Sonego, Maicon Analysis of PDEs We consider the heat equation in a smooth bounded convex domain $Ω\subset \mathbb{R}^2$ with nonlinear Neumann boundary condition $\partial_νu = λ(u - u^3)$. Stable non-constant stationary solutions do not exist when $Ω$ is a ball. We show that this behavior is not a consequence of convexity alone. More precisely, if the inradius of $Ω$ is fixed and its diameter is sufficiently large, then there exists $λ>0$ for which the problem admits such a solution. The result reveals a geometric mechanism for the emergence of stable non-constant stationary solutions in elongated convex domains. |
| title | Boundary-driven patterns in elongated convex domains |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2602.20938 |