Elliptic unique continuation below the Lipschitz threshold
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908623937994752 |
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| author | Jeznach, Cole |
| author_facet | Jeznach, Cole |
| contents | In this article, we investigate unique continuation principles for solutions $u$ of uniformly elliptic equations of the form $-\mathrm{div}(A \nabla u) = 0$ when $A$ is less regular than Lipschitz. For general matrices $A$, we prove that strong unique continuation holds provided that $A$ has modulus of continuity $ω$ satisfying the Osgood condition $\int_0^1 ω(t)^{-1}dt = \infty$, plus some other mild hypotheses. Along with the counterexamples of Mandache, this shows that the sharp condition on $A$ that guarantees unique continuation is essentially that $A$ is log-Lipschitz. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_23614 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Elliptic unique continuation below the Lipschitz threshold Jeznach, Cole Analysis of PDEs 35A02, 35J15, 35J10 In this article, we investigate unique continuation principles for solutions $u$ of uniformly elliptic equations of the form $-\mathrm{div}(A \nabla u) = 0$ when $A$ is less regular than Lipschitz. For general matrices $A$, we prove that strong unique continuation holds provided that $A$ has modulus of continuity $ω$ satisfying the Osgood condition $\int_0^1 ω(t)^{-1}dt = \infty$, plus some other mild hypotheses. Along with the counterexamples of Mandache, this shows that the sharp condition on $A$ that guarantees unique continuation is essentially that $A$ is log-Lipschitz. |
| title | Elliptic unique continuation below the Lipschitz threshold |
| topic | Analysis of PDEs 35A02, 35J15, 35J10 |
| url | https://arxiv.org/abs/2507.23614 |