Refined regularity at critical points for linear elliptic equations
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908401963892736 |
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| author | Choi, Jongkeun Dong, Hongjie Kim, Seick |
| author_facet | Choi, Jongkeun Dong, Hongjie Kim, Seick |
| contents | We investigate the regularity of solutions to linear elliptic equations in both divergence and non-divergence forms, particularly when the principal coefficients have Dini mean oscillation. We show that if a solution $u$ to a divergence-form equation satisfies $Du(x^o)=0$ at a point, then the second derivative $D^2u(x^o)$ exists and satisfies sharp continuity estimates. As a consequence, we obtain ``$C^{2,α}$ regularity'' at critical points when the coefficients of $L$ are $C^α$. This result refines a theorem of Teixeira (Math. Ann. 358 (2014), no. 1--2, 241--256) in the linear setting, where both linear and nonlinear equations were considered. We also establish an analogous result for equations in non-divergence form. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_08281 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Refined regularity at critical points for linear elliptic equations Choi, Jongkeun Dong, Hongjie Kim, Seick Analysis of PDEs We investigate the regularity of solutions to linear elliptic equations in both divergence and non-divergence forms, particularly when the principal coefficients have Dini mean oscillation. We show that if a solution $u$ to a divergence-form equation satisfies $Du(x^o)=0$ at a point, then the second derivative $D^2u(x^o)$ exists and satisfies sharp continuity estimates. As a consequence, we obtain ``$C^{2,α}$ regularity'' at critical points when the coefficients of $L$ are $C^α$. This result refines a theorem of Teixeira (Math. Ann. 358 (2014), no. 1--2, 241--256) in the linear setting, where both linear and nonlinear equations were considered. We also establish an analogous result for equations in non-divergence form. |
| title | Refined regularity at critical points for linear elliptic equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2506.08281 |