The Dirichlet problem for second-order elliptic equations in non-divergence form with continuous coefficients: The two-dimensional case
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915267988160512 |
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| author | Dong, Hongjie Kim, Dong-ha Kim, Seick |
| author_facet | Dong, Hongjie Kim, Dong-ha Kim, Seick |
| contents | This paper investigates the Dirichlet problem for a non-divergence form elliptic operator $L$ in a bounded domain of $\mathbb{R}^2$. Assuming that the principal coefficients satisfy the Dini mean oscillation condition, we establish the equivalence between regular points for $L$ and those for the Laplace operator. This result closes a gap left in the authors' recent work on higher-dimensional cases (Math. Ann. 392(1): 573--618, 2025). Furthermore, we construct the Green's function for $L$ in regular two-dimensional domains, extending a result by Dong and Kim (SIAM J. Math. Anal. 53(4): 4637--4656, 2021). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_00190 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Dirichlet problem for second-order elliptic equations in non-divergence form with continuous coefficients: The two-dimensional case Dong, Hongjie Kim, Dong-ha Kim, Seick Analysis of PDEs This paper investigates the Dirichlet problem for a non-divergence form elliptic operator $L$ in a bounded domain of $\mathbb{R}^2$. Assuming that the principal coefficients satisfy the Dini mean oscillation condition, we establish the equivalence between regular points for $L$ and those for the Laplace operator. This result closes a gap left in the authors' recent work on higher-dimensional cases (Math. Ann. 392(1): 573--618, 2025). Furthermore, we construct the Green's function for $L$ in regular two-dimensional domains, extending a result by Dong and Kim (SIAM J. Math. Anal. 53(4): 4637--4656, 2021). |
| title | The Dirichlet problem for second-order elliptic equations in non-divergence form with continuous coefficients: The two-dimensional case |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.00190 |