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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Online-Zugang: | https://arxiv.org/abs/2401.11486 |
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| _version_ | 1866910304498089984 |
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| author | Cao, Daomin Wan, Jie |
| author_facet | Cao, Daomin Wan, Jie |
| contents | We consider Green's function $ G_K $ of the elliptic operator in divergence form $ \mathcal{L}_K=-\text{div}(K(x)\nabla ) $ on a bounded smooth domain $ Ω\subseteq\mathbb{R}^n (n\geq 2) $ with zero Dirichlet boundary condition, where $ K $ is a smooth positively definite matrix-valued function on $ Ω$. We obtain a high-order asymptotic expansion of $ G_K(x, y) $, which defines uniquely a regular part $ H_K(x, y) $. Moreover, we prove that the associated Robin's function $ R_K(x) = H_K(x, x) $ is smooth in $ Ω$, despite the regular part $ H_K\notin C^1(Ω\timesΩ) $ in general. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_11486 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Expansion of Green's function and regularity of Robin's function for elliptic operators in divergence form Cao, Daomin Wan, Jie Analysis of PDEs We consider Green's function $ G_K $ of the elliptic operator in divergence form $ \mathcal{L}_K=-\text{div}(K(x)\nabla ) $ on a bounded smooth domain $ Ω\subseteq\mathbb{R}^n (n\geq 2) $ with zero Dirichlet boundary condition, where $ K $ is a smooth positively definite matrix-valued function on $ Ω$. We obtain a high-order asymptotic expansion of $ G_K(x, y) $, which defines uniquely a regular part $ H_K(x, y) $. Moreover, we prove that the associated Robin's function $ R_K(x) = H_K(x, x) $ is smooth in $ Ω$, despite the regular part $ H_K\notin C^1(Ω\timesΩ) $ in general. |
| title | Expansion of Green's function and regularity of Robin's function for elliptic operators in divergence form |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2401.11486 |