Solvability of the Dirichlet problem for a new class of elliptic operators
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909716851982336 |
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| author | Ulmer, Martin |
| author_facet | Ulmer, Martin |
| contents | We study an elliptic operator $L:=\mathrm{div}(A\nabla \cdot)$ on the upper half space. It is known that if the matrix $A$ is independent in the transversal $t$-direction, then we have $ω\in A_\infty(σ)$. In the present paper we improve on the $t$-independence condition by introducing a mixed $L^1-L^\infty$ Carleson type condition that only depends on $\partial_t A$ and show $ω\in A_\infty(σ)$ under this condition. This condition is different from other conditions in the literature.
In the case of the upper half plane, we obtain the improvement that an $L^1$-Carleson condition on $|\partial_tA|$ implies $ω\in A_\infty(σ)$. In particular, this condition is similar to an $L^1$-version of the DKP condition with derivative in only the transversal direction. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_00614 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Solvability of the Dirichlet problem for a new class of elliptic operators Ulmer, Martin Analysis of PDEs 35J25, 35J15, 42B37, 47D03 We study an elliptic operator $L:=\mathrm{div}(A\nabla \cdot)$ on the upper half space. It is known that if the matrix $A$ is independent in the transversal $t$-direction, then we have $ω\in A_\infty(σ)$. In the present paper we improve on the $t$-independence condition by introducing a mixed $L^1-L^\infty$ Carleson type condition that only depends on $\partial_t A$ and show $ω\in A_\infty(σ)$ under this condition. This condition is different from other conditions in the literature. In the case of the upper half plane, we obtain the improvement that an $L^1$-Carleson condition on $|\partial_tA|$ implies $ω\in A_\infty(σ)$. In particular, this condition is similar to an $L^1$-version of the DKP condition with derivative in only the transversal direction. |
| title | Solvability of the Dirichlet problem for a new class of elliptic operators |
| topic | Analysis of PDEs 35J25, 35J15, 42B37, 47D03 |
| url | https://arxiv.org/abs/2311.00614 |