Solvability of the Dirichlet problem using a weaker Carleson condition in the upper half plane
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913969745166336 |
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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 plane $\mathbb{R}^2_+$. There are several conditions on the behavior of the matrix $A$ in the transversal $t$-direction that yield $ω\in A_\infty(σ)$. These include the $t$-independence condition, a mixed $L^1-L^\infty$ condition on $\partial_t A$, and Dini-type conditions. We introduce an $L^1$ Carleson condition on $\partial_t A(x,t)$ that extends the class of elliptic operators for which we have $ω\in A_\infty(σ)$, i.e. solvability of the $L^p$ Dirichlet problem for some $1<p<\infty$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_19106 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Solvability of the Dirichlet problem using a weaker Carleson condition in the upper half plane Ulmer, Martin Analysis of PDEs 35J25, 35J15, 42B37, 47D03 We study an elliptic operator $L:=\mathrm{div}(A\nabla \cdot)$ on the upper half plane $\mathbb{R}^2_+$. There are several conditions on the behavior of the matrix $A$ in the transversal $t$-direction that yield $ω\in A_\infty(σ)$. These include the $t$-independence condition, a mixed $L^1-L^\infty$ condition on $\partial_t A$, and Dini-type conditions. We introduce an $L^1$ Carleson condition on $\partial_t A(x,t)$ that extends the class of elliptic operators for which we have $ω\in A_\infty(σ)$, i.e. solvability of the $L^p$ Dirichlet problem for some $1<p<\infty$. |
| title | Solvability of the Dirichlet problem using a weaker Carleson condition in the upper half plane |
| topic | Analysis of PDEs 35J25, 35J15, 42B37, 47D03 |
| url | https://arxiv.org/abs/2503.19106 |