The regularity problem with a weaker condition on only the transversal direction
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866917976027955200 |
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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 the regularity boundary value problem is solvable with data in a Sobolev space. In the present paper we improve on the $t$-independence condition by introducing a mixed $L^1-L^\infty$ condition that only depends on $\partial_t A$, the derivative of $A$ in transversal direction. This condition is different from other conditions in the literature and has already been proven to imply solvability of the Dirichlet boundary value problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_20453 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The regularity problem with a weaker condition on only the transversal direction 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 the regularity boundary value problem is solvable with data in a Sobolev space. In the present paper we improve on the $t$-independence condition by introducing a mixed $L^1-L^\infty$ condition that only depends on $\partial_t A$, the derivative of $A$ in transversal direction. This condition is different from other conditions in the literature and has already been proven to imply solvability of the Dirichlet boundary value problem. |
| title | The regularity problem with a weaker condition on only the transversal direction |
| topic | Analysis of PDEs 35J25, 35J15, 42B37, 47D03 |
| url | https://arxiv.org/abs/2503.20453 |