The regularity problem for the laplace equation in rough domains
Let Ω ⊂ ℝ+, n ≥ 2 be a bounded open and connected set satisfying the corkscrew condition with uniformly n-rectifiable boundary. In this paper we study the connection among the solvability of (D́), the Dirichlet problem for the Laplacian with boundary data in L(∂Ω), and (R) (resp., (R͂)), the regular...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2024 |
| País: | España |
| Institución: | Universitat Autònoma de Barcelona |
| Repositorio: | Dipòsit Digital de Documents de la UAB |
| Idioma: | inglés |
| OAI Identifier: | oai:ddd.uab.cat:325487 |
| Acceso en línea: | https://ddd.uab.cat/record/325487 https://dx.doi.org/urn:doi:10.1215/00127094-2023-0044 |
| Access Level: | acceso abierto |
| Palabra clave: | Chord-arc domain Dirichlet problem Laplace equation Regularity problem |
| Sumario: | Let Ω ⊂ ℝ+, n ≥ 2 be a bounded open and connected set satisfying the corkscrew condition with uniformly n-rectifiable boundary. In this paper we study the connection among the solvability of (D́), the Dirichlet problem for the Laplacian with boundary data in L(∂Ω), and (R) (resp., (R͂)), the regularity problem for the Laplacian with boundary data in the Hajłasz Sobolev space W (∂Ω) (resp., W͂(∂Ω), the usual Sobolev space in terms of the tangential derivative), where p ∈ (1, 2 + ε) and 1/p + 1/p = 1. Our main result shows that (D́) is solvable if and only if (R) also is. Under additional geometric assumptions (two-sided local John condition or weak Poincaré inequality on the boundary), we prove that (D́) ⇒ (R͂). In particular, we deduce that in bounded chord-arc domains (resp., two-sided chord-arc domains), there exists p ∈ (1, 2 + ε) so that (R) (resp., (R͂)) is solvable. We also extend the results to unbounded domains with compact boundary and show that in two-sided corkscrew domains with n-Ahlfors-David regular boundaries, the single-layer potential operator is invertible from L(∂Ω) to the inhomogeneous Sobolev space W (∂Ω). Finally, we provide a counterexample of a chord-arc domain Ω ⊂ ℝ+, n ≥ 3, so that (R͂) is not solvable for any p ∈ [1, ∞). |
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