The Dirichlet problem on lower dimensional boundaries: Schauder estimates via perforated domains
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| Format: | Preprint |
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2024
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| author | Fioravanti, Gabriele |
| author_facet | Fioravanti, Gabriele |
| contents | In this paper, we investigate the Dirichlet problem on lower dimensional manifolds for a class of weighted elliptic equations with coefficients that are singular on such sets. Specifically, we study the problem \[\begin{cases} -{\rm div}(|y|^a A(x,y) \nabla u) = |y|^a f + {\rm div}(|y|^a F), \\ u = ψ, \quad \text{ on } Σ_0, \end{cases} \] where $(x,y) \in \mathbb{R}^{d-n} \times \mathbb{R}^n$, $2 \leq n \leq d$, $a + n \in (0,2)$, and $Σ_0 = \{|y| = 0\}$ is the lower dimensional manifold where the equation loses uniform ellipticity. Our primary objective is to establish $C^{0,α}$ and $C^{1,α}$ regularity estimates up to $Σ_0$, under suitable assumptions on the coefficients and the data. Our approach combines perforated domain approximations, Liouville-type theorems and a fine blow-up argument. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_11294 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Dirichlet problem on lower dimensional boundaries: Schauder estimates via perforated domains Fioravanti, Gabriele Analysis of PDEs 35B65 (primary), 35J75, 35J25, 35B44, 35B53 (secondary) In this paper, we investigate the Dirichlet problem on lower dimensional manifolds for a class of weighted elliptic equations with coefficients that are singular on such sets. Specifically, we study the problem \[\begin{cases} -{\rm div}(|y|^a A(x,y) \nabla u) = |y|^a f + {\rm div}(|y|^a F), \\ u = ψ, \quad \text{ on } Σ_0, \end{cases} \] where $(x,y) \in \mathbb{R}^{d-n} \times \mathbb{R}^n$, $2 \leq n \leq d$, $a + n \in (0,2)$, and $Σ_0 = \{|y| = 0\}$ is the lower dimensional manifold where the equation loses uniform ellipticity. Our primary objective is to establish $C^{0,α}$ and $C^{1,α}$ regularity estimates up to $Σ_0$, under suitable assumptions on the coefficients and the data. Our approach combines perforated domain approximations, Liouville-type theorems and a fine blow-up argument. |
| title | The Dirichlet problem on lower dimensional boundaries: Schauder estimates via perforated domains |
| topic | Analysis of PDEs 35B65 (primary), 35J75, 35J25, 35B44, 35B53 (secondary) |
| url | https://arxiv.org/abs/2412.11294 |