Schauder estimates for elliptic equations degenerating on lower dimensional manifolds
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| Format: | Preprint |
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2025
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| _version_ | 1866913840535437312 |
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| author | Cora, Gabriele Fioravanti, Gabriele Vita, Stefano |
| author_facet | Cora, Gabriele Fioravanti, Gabriele Vita, Stefano |
| contents | In this paper we begin exploring a local regularity theory for elliptic equations having coefficients which are degenerate or singular on some lower dimensional manifold $$ -\mathrm{div}(|y|^aA(x,y)\nabla u)=|y|^af+\mathrm{div}(|y|^aF)\qquad\mathrm{in \ } B_1\subset\mathbb R^d, $$ where $z=(x,y)\in\mathbb R^{d-n}\times\mathbb R^n$, $2\leq n\leq d$ are two integers and $a\in\mathbb R$. Such equations are a prototypical example of elliptic equations spoiling their uniform ellipticity on the (possibly very) thin characteristic manifold $Σ_0=\{|y|=0\}$ of dimension $0\leq d-n\leq d-2$, having $$λ|y|^a|ξ|^2\leq |y|^aA(x,y)ξ\cdotξ\leqΛ|y|^a|ξ|^2.$$ Whenever $a+n>0$, the weak solutions with a homogeneous conormal boundary condition at $Σ_0$ are provided to be $C^{0,α}$ or even $C^{1,α}$ regular up to $Σ_0$. Our approach relies on a regularization-approximation scheme which employs domain perforation, very fine blow-up procedures, and a new Liouville theorem in the perforated space. Our theory extends to the case of equations degenerating on suitably smooth curved manifolds. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_19033 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Schauder estimates for elliptic equations degenerating on lower dimensional manifolds Cora, Gabriele Fioravanti, Gabriele Vita, Stefano Analysis of PDEs 35B65, 35J70, 35J75, 35B40, 35B44, 35B45, 35B53 In this paper we begin exploring a local regularity theory for elliptic equations having coefficients which are degenerate or singular on some lower dimensional manifold $$ -\mathrm{div}(|y|^aA(x,y)\nabla u)=|y|^af+\mathrm{div}(|y|^aF)\qquad\mathrm{in \ } B_1\subset\mathbb R^d, $$ where $z=(x,y)\in\mathbb R^{d-n}\times\mathbb R^n$, $2\leq n\leq d$ are two integers and $a\in\mathbb R$. Such equations are a prototypical example of elliptic equations spoiling their uniform ellipticity on the (possibly very) thin characteristic manifold $Σ_0=\{|y|=0\}$ of dimension $0\leq d-n\leq d-2$, having $$λ|y|^a|ξ|^2\leq |y|^aA(x,y)ξ\cdotξ\leqΛ|y|^a|ξ|^2.$$ Whenever $a+n>0$, the weak solutions with a homogeneous conormal boundary condition at $Σ_0$ are provided to be $C^{0,α}$ or even $C^{1,α}$ regular up to $Σ_0$. Our approach relies on a regularization-approximation scheme which employs domain perforation, very fine blow-up procedures, and a new Liouville theorem in the perforated space. Our theory extends to the case of equations degenerating on suitably smooth curved manifolds. |
| title | Schauder estimates for elliptic equations degenerating on lower dimensional manifolds |
| topic | Analysis of PDEs 35B65, 35J70, 35J75, 35B40, 35B44, 35B45, 35B53 |
| url | https://arxiv.org/abs/2501.19033 |