Fine Boundary Regularity For The Fractional (p,q)-Laplacian
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866908372512538624 |
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| author | Dhanya, R. Jana, Ritabrata Kumar, Uttam Tiwari, Sweta |
| author_facet | Dhanya, R. Jana, Ritabrata Kumar, Uttam Tiwari, Sweta |
| contents | In this article, we deal with the fine boundary regularity, a weighted Hölder regularity of weak solutions to the problem involving the fractional $(p,q)$ Laplacian denoted by $(-Δ)_{p}^{s} u + (-Δ)_{q}^{s} u = f(x)$ in $Ω,$ and $u=0$ in $\mathbb{R}^N\setminusΩ;$ where $Ω$ is a $C^{1,1}$ bounded domain and $2 \leq p \leq q <\infty.$ For $0<s<1$ and for non-negative data $f\in L^{\infty}(Ω),$ we employ the nonlocal analogue of the boundary Harnack method to establish that $u/{d_Ω^{s}} \in C^α(\BarΩ)$ for some $α\in (0,1),$ where $d_Ω(x)$ is the distance of $x$ from the boundary. A novel barrier construction allows us to analyse the regularity theory even in the absence of the scaling or the homogeneity properties of the operator. Additionally, we extend our idea to sign changing bounded $f$ as well and prove a fine boundary regularity for fractional $(p,q)$ Laplacian for some range of $s.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_07995 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fine Boundary Regularity For The Fractional (p,q)-Laplacian Dhanya, R. Jana, Ritabrata Kumar, Uttam Tiwari, Sweta Analysis of PDEs 35R11, 35J60, 35D30, 35B65 In this article, we deal with the fine boundary regularity, a weighted Hölder regularity of weak solutions to the problem involving the fractional $(p,q)$ Laplacian denoted by $(-Δ)_{p}^{s} u + (-Δ)_{q}^{s} u = f(x)$ in $Ω,$ and $u=0$ in $\mathbb{R}^N\setminusΩ;$ where $Ω$ is a $C^{1,1}$ bounded domain and $2 \leq p \leq q <\infty.$ For $0<s<1$ and for non-negative data $f\in L^{\infty}(Ω),$ we employ the nonlocal analogue of the boundary Harnack method to establish that $u/{d_Ω^{s}} \in C^α(\BarΩ)$ for some $α\in (0,1),$ where $d_Ω(x)$ is the distance of $x$ from the boundary. A novel barrier construction allows us to analyse the regularity theory even in the absence of the scaling or the homogeneity properties of the operator. Additionally, we extend our idea to sign changing bounded $f$ as well and prove a fine boundary regularity for fractional $(p,q)$ Laplacian for some range of $s.$ |
| title | Fine Boundary Regularity For The Fractional (p,q)-Laplacian |
| topic | Analysis of PDEs 35R11, 35J60, 35D30, 35B65 |
| url | https://arxiv.org/abs/2406.07995 |