Regularity results for a class of mixed local and nonlocal singular problems involving distance function
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| Format: | Preprint |
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2024
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| _version_ | 1866910791625605120 |
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| author | Bal, Kaushik Das, Stuti |
| author_facet | Bal, Kaushik Das, Stuti |
| contents | We investigate the following mixed local and nonlocal quasilinear equation with singularity given by \begin{eqnarray*} \begin{split} -Δ_pu+(-Δ)_q^s u&=\frac{f(x)}{u^δ}\text { in } Ω, \\u&>0 \text{ in } Ω,\\u&=0 \text { in }\mathbb{R}^n \backslash Ω; \end{split} \end{eqnarray*} where, \begin{equation*} (-Δ)_q^s u(x):= c_{n,s}\operatorname{P.V.}\int_{\mathbb{R}^n}\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{n+sq}} d y, \end{equation*} with $Ω$ being a bounded domain in $\mathbb{R}^{n}$ with $C^2$ boundary, $1<q\leq p<\infty$, $s\in(0,1)$, $δ>0$ and $f\in L^\infty_{\mathrm{loc}}(Ω)$ is a non-negative function which behaves like $\mathbf{dist(x,\partial Ω)^{-β}}$, $β\geq 0$ near $\partial Ω$. We start by proving several Hölder and gradient Hölder regularity results for a more general class of quasilinear operators when $δ=0$. Using the regularity results we deduce existence, uniqueness and Hölder regularity of a weak solution of the singular problem in $W_{\mathrm{loc}}^{1,p}(Ω)$ and its behavior near $\partial Ω$ albeit with different exponents depending on $β+δ$. Boundedness and Hölder regularity result to the singular equation with critical exponent were also discussed. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_14217 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Regularity results for a class of mixed local and nonlocal singular problems involving distance function Bal, Kaushik Das, Stuti Analysis of PDEs 35J75, 35M10, 35R11 We investigate the following mixed local and nonlocal quasilinear equation with singularity given by \begin{eqnarray*} \begin{split} -Δ_pu+(-Δ)_q^s u&=\frac{f(x)}{u^δ}\text { in } Ω, \\u&>0 \text{ in } Ω,\\u&=0 \text { in }\mathbb{R}^n \backslash Ω; \end{split} \end{eqnarray*} where, \begin{equation*} (-Δ)_q^s u(x):= c_{n,s}\operatorname{P.V.}\int_{\mathbb{R}^n}\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{n+sq}} d y, \end{equation*} with $Ω$ being a bounded domain in $\mathbb{R}^{n}$ with $C^2$ boundary, $1<q\leq p<\infty$, $s\in(0,1)$, $δ>0$ and $f\in L^\infty_{\mathrm{loc}}(Ω)$ is a non-negative function which behaves like $\mathbf{dist(x,\partial Ω)^{-β}}$, $β\geq 0$ near $\partial Ω$. We start by proving several Hölder and gradient Hölder regularity results for a more general class of quasilinear operators when $δ=0$. Using the regularity results we deduce existence, uniqueness and Hölder regularity of a weak solution of the singular problem in $W_{\mathrm{loc}}^{1,p}(Ω)$ and its behavior near $\partial Ω$ albeit with different exponents depending on $β+δ$. Boundedness and Hölder regularity result to the singular equation with critical exponent were also discussed. |
| title | Regularity results for a class of mixed local and nonlocal singular problems involving distance function |
| topic | Analysis of PDEs 35J75, 35M10, 35R11 |
| url | https://arxiv.org/abs/2411.14217 |