A nonlinear Calderón-Zygmund $ L^2$-theory for the Dirichlet problem involving $ -|Du|^γΔ^N_p u=f$
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| Format: | Preprint |
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2024
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| _version_ | 1866929585491279872 |
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| author | Miao, Qianyun Peng, Fa Zhou, Yuan |
| author_facet | Miao, Qianyun Peng, Fa Zhou, Yuan |
| contents | We establish a nonlinear Calderón-Zygmund $L^2$-theory to the Dirichlet problem $$-|Du|^γΔ^N_p u=f\in L^2(Ω)\quad {\rm in}\quad Ω; \quad u=0 \ \mbox{on $\partialΩ$} $$ for $n\ge2$, $ p>1$ and a large range of $γ>-1$, in particular, for all $p>1$ and all $ γ>-1$ when $n=2$. Here $Ω\subset \mathbb{R}^n$ is a bounded convex domain, or a bounded Lipschitz domain whose boundary has small weak second fundamental form in the sense of Cianchi-Maz'ya (2018).
The proof relies on an extension of an Miranda-Talenti \& Cianchi-Maz'ya type inequality, that is, for any $v\in C^\infty_0(Ω)$ in any bounded smooth domain $Ω$, $\|D[(|Dv|^2+ε)^{\fracγ2}Dv]\|_{L^2(Ω)}$ is bounded via $\|(|Dv|^2+ε)^{\fracγ2} Δ^N_{p,ε}v \|_{L^2(Ω)}$, where $Δ^N_{p,ε}v$ is the $ε$-regularization of normalized $p$-Laplacian.
Our results extend the well-known Calderón-Zygmund $L^2$-estimate for the Poisson equation, a nonlinear global second order Sobolev estimate for inhomogeneous $p$-Laplace equation by Cianchi-Maz'ya (2018), and a local $W^{2,2}$-estimate for inhomogeneous normalized $p$-Laplace equation by Attouchi-Ruosteenoja (2018). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_03796 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A nonlinear Calderón-Zygmund $ L^2$-theory for the Dirichlet problem involving $ -|Du|^γΔ^N_p u=f$ Miao, Qianyun Peng, Fa Zhou, Yuan Analysis of PDEs 35J25, 35J60, 35B65 We establish a nonlinear Calderón-Zygmund $L^2$-theory to the Dirichlet problem $$-|Du|^γΔ^N_p u=f\in L^2(Ω)\quad {\rm in}\quad Ω; \quad u=0 \ \mbox{on $\partialΩ$} $$ for $n\ge2$, $ p>1$ and a large range of $γ>-1$, in particular, for all $p>1$ and all $ γ>-1$ when $n=2$. Here $Ω\subset \mathbb{R}^n$ is a bounded convex domain, or a bounded Lipschitz domain whose boundary has small weak second fundamental form in the sense of Cianchi-Maz'ya (2018). The proof relies on an extension of an Miranda-Talenti \& Cianchi-Maz'ya type inequality, that is, for any $v\in C^\infty_0(Ω)$ in any bounded smooth domain $Ω$, $\|D[(|Dv|^2+ε)^{\fracγ2}Dv]\|_{L^2(Ω)}$ is bounded via $\|(|Dv|^2+ε)^{\fracγ2} Δ^N_{p,ε}v \|_{L^2(Ω)}$, where $Δ^N_{p,ε}v$ is the $ε$-regularization of normalized $p$-Laplacian. Our results extend the well-known Calderón-Zygmund $L^2$-estimate for the Poisson equation, a nonlinear global second order Sobolev estimate for inhomogeneous $p$-Laplace equation by Cianchi-Maz'ya (2018), and a local $W^{2,2}$-estimate for inhomogeneous normalized $p$-Laplace equation by Attouchi-Ruosteenoja (2018). |
| title | A nonlinear Calderón-Zygmund $ L^2$-theory for the Dirichlet problem involving $ -|Du|^γΔ^N_p u=f$ |
| topic | Analysis of PDEs 35J25, 35J60, 35B65 |
| url | https://arxiv.org/abs/2411.03796 |