Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A nonlinear Calderón-Zygmund $ L^2$-theory for the Dirichlet problem involving $ -|Du|^γΔ^N_p u=f$

Qianyun Miao, Fa Peng, Yuan Zhou

Abstract

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).

A nonlinear Calderón-Zygmund $ L^2$-theory for the Dirichlet problem involving $ -|Du|^γΔ^N_p u=f$

Abstract

We establish a nonlinear Calderón-Zygmund -theory to the Dirichlet problem for , and a large range of , in particular, for all and all when . Here 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 in any bounded smooth domain , is bounded via , where is the -regularization of normalized -Laplacian. Our results extend the well-known Calderón-Zygmund -estimate for the Poisson equation, a nonlinear global second order Sobolev estimate for inhomogeneous -Laplace equation by Cianchi-Maz'ya (2018), and a local -estimate for inhomogeneous normalized -Laplace equation by Attouchi-Ruosteenoja (2018).

Paper Structure

This paper contains 14 sections, 23 theorems, 229 equations.

Table of Contents

  1. Introduction
  2. Motivation and related studies in the literature
  3. Sketch the ideas to prove Theorems \ref{['th-re1']}-\ref{['thm-v']}
  4. Viscosity solutions
  5. A key inequality involving the operator $\Delta^N_{p,\epsilon}$
  6. Global Sobolev estimates and proof of Theorem \ref{['thm-v']}
  7. Proof of Lemma \ref{['sd-es']}
  8. Proof of Lemma \ref{['Lq-es']}
  9. Some Hölder estimates
  10. Proof of Lemma \ref{['holder-2']}
  11. Proof of Proposition \ref{['example']}
  12. Existence of viscosity solutions under $f\in L^\infty(\Omega)\cap C^0(\Omega)$
  13. Proof of Theorem \ref{['th-re0']}
  14. Proofs of Theorem \ref{['th-re1']} and Theorem \ref{['th-re1a']}

Key Result

Theorem 1.1

Suppose that the triplet $(n,p,\gamma)$ is admissible and $\gamma>-\frac{4}{n+2}$. Let $\Omega\subset{{{\mathbb R}}^n}$ be a bounded convex domain. For any $f\in L^2(\Omega)$, the problem plapgz admits an approximation solution $u\in W^{1,2}_0(\Omega)$, which satisfies and has the upper bound for some constant $C=C(p,n,{\gamma})$. If $f\in C^0(\Omega)$ in addition, then $u\in C^0(\Omega)$ is in

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • Lemma 3.1
  • Lemma 3.2
  • ...and 41 more