Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Global Calderòn & Zygmund theory for nonlinear parabolic systems

Verena Bögelein

Abstract

We establish a global Calderón & Zygmund theory for solutions of a huge class of nonlinear parabolic systems whose model is the inhomogeneous parabolic $p$-Laplacian system \begin{equation*} \left\{\begin{array}{cc} \partial_t u - \Div (|Du|^{p-2}Du) = \Div (|F|^{p-2}F) &\mbox{in $Ω_T:=Ω\times(0,T)$} \\[5pt] u=g &\mbox{on $\partialΩ\times(0,T)\cup \barΩ\times\{0\}$} \end{array}\right. \end{equation*} with given functions $F$ and $g$. Our main result states that the spatial gradient of the solution is as integrable as the data $F$ and $g$ up to the lateral boundary of $Ω_T$, i.e. \begin{equation*} F,Dg\in L^q(Ω_T),\ \ \partial_t g\in L^{\frac{q(n+2)}{p(n+2)-n}}(Ω_T) \quad\Rightarrow \quad Du\in L^q(Ω\times(δ,T)) \end{equation*} for any $q>p$ and $δ\in(0,T)$, together with quantitative estimates. This result is proved in a much more general setting, i.e. for asymptotically regular parabolic systems.

Global Calderòn & Zygmund theory for nonlinear parabolic systems

Abstract

We establish a global Calderón & Zygmund theory for solutions of a huge class of nonlinear parabolic systems whose model is the inhomogeneous parabolic -Laplacian system \begin{equation*} \left\{\begin{array}{cc} \partial_t u - \Div (|Du|^{p-2}Du) = \Div (|F|^{p-2}F) &\mbox{in } \\[5pt] u=g &\mbox{on } \end{array}\right. \end{equation*} with given functions and . Our main result states that the spatial gradient of the solution is as integrable as the data and up to the lateral boundary of , i.e. \begin{equation*} F,Dg\in L^q(Ω_T),\ \ \partial_t g\in L^{\frac{q(n+2)}{p(n+2)-n}}(Ω_T) \quad\Rightarrow \quad Du\in L^q(Ω\times(δ,T)) \end{equation*} for any and , together with quantitative estimates. This result is proved in a much more general setting, i.e. for asymptotically regular parabolic systems.

Paper Structure

This paper contains 16 sections, 15 theorems, 223 equations.

Table of Contents

  1. Introduction
  2. Statement of the results
  3. Asymptotically $p$-Laplacian systems
  4. General non-linear parabolic systems
  5. The general setting
  6. Preliminaries
  7. Some notation
  8. Auxiliary tools
  9. Higher integrability
  10. A priori gradient estimates
  11. Proof of the Calderón & Zygmund estimates
  12. Transformation to the model situation
  13. Exit cylinders and covering
  14. Comparison estimates
  15. Estimates on intrinsic cylinders
  16. ...and 1 more sections

Key Result

Theorem 1

Let $p>\frac{2n}{n+2}$ and suppose that is a weak solution to the Cauchy-Dirichlet problem where $\Omega_T:=\Omega\times(0,T)$, $\Omega$ is a bounded $C^1$ domain and $c\colon\Omega_T\to [\nu,L]$, $0<\nu\le L$ is VMO with respect to $x$ (see VMO-c below) and measurable with respect to $t$. Further, assume that for some $q>p$. Then and there holds the quantitative $L^q$-estimate CZ-est below.

Theorems & Definitions (22)

  • Theorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Definition \oldthetheorem
  • ...and 12 more