Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Higher integrability for doubly nonlinear parabolic systems

Verena Bögelein, Frank Duzaar, Juha Kinnunen, Christoph Scheven

Abstract

This paper proves a local higher integrability result for the spatial gradient of weak solutions to doubly nonlinear parabolic systems. The new feature of the argument is that the intrinsic geometry involves the solution as well as its spatial gradient. The main result holds true for a range of parameters suggested by other nonlinear parabolic systems.

Higher integrability for doubly nonlinear parabolic systems

Abstract

This paper proves a local higher integrability result for the spatial gradient of weak solutions to doubly nonlinear parabolic systems. The new feature of the argument is that the intrinsic geometry involves the solution as well as its spatial gradient. The main result holds true for a range of parameters suggested by other nonlinear parabolic systems.

Paper Structure

This paper contains 17 sections, 17 theorems, 208 equations.

Table of Contents

  1. Introduction
  2. Notation and the main result
  3. Notation
  4. Assumptions and the main result
  5. Auxiliary Material
  6. Energy bounds
  7. Parabolic Sobolev-Poincaré type inequalities
  8. The case $\pmb{\max\{\frac{2n}{n+2},1\}<p\le 2}$
  9. The case $\pmb{p\ge 2}$
  10. Reverse Hölder inequality
  11. Higher integrability: Proof of Theorem \ref{['thm:higherint']}
  12. Construction of a non-uniform system of cylinders
  13. Covering property
  14. Stopping time argument
  15. A Reverse Hölder Inequality
  16. ...and 2 more sections

Key Result

Theorem \oldthetheorem

Let where the right-hand side is interpreted as $\infty$ for the dimensions $n=1$ and $n=2$, and assume that $\sigma>p$. Then, there exists $\varepsilon_o=\varepsilon_o(n,p,\nu,L)\in (0,1]$ such that whenever $F\in L^\sigma(\Omega_T,\mathbb{R}^N)$ and $u$ is a weak solution to eq-doubly in the sense of where $\varepsilon_1:=\min\{\varepsilon_o,\frac{\sigma}{p}-1\}$. Moreover, for every $\varepsi

Theorems & Definitions (30)

  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • ...and 20 more