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Maximal Sobolev regularity of the stress tensor for the symmetric gradient p-Laplace system

Linus Behn, Andrea Cianchi, Lars Diening, Fa Peng

Abstract

The symmetric $p$-Laplace operator enters various models in mathematical physics, such as incompressible materials with power-type hardening and non-Newtonian fluids. In this work, second-order differentiability properties of solutions to the symmetric $p$-Laplace system are established. They are formulated as maximal Sobolev regularity of the nonlinear stress tensor for locally square integrable right-hand sides.

Maximal Sobolev regularity of the stress tensor for the symmetric gradient p-Laplace system

Abstract

The symmetric -Laplace operator enters various models in mathematical physics, such as incompressible materials with power-type hardening and non-Newtonian fluids. In this work, second-order differentiability properties of solutions to the symmetric -Laplace system are established. They are formulated as maximal Sobolev regularity of the nonlinear stress tensor for locally square integrable right-hand sides.
Paper Structure (7 sections, 13 theorems, 223 equations, 1 table)

This paper contains 7 sections, 13 theorems, 223 equations, 1 table.

Table of Contents

  1. Introduction and main results
  2. Young functions associated with the symmetric gradient p-Laplace system
  3. Pointwise differential inequalities
  4. Function spaces
  5. Solutions to symmetric gradient systems
  6. Nonlinear systems with quadratic growth
  7. Proof of Theorem \ref{['thm:local']}

Key Result

theorem 1

Assume that $p^-(n) <p< p^+(n)$. Let $f \in L^2_{\loc}(\Omega)$ and let $u$ be a local approximable solution to the system eq:system. Then, and there exists a constant $c=c(p,n)$ such that for every ball $B_R$ such that $B_{2R}\subset\subset \Omega$.

Theorems & Definitions (28)

  • theorem 1
  • remark 1
  • lemma 1
  • proof
  • lemma 2
  • proof
  • lemma 3: First pointwise identity
  • proof : Proof of Lemma \ref{['lem:pw_identity_lowdim']}
  • lemma 4: Second pointwise identity
  • proof
  • ...and 18 more