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Global regularity for a physically nonlinear version of the relaxed micromorphic model on Lipschitz domains

Dorothee Knees, Sebastian Owczarek, Patrizio Neff

Abstract

In this paper, we investigate the global higher regularity properties of weak solutions for a linear elliptic system coupled with a nonlinear Maxwell-type system defined on Lipschitz domains. The regularity result is established using a modified finite difference approach. These adjusted finite differences involve inner variations in conjunction with a Piola-type transformation to preserve the curl-structure within the matrix Maxwell system. The proposed method is further applied to the linear relaxed micromorphic model. As a result, for a physically nonlinear version of the relaxed micromorphic model, we demonstrate that for arbitrary $ε> 0$, the displacement vector $u$ belongs to $H^{\frac{3}{2}-ε}(Ω)$, and the microdistortion tensor $P$ belongs to $H^{\frac{1}{2}-ε}(Ω)$ while $\Curl P$ belongs to $H^{\frac{1}{2}-ε}(Ω)$.

Global regularity for a physically nonlinear version of the relaxed micromorphic model on Lipschitz domains

Abstract

In this paper, we investigate the global higher regularity properties of weak solutions for a linear elliptic system coupled with a nonlinear Maxwell-type system defined on Lipschitz domains. The regularity result is established using a modified finite difference approach. These adjusted finite differences involve inner variations in conjunction with a Piola-type transformation to preserve the curl-structure within the matrix Maxwell system. The proposed method is further applied to the linear relaxed micromorphic model. As a result, for a physically nonlinear version of the relaxed micromorphic model, we demonstrate that for arbitrary , the displacement vector belongs to , and the microdistortion tensor belongs to while belongs to .
Paper Structure (7 sections, 9 theorems, 75 equations)

This paper contains 7 sections, 9 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. A nonlinear version of the relaxed micromorphic model
  3. Background from function space theory
  4. Assumptions on the energy density $W$
  5. Global regularity on Lipschitz domains
  6. Inner variations and preliminary results
  7. Proof of Theorem \ref{['thm:globlipnonlin']}

Key Result

theorem 1

Let $\Omega\subset\mathbb{R}^3$ be a bounded domain with a Lipschitz boundary. Then and hence, for every $p\in L^2(\Omega;\mathbb{R}^3)$ there exist a unique $v\in H_0^1(\Omega)$ and $q\in H(\mathop{\mathrm{div}}\nolimits,0;\Omega)$ such that $p=\mathrm{D} v + q$.

Theorems & Definitions (18)

  • definition 1
  • theorem 1: Helmholtz decomposition
  • proposition 1
  • theorem 2: Embedding Theorem
  • remark 1
  • remark 2
  • remark 3
  • theorem 3
  • lemma 1
  • proof
  • ...and 8 more