Regularity for a geometrically nonlinear flat Cosserat micropolar membrane shell with curvature

TL;DR

This paper establishes full interior regularity for a nonlinear Cosserat membrane shell model, coupling harmonic maps to SO(3) with a linear deformation equation, under minimal curvature energy assumptions.

Contribution

It rigorously derives the membrane limit of a 3D Cosserat model and proves interior regularity using harmonic map techniques, with minimal structural assumptions.

Findings

Full interior regularity of deformation and microrotation fields

Coupling harmonic maps with linear equations handled via special structure

Regularity results under minimal curvature energy dependence

Abstract

We consider the rigorously derived thin shell membrane $\Gamma$-limit of a three-dimensional isotropic geometrically nonlinear Cosserat micropolar model and deduce full interior regularity of both the midsurface deformation $m:\omega\subset{\mathbb R}^2\to{\mathbb R}^3$ and the orthogonal microrotation tensor field $R:\omega\subset{\mathbb R}^2\to SO(3)$. The only further structural assumption is that the curvature energy depends solely on the uni-constant isotropic Dirichlet type energy term $|DR|^2$. We use Rivi\`ere's regularity techniques of harmonic map type systems for our system which couples harmonic maps to $SO(3)$ with a linear equation for $m$. The additional coupling term in the harmonic map equation is of critical integrability and can only be handled because of its special structure.