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

Abstract

We consider the rigorously derived thin shell membrane $\hat{I}“$-limit of a three-dimensional isotropic geometrically nonlinear Cosserat micropolar model and deduce full interior regularity of both the midsurface deformation $m:\ddot{I}‰\subset {\mathbb{R}}^2\to {\mathbb{R}}^3$ and the orthogonal microrotation tensor field $R:\ddot{I}‰\subset {\mathbb{R}}^2\to SO(3)$. 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 Rivière'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.