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

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 $R:\omega \subset {\mathbb{R}}^2\to \mathrm{SO}(3)$. The only further structural assumption is that the curvature energy depends solely on the uni-constant isotropic Dirichlet-type energy term $|\mathrm{D}R{|}^2$. We use Rivière’s regularity techniques of harmonic-map-type systems for our system which couples harmonic maps to $\mathrm{SO}(3)$ 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.