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\colon \omega\subset \mathbb{R}^{2}\to \mathbb{R}^{3} and the orthogonal microrotation tensor field R\colon \omega\subset \mathbb{R}^{2}\to \operatorname{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 \operatorname{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.