# Regularity issues for Cosserat continua and $p$-harmonic maps

## TL;DR

This paper investigates the regularity of minimizers in a nonlinear Cosserat elasticity model, establishing interior Hölder regularity except at isolated singular points related to $p$-harmonic maps to $S^3$, with improvements on existing regularity results.

## Contribution

It proves interior Hölder regularity for Cosserat model minimizers, identifying singularities linked to $p$-harmonic maps and enhancing regularity theorems for these maps.

## Key findings

- Regularity holds except at isolated singular points.
- Singularities are connected to homogeneous $p$-harmonic maps to $S^3$.
- Improved regularity theorems for $p$-harmonic maps facilitate the main results.

## Abstract

For minimizers in a geometrically nonlinear Cosserat model for micropolar elasticity of continua, we prove interior H\"older regularity, up to isolated singular points that may be possible if the exponent $p$ from the model is $2$ or in $(\frac{32}{15},3)$. The obstacle to full continuity turns out to be the existence of certain minimizing homogeneous $p$-harmonic maps to $S^3$. For those, we slightly improve existing regularity theorems in order to achieve our result on the Cosserat model.