Regularity of weak solution of variational problems modeling the Cosserat micropolar elasticity

TL;DR

This paper investigates the regularity and singularities of weak solutions to a variational model of Cosserat micropolar elasticity, revealing conditions under which solutions are smooth or have discrete singularities.

Contribution

It establishes new regularity results for weak solutions, showing singular sets are discrete or measure zero, and identifies parameter ranges for full regularity.

Findings

Singular set is discrete for 2<p<3.

Singular set has zero 1-dimensional Hausdorff measure at p=2.

Stable solutions are regular for p in [2, 32/15].

Abstract

In this paper, we consider weak solutions of the Euler-Lagrange equation to a variational energy functional modeling the geometrically nonlinear Cosserat micropolar elasticity of continua in dimension three, which is a system coupling between the Poisson equation and the equation of $p$-harmonic maps ($2\le p\le 3$). We show that if a weak solutions is stationary, then its singular set is discrete for $2<p<3$ and has zero $1$-dimensional Hausdorff measure for $p=2$. If, in addition, it is a stable-stationary weak solution, then it is regular everywhere when $p\in [2, \frac{32}{15}]$.