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

A first regularity result for the Armstrong-Frederick cyclic hardening plasticity model with Cosserat effects

The purpose of this article is to prove the Hölder continuity up to the boundary of the displacement vector and the microrotation matrix for the quasistatic, rate-independent Armstrong-Frederick cyclic hardening plasticity model with Cosserat effects. This model is of non-monotone and non-associated type. In the case of two space dimensions we use the hole-filling technique of Widman and the Morrey's Dirichlet growth theorem.

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

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

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

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 $p\in [2, \frac{32}{15}]$.

超临界维度下几何非线性系统的$L^p$正则性

$L^p$-regularity of a geometrically nonlinear system in supercritical dimensions

本文研究了源自几何非线性平坦Cosserat微极模型的系统，在超临界维度下建立了内部正则性和精确的$L^p$正则性，扩展了Gastel和Neff在临界维度下的工作。

The Armstrong-Frederick cyclic hardening plasticity model with Cosserat effects

We propose an extension of the cyclic hardening plasticity model formulated by Armstrong and Frederick which includes micropolar effects. Our micropolar extension establishes coercivity of the model which is otherwise not present. We study then existence of solutions to the quasistatic, rate-independent Armstrong-Frederick model with Cosserat effects which is, however, still of non-monotone, non-associated type. In order to do this, we need to relax the pointwise definition of the flow rule into a suitable weak energy-type inequality. It is shown that the limit in the Yosida approximation proc…

抛物型Signorini问题中几乎极小化子的自由边界正则性

Free boundary regularity for almost minimizers of the parabolic Signorini problem

本文针对抛物型Signorini问题中的几乎极小化子，通过建立Weiss型单调性公式和Almgren型频率公式，结合epiperimetric不等式，证明了正则自由边界点的优化增长及正则集的正则性。