Geometric nonlinearities in PDEs for Cosserat elasticity:how they affect the regularity of solutions

Cosserat微极弹性中的壳或固体模型涉及对能量有贡献的微旋转。如果方程中的相应项不线性化，则模型的欧拉-拉格朗日方程是一个临界非线性偏微分方程组。它由描述力平衡的方程和描述角动量平衡的方程组成。后者与到SO（3）的调和映射系统（从几何分析和数学物理中已知）相一致，加上一个额外的耦合项。对于调和映射，有丰富的正则性理论，包括关于奇点结构的有趣结果。由于调和映射系统在上面提到的Cosserat模型中起着主导作用，因此Cosserat固体和壳中可能存在的奇异性有着有趣的意义。基于申请人和内夫关于正则性的第一个定理，以及申请人和Hüsken在优先计划第一阶段对Cosserat固体奇异行为的示例，我们将把正则性方法扩展到更现实的设置。这涉及到允许更多参数的模型，而不是迄今为止使用的“唯一近似”，这将需要新的参数，因为调和映射类型系统的结构相似性将比以前弱得多。目前尚不清楚现有的方法是否足够强大，以涵盖更一般的方程。这也涉及到对应用程序有用的边界条件的研究，这些应用程序涉及固定和自由边界部分。即使是调和映射，边界正则性也没有在这种情况下得到证明。最后，我们希望正则性估计可以帮助建立在Cosserat理论的参数选择，迄今尚未解决的存在极小。

Models for shells or solids in Cosserat micropolar elasticity involve micro-rotations which contribute to the energy. If the corresponding term in the equations is not linearized, the Euler-Lagrange equations of the model are a critically nonlinear system of partial differential equations. It consists of equations describing the balance of forces, coupled with equations describing the balance of angular momentum. The latter coincide with the system for harmonic maps to SO(3) (known from geometric analysis and mathematical physics), plus an additional coupling term. For harmonic maps, there is a rich regularity theory, including interesting results about the structure of singularities. Since the harmonic maps system plays a dominating role in the Cosserat models mentioned above, there are interesting implications to be expected about possible singularities in Cosserat solids and shells. Based on first theorems about regularity by the applicant and Neff, and on examples for singular behavior of Cosserat solids by the applicant and Hüsken from the first phase of the priority program, we will extend regularity methods towards more realistic settings. This involves models allowing for more parameters instead of the "uniconstant approximations" used so far, which will require new arguments, because the structural similarity to harmonic map type systems will be much weaker than before. It is not clear if existing methods are strong enough to cover the more general equations. This also involves the study of boundary conditions useful for applications, which involve both fixed and free boundary parts. Even for harmonic maps, boundary regularity has not been proven in this setting. Finally, we hope that regularity estimates can help to establish the existence of minimizers in Cosserat theory for choices of parameters for which that has not been settled so far.