Project Details
Projekt
Geometric nonlinearities in PDEs for Cosserat elasticity: how they affect the regularity of solutions
Applicant
Professor Dr. Andreas Gastel
Subject Area
Mathematics
Term
since 2020
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 441380936
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.
DFG Programme
Priority Programmes
