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Description:
In this thesis we consider systems of partial differential equations of continuum mechanics and analyze regularity properties of their weak solutions. The first chapter contains a detailed introduction and reviews the contents of chapter two, three and four. We start in chapter 2 with the local regularity problem related to the equations modelling the mechanical behaviour of elasto-perfect plastic materials respectively to an elasto-viscoplastic approximation of these materials, i. e. we consider the Norton-Hoff approximation to Hencky's law. These equations form a nonlinear systems of partial differential equations of second order and of elliptic type in the usual primal formulation, where one is interested in the displacement vector u=u(x) respectively the strain tensor ε(u) = ½ (∇u + (∇u) T ) . We study these systems via a dual approach which was developed by A. Bensoussan and J. Frehse. In this approach we look for the stress tensor σ = σ(x) which solves the system of equations: A σ + | σ D | p-2 σ D = ε(u) div σ + f = 0 in the weak sense with mixed boundary conditions. We show local Hölder continuity of the stress tensor in two dimensions for the Norton-Hoff approximation of the Hencky law in plasticity theory and deduce also corresponding results for the strain tensor ε(u) . The main tool to achieve this result is a logarithmic Morrey estimate, which was developed by J. Frehse together with A. Bensoussan and G. Seregin in the here considered context of the dual theory of elliptic systems. These logarithmic Morrey estimates combined with a suitable adapted estimate on higher integrability a la Meyers-Necas-Gehring-Giaquinta-Modica give the final result. We also deal with a system of partial differential equations describing a steady motion of an incompressible fluid with shear-dependent viscosity and present a new global existence result for p > 2d / d+2 . Here p is the coercivity parameter of the nonlinear elliptic operator related to the stress tensor and d is the dimension of the space. Lipschitz test ...