Variational Methods for Problems from Plasticity Theory and for Generalized Newtonian Fluids

Introduction1 Weak solutions to boundary value problems in the deformation theory of perfect elastoplasticity1.0. Preliminaries1.1. The classical boundary value problem for the equilibrium state of a perfect elastoplastic body and its primary functional formulation1.2. Relaxation of convex variational problems in non reflexive spaces.General construction1.3. Weak solutions to variational problems of perfect elastoplasticity2 Differentiability properties of weak solutions to boundary value problems in the deformation theory of plasticity2.0. Preliminaries2.1. Formulation of the main results2.2. Approximation and proof of Lemma 2.1.12.3. Proof of Theorem 2.1.1 and local estimate of Caccioppoli-type for the stress tensor2.4. Estimates for solutions of certain systems of PDE's with constant coeffcients2.5. The main lemma and its iteration2.6. Proof of Theorem 2.1.22.7. Open Problems2.8. Remarks on the regularity of minimizers of variational functionals from the deformation theory of plasticity with power hardeningAppendix AA.1 Density of smooth functions in spaces of tensor-valued functionsA.2 Density of smooth functions in spaces of vector-valued functionsA.3 Some properties of the space BDA.4 Jensen's inequality3 Quasi-static fluids of generalized Newtonian type3.0. Preliminaries3.1. Partial C1 regularity in the variational setting3.2. Local boundedness of the strain velocity3.3. The two-dimensional case3.4. The Bingham variational inequality in dimensions two and three3.5. Some open problems and comments concerning extensions4 Fluids of Prandtl-Eyring type and plastic materials with logarithmic hardening law4.0. Preliminaries4.1. Some functions spaces related to the Prandtl-Eyring fluid model4.2. Existence of higher order weak derivatives and a Caccioppoli-type inequality4.3. Blow-up: the proof of Theorem 4.1.1 for n=34.4. The two-dimensional case4.5. Partial regularity for plastic materials with logarithmic hardening4.6. A general class of constitutive relationsAppendix BB.1 Density resultsNotation and tools from functional analysis