In this contribution, we completely and explicitly characterize Young measures generated by gradients of quasiconformal maps in the plane. By doing so, we generalize the results of Astala and Faraco [5] who provided a similar result for quasiregular maps and Benešová and Kružík [14] who characterized Young measures generated by gradients of bi-Lipschitz maps. Our results are motivated by non-linear elasticity where injectivity of the functions in the generating sequence is essential in order to assure non-interpenetration of matter.
We introduce a two time-scale scheme which allows to extend the method of minimizing movements to hyperbolic problems. This method is used to show the existence of weak solutions to a fluid-structure interaction problem between a nonlinear, visco-elastic, ndimensional bulk solid governed by a hyperbolic evolution and an incompressible fluid governed by the (n-dimensional) Navier-Stokes equations for n ≥ 2. Contents 1. Introduction 1 1.1. Setup 3 1.2. Main result 5 1.3. Mechanical and analytical restrictions on the energy/dissipation functional 9 1.4. Outlook and further applicability 11 1.5. Notation 12 2. Minimizing movements for fluid-structure interactions. 13 2.1. Preliminary analysis 14 2.2. Proof of Theorem 2.2 16 2.3. The example Energy-Dissipation pair 26 3. Minimizing movements for hyperbolic evolutions 28 3.1. The time-delayed problem 30 3.2. Proof of Theorem 3.2 33 3.3. Remarks 37 4. The unsteady fluid-structure interaction problem 39 4.1. An intermediate, time delayed model 40 4.2. Proof of Theorem 1.2 46 4.3. Remarks on the full problem 56 Appendix A. 57 A.1. Some technical lemmata 57 A.2. Proof of Proposition 2.22 59 Acknowledgments 62 References 62
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