harmonic homemorphism; minimal conformal energy; deformation gradient; Jacobian; Nitsche conjecture; Hopf differential; diffeomorphic approximation of Sobolev homeomorphism; Schottky theorem
These notes were prepared for International School on Nonlinear Analysis, Function Spaces and Applications 9 in Třešť (Czech Republic), September 11-17, 2010. They give an account of some recent developments in which quasiconformal theory and nonlinear elasticity share common problems of compelling mathematical interest. As this interplay developed homeomorphisms with smallest conformal energy became valid and well acknowledged as generalization of conformal mappings in $\R^n$. The main interest lies on two type of mapping problems: i) the existence of homeomorphisms that minimizes the conformal energy; ii) the existence of harmonic homeomorphisms. Here no boundary conditions are imposed. In presenting these topics I will rely on a few recent joint articles with Tadeusz Iwaniec and Leonid V. Kovalev as well as with Kari Astala, Ngin-Tee Koh and Gaven Martin.