Quasiconvexity conditions when minimizing over homeomorphisms in the plane

In this talk we characterize necessary and sufficient conditions on the stored energy density in order to assure weak* lower semicontinuity on the set of bi-Lipschitz functions in the plane. This problem is motivated by variational problems in nonlinear elasticity where the orientation preservation and injectivity of the admissible deformations are key requirements. Generally speaking, the main difficulty in finding such conditions is that the set of bi-Lipschitz functions is non-convex. Thus, standard cut-off techniques that modify the generating sequence to have the same boundary conditions as the limit generally fail; however, the standard proofs of in calculus of variations rely on such methods. We obtain this cut-off by following a strategy inspired by Daneri&Pratelli, i.e. we modify the generating sequence (on a set of gradually vanishing measure near the boundary) first on a one dimensional grid and then rely on bi-Lipschitz extension theorems. We also present method of modifying the sequence on the grid that could be extended to more general classes of mappings. This is joint work with Martin Kruzik (Prague) and Malte Kampschulte (Aachen).