Geometric Function theory, Quasiconvexity and lower Semicontinuity

Abstract: A central issue in the vectorial Calculus of Variations is to characterize sequentially weakly lower semicontinuous functionals, which would allow to use the direct method of the Calculus of Variations. For scalar problems the relevant condition is convexity. For vectorial problems, the abstract notion of quasiconvexity is the relevant case but it is not easy to work with. I will explain how ideas from geometric function theory, such as holomorphic motions or Stoilow factorization, have indeed allowed to make substancial progress in this set of problems. This is joint research with K.Astala (Helsinki), A.Guerra (ETH), J.Kristensen (Oxford) and A.Koski (Helsinki).