Lower semicontinuity, Stoilow factorization and principal maps

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Abstract
We consider a refinement of the usual quasiconvexity condition of Morrey in two dimensions that allows us to prove lower semicontinuity and existence of minimizers for a class of functionals which are unbounded as the determinant vanishes and are non-polyconvex in general. This notion, that we call principal quasiconvexity, arose from the planar theory of quasiconformal mappings and mappings of finite distortion. We compare it with other quasiconvexity conditions that have appeared in the literature and provide a number of concrete examples of principally quasiconvex functionals that are not polyconvex. We also describe local conditions which combined with quasiconvexity yield principal quasiconvexity. The Stoilow factorization, that in the context of maps of integrable distortion was developed by Iwaniec and Šverák, plays a prominent role in our approach.

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