Sobolev Homeomorphic Extensions in Geometric Analysis and Nonlinear Elasticity

Explore fundamental problems in Geometric Analysis through this lecture on Sobolev homeomorphic extensions. Delve into the complexities of understanding which properties of boundary maps allow for homeomorphic extensions with specific geometric and analytic properties. Examine the importance of Sobolev homeomorphisms between given shapes in Nonlinear Elasticity and their role in well-defined minimization problems. Review classical results like the Beurling-Ahlfors extension theorem and the Radó-Kneser-Choquet theorem, discussing their limitations in irregular domains and higher dimensions. Discover recent methods developed by the speaker and collaborators to address these challenges. Focus on the Sobolev Jordan-Schönflies problem of extending boundary maps between planar Jordan domains as Sobolev homeomorphisms, and explore its challenging higher-dimensional alternatives. Gain insights into cutting-edge research in this field of mathematics presented at the Workshop on "Between Regularity and Defects: Variational and Geometrical Methods in Materials Science" held at the Erwin Schrödinger International Institute for Mathematics and Physics.