Jani Onninen
Syracuse University
Title
Sobolev Homeomorphic Extensions
Abstract
The classical Jordan–Schönflies theorem states that every topological embedding of the unit circle extends to a homeomorphism of the unit disk. In this talk, I will discuss its Sobolev counterpart. Foundational results such as the Beurling–Ahlfors extension and the Radó–Kneser–Choquet theorem provide key building blocks in the Sobolev setting, but their applicability breaks down for irregular target domains. I will present a complete characterization of circle embeddings that admit Sobolev homeomorphic extensions to the disk, thereby establishing a Sobolev analogue of the Jordan–Schönflies theorem. This result is pivotal for the well-posedness of variational problems in Nonlinear Elasticity and Geometric Function Theory. This talk is based on joint work with Aleksis Koski, and Haiqing Xu.
