Homeomorphic Sobolev Extensions of Parametrizations of Jordan Curves

Explore homeomorphic Sobolev extensions of Jordan curve parametrizations in this 32-minute lecture by Pekka Koskela at the Erwin Schrödinger International Institute for Mathematics and Physics. Delve into the intriguing question of when homeomorphic parametrizations of Jordan curves can be extended to homeomorphisms of the entire plane with Sobolev regularity. Examine the simplified problem of finding homeomorphisms from the unit disk with p-integrable first-order distributional derivatives, given a homeomorphic embedding of the unit circle into the plane. Understand the motivation behind this research, including its relevance to deformations with finite energy. Discover recent findings by Koskela and his collaborators on this mathematical challenge, presented as part of the Workshop on "Between Regularity and Defects: Variational and Geometrical Methods in Materials Science" held at ESI in February 2023.