2020/08/23 by Aleksis Koski, Koski, Aleksis, Jani Onninen +1 · 1 citation
Biochemistry, Genetics and Molecular Biology · Engineering · Mathematics · #46E35 #58E20 #Analytic and geometric function theory #Complex Variables (math.CV) #Dermatological and Skeletal Disorders #Elasticity and Material Modeling #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2008.09947
openalex publication_date 2020/08/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the planar unit disk \mathbb D as the reference configuration and a Jordan domain \mathbb Y as the deformed configuration, and study the problem of extending a given boundary homeomorphism φ\colon ∂ \mathbb D → ∂ \mathbb Y as a Sobolev homeomorphism of the complex plane. Investigating such a Sobolev variant of the classical Jordan-Schönflies theorem is motivated by the well-posedness of the related pure displacement variational questions in the theory of Nonlinear Elasticity (NE) and Geometric Function Theory (GFT). Clearly, the necessary condition for the boundary mapping φ to admit a W1,p-Sobolev homeomorphic extension is that it first admits a continuous W1,p-Sobolev extension. For an arbitrary target domain \mathbb Y this, however, is not sufficent.