Orateur
Heiner Olbermann
(Université de Leipzig, Allemagne)
Description
We reconsider the proof of uniqueness of isometric immersions of two-dimensional spheres with positive Gauss curvature, with derivatives in a certain Hölder class. We observe that an understanding of the integrability properties of the Brouwer degree is crucial to extend the range of validity for the uniqueness statement. We take this as a motivation to state and prove a theorem about the integrability of the Brouwer degree with irregular arguments. Furthermore, we show how these questions are linked to the validity of the chain rule for distributional Jacobian determinants $[Ju]$ of maps $u:\Omega\to\mathbf{R}^n$ in certain fractional Sobolev spaces. We prove the so-called weak chain rule for $u\in W^{s,n}(\Omega,\mathbf{R}^n)$, where $\Omega\subset \mathbf{R}^n$ and $s>(n-1)/n$, and the so-called strong chain rule for $u\in W^{s,n+1}(\Omega,\mathbf{R}^n)$ where $s>n/(n+1)$.
Auteur principal
Heiner Olbermann
(Université de Leipzig, Allemangne)
Co-auteur
Peter Gladbach
(Université de Leipzig, Allemagne)
