# Transitions de phase et équations non locales

25-27 April 2018
Institut de mathématique Simion Stoilow de l'Académie Roumaine
Europe/Bucharest timezone

## Intégrabilité du degré de Brouwer et règle de la chaîne pour les Jacobiens au sens des distributions/Integrability of the Brouwer degree and chain rules for distributional Jacobians

26 Apr 2018, 16:30
1h

### Speaker

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 $s>n/(n+1)$.

### Primary author

Heiner Olbermann (Université de Leipzig, Allemangne)

### Co-author

Peter Gladbach (Université de Leipzig, Allemagne)

### Presentation Materials

 Transparents