Absolutely continuous representatives on curves for Sobolev functions

Absolutely continuous representatives on curves for Sobolev functions

We consider a class of Lipschitz vector fields S :Ω →Rn whose values lie in a suitable cone and we show that the trajectories of the system x′ = S(x) admit a parametrization that is invertible and Lipschitz with its inverse. As a consequence, every v in W1,1(Ω) admits a representative that is absolutely continuous on almost every trajectory of x′ = S(x). If S is an arbritrary Lipschitz field th...

On Absolutely Continuous Transformations

Removability of Hölder Graphs for Continuous Sobolev Functions

We characterize the removability of Hölder-α graphs with respect to continuous Sobolev W 1,2 functions. For α > 2/3 these graphs are removable, while for α < 2/3 there exist graphs which are not removable.

Absolutely Continuous Functions with Values in Metric Spaces

(see e.g. [1, Lemma 1.1]), we could without any loss of generality work with Banach spaces only. The main obstacle in dealing with metric spaces (or arbitrary Banach spaces) is the absence of the Radon-Nikodým property and the resulting non-existence of derivatives. Thus, instead of the “usual” derivative, we have to employ the notion of a “metric derivative” (which was introduced by Kirchheim ...

On Essentially Absolutely Continuous Plane Transformations