Finite distortion curves: Continuity, Differentiability and Lusin's (N) property

TL;DR

This paper introduces finite distortion ω-curves, demonstrating their continuity, differentiability, and Lusin's (N) property under certain integrability conditions, with sharpness results and implications of weak monotonicity.

Contribution

It establishes new regularity results for finite distortion ω-curves, including conditions for continuity and differentiability, and explores the sharpness and monotonicity effects.

Findings

Finite distortion ω-curves are continuous under exponential integrability.

Almost everywhere differentiability of these curves is proven.

Lusin's (N) property holds for the curves under certain conditions.

Abstract

We define finite distortion $\omega$-curves and we show that for some forms $\omega$ and when the distortion function is sufficiently exponentially integrable the map is continuous, differentiable almost everywhere and has Lusin's (N) property. This is achieved through some higher integrability results about finite distortion $\omega$-curves. It is also shown that this result is sharp both for continuity and for Lusin's (N) property. We also show that if we assume weak monotonicity for the coordinates of a finite distortion $\omega$-curve we obtain continuity.