Calculus of Variations and Geometric Measure Theory

A. Marchese

Residually many BV homeomorphisms map a null set in a set of full measure

created by marchese on 20 Jul 2017
modified on 08 Nov 2018


Accepted Paper

Inserted: 20 jul 2017
Last Updated: 8 nov 2018

Journal: Proc. Roy. Soc. Edinburgh Sect. A
Year: 2017


Let $Q$ be the open unit square in $\mathbb{R}^2$. We prove that in a natural complete metric space of $BV$ homeomorphisms $f:Q\rightarrow Q$ with $f_{
\partial Q}=Id$, residually many homeomorphisms (in the sense of Baire categories) map a null set in a set of full measure, and vice versa. Moreover we observe that, for $1\leq p<2$, the family of $W^{1,p}$ homemomorphisms satisfying the above property is of first category.

Keywords: Sobolev homeomorphism, Baire categories, piecewise affine homeomorphism