Calculus of Variations and Geometric Measure Theory

A. Arroyo-Rabasa

Slicing and fine properties of functions with bounded $\mathcal A$-variation

created by arroyorabasa on 29 Sep 2020



Inserted: 29 sep 2020
Last Updated: 29 sep 2020

Pages: 45
Year: 2020


We study the slicing and fine properties of functions in $\mathrm{BV}^{\mathcal A}$, the space of functions of bounded $\mathcal A$-variation. Here, $\mathcal A$ is a homogeneous linear differential operator with constant coefficients (of arbitrary order). Our main result is the characterization of all $\mathcal A$ satifying the following one-dimensional structure theorem: every $u \in \mathrm{BV}^{\mathcal A}$ can be sliced into one-dimensional $BV$-sections. Moreover, decomposing $\mathcal A u$ into an absolutely continuous part $\mathcal A^a u$, a Cantor part $\mathcal A^c u$ and a jump part $\mathcal A^j u$, each of these measures can be recovered from the corresponding classical $D^a,D^c$ and $D^j$ $BV$-derivatives of its one-dimensional sections. By means of this result, we are able to analyze the set of Lebesgue points as well as the set of jump points where these functions have approximate one-sided limits. Thus, proving a structure and fine properties theorem in $\mathrm{BV}^{\mathcal A}$. Our results extend most of the classical fine properties of $\mathrm{BV}$ (and all of those known for $\mathrm{BD}$) to $\mathrm{BV}^{\mathcal A}$. Applications of our results are discussed for operators that are not covered by the existing theory.

Keywords: fine properties, structure theorem, bounded $\mathcal A$-variation, slicing