Calculus of Variations and Geometric Measure Theory

Lecture Series "Slicing and fine properties of $BV^{\mathcal A}$ functions"

created by arroyorabasa on 01 Apr 2021
modified on 07 Apr 2021

7 apr 2021 - 21 apr 2021   [open in google calendar]

This will be a series of three lectures via Zoom (link and pass below)

There was a small error in the starting time

07.04 9:30-11:30 CET

14.04 9:30-11:30 CET

21.04 9:30-11:30 CET


Let $\Omega \subset \mathbb R^n$ be an open domain. We study the slicing and fine properties of functions of anisotropic BV-spaces. Namely, for the space \[ BV^{\mathcal A}(\Omega) = \{u \in L^1(\Omega;V) : \mathcal A u \in \mathcal M(\Omega;W)\}, \] of functions with bounded $\mathcal A$-variation. Here, $V$ and $W$ are finite dimensional euclidean spaces which, up to a linear isomorphism may be thought of as $\mathbb R^N$ and $\mathbb R^M$. In order to keep the exposition as simple as possible, I will restrict to the slicing and fine properties of the spaces where $\mathcal A$ is a constant coefficient first-order homogeneous linear differential operator of the form \[ \mathcal A = \sum_{j = 1}^n A_j \partial_j\,, \qquad A_j \in \mathrm{Lin}(V;W). \] The purpose of this work is to give a comprehensive determination of the structural and fine properties of functions in $BV^{\mathcal A}(\Omega)$, very much in the fashion of what is known for $BV(\Omega;\mathbb R^N)$. Our main result is a characterization of all operators $\mathcal A$ (in the form of a closed algebraic property) satisfying the following one-dimensional structure theorem: every $u \in {BV}^{\mathcal A}$ 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 ${BV}^{\mathcal A}$. Our results extend most of the classical fine properties of ${BV}$ (and all of those known for $\mathrm{BD}$). In particular, we establish a slicing theory and fine properties for $\mathscr {BV}^k$, $BD^k$ and a whole class of $\mathrm{BV}^{\mathcal A}$-spaces that is not covered by the existing theory.

Organizers: Daniel Spector.

Speakers: Adolfo Arroyo-Rabasa.