A Lebesgue-Lusin property for linear operators of first and second order

TL;DR

This paper establishes a Lebesgue-Lusin property for certain linear PDE operators of order up to two, showing that integrable maps in their range can be realized as derivatives of functions with special bounded variation.

Contribution

It extends Alberti's result on gradients to a broader class of first and second order linear PDE operators, linking integrable maps to functions of special bounded variation.

Findings

Existence of solutions of special bounded variation for given PDE data

Extension of Alberti's gradient result to higher order operators

Every integrable m-vector field is the boundary of a normal (m+1)-current

Abstract

We prove that for a homogeneous linear partial differential operator $\mathcal A$ of order $k \le 2$ and an integrable map $f$ taking values in the essential range of that operator, there exists a function $u$ of special bounded variation satisfying \[ \mathcal A u(x)= f(x) \qquad \text{almost everywhere}. \] This extends a result of G. Alberti for gradients on $\mathbf R^N$. In particular, for $0 \le m < N$, it is shown that every integrable $m$-vector field is the absolutely continuous part of the boundary of a normal $(m+1)$-current.