A Lebesgue-Lusin property for gradients of first-order linear operators
(2022) Cornell University — Vol. np, n° np, p. 7 (2022)
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- Abstract
- We prove that for a first-order homogeneous linear partial differential operator $\mathcal{A}$ and a map $f$ taking values in the essential range of the operator, there exists a function $u$ of special bounded variation satisfying $\mathcal{A}u(x) = f(x)$ almost everywhere. This extends a result of G. Alberti for gradients on $\mathbb{R}^nN$. In particular, for $m < N$, it is shown that every integrable $m$-form field on $\mathbb{R}^nN$ is the absolutely continuous part of the boundary of some locally normal current with finite mass.
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Arroyo Rabasa, A. (2022). A Lebesgue-Lusin property for gradients of first-order linear operators. Cornell University, np(np), 7. https://doi.org/10.48550/arXiv.2209.14062 (Original work published 2022)