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

Part of: Manifolds Miscellaneous topics - Partial differential equations General theory of linear operators

Published online by Cambridge University Press:  06 November 2023

Adolfo Arroyo-Rabasa Open the ORCID record for Adolfo Arroyo-Rabasa [Opens in a new window]
Adolfo Arroyo-Rabasa*
Affiliation:
Université catholique de Louvain, Louvain La Neuve, Belgium (adolforabasa@gmail.com)
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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.

Keywords

bounded variationLusin propertyfirst-ordersecond-orderdifferential operator

MSC classification

Primary: 35R06: Partial differential equations with measure
Secondary: 47A50: Equations and inequalities involving linear operators, with vector unknowns 49Q15: Geometric measure and integration theory, integral and normal currents 49Q20: Variational problems in a geometric measure-theoretic setting

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 155 , Issue 3 , June 2025 , pp. 858 - 868
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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