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The identity G(D)f = F for a linear partial differential operator G(D). Lusin type and structure results in the non-integrable case

Part of: Partial differential equations Classical measure theory

Published online by Cambridge University Press:  26 November 2020

Silvano Delladio
Silvano Delladio*
Affiliation:
Dipartimento di Matematica, University of Trento, via Sommarive 14, Povo, I-38123Trento, Italy (silvano.delladio@unitn.it)
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Abstract

We prove a Lusin type theorem for a certain class of linear partial differential operators G(D), reducing to [1, Theorem 1] when G(D) is the gradient. Moreover, we describe the structure of the set {G(D)f = F}, under assumptions of non-integrability on F, in terms of lower dimensional rectifiability and superdensity. Applications to Maxwell type system and to multivariable Cauchy–Riemann system are provided.

Keywords

Partial differential operatorsintegrability conditionpoints of Lebesgue densitysuperdensityfine properties of sets

MSC classification

Secondary: 35A: General topics 28A75: Length, area, volume, other geometric measure theory
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 6 , December 2021 , pp. 1893 - 1919
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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