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2 results

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
Silvano Delladio
Journal:

Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 151 / Issue 6 / December 2021

Published online by Cambridge University Press:

26 November 2020, pp. 1893-1919

Print publication:

December 2021
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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.

Poincaré Lemma on Quaternion-like Heisenberg Groups
Der-Chen Chang, Nanping Yang, Hsi-Chun Wu
Journal:

Canadian Mathematical Bulletin / Volume 61 / Issue 3 / 01 September 2018

Published online by Cambridge University Press:

20 November 2018, pp. 495-508

Print publication:

01 September 2018
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For smooth functions ${{a}_{1}}\,,\,{{a}_{2}}\,,\,{{a}_{3}}\,,\,{{a}_{4}}\,$ on a quaternion Heisenberg group, we characterize the existence of solutions of the partial differential operator system ${{X}_{1}}f\,=\,{{a}_{1}},\,{{X}_{2}}f=\,{{a}_{2}},\,{{X}_{3}}f\,=\,{{a}_{3}},\,\text{and}\,{{X}_{4}}f\,=\,{{a}_{4}}$. In addition, a formula for the solution function $f$ is deduced, assuming solvability of the system.