A note on Alberti’s Luzin-type theorem for gradients

Li, Siran

doi:10.1007/s11587-020-00485-w

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2022

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“…Then, by the same arguments as in Section 5 of [10] with Theorem 4.1 in place of [10,Theorem 4.1], we obtain the following well known result (cf. [2,14,16]).…”

Section: Alberti's Theoremmentioning

confidence: 99%

Some Results About the Structure of Primitivity Domains for Linear Partial Differential Operators with Constant Coefficients

Delladio

2022

Mediterr. J. Math.

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Let G(D) be a linear partial differential operator on $${\mathbb {R}}^n$$ R n , with constant coefficients. Moreover let $$\Omega \subset {\mathbb {R}}^n$$ Ω ⊂ R n be open and $$F\in L^1_{\text {loc}} (\Omega , {\mathbb {C}}^N)$$ F ∈ L loc 1 ( Ω , C N ) . Then any set of the form $$\begin{aligned} A_{f,F}:= \{ x\in \Omega \, \vert \, (G(D)f)(x)=F(x)\}, \text { with }f\in W^{g,1}_{\text {loc}}(\Omega , {\mathbb {C}}^k) \end{aligned}$$ A f , F : = { x ∈ Ω | ( G ( D ) f ) ( x ) = F ( x ) } , with f ∈ W loc g , 1 ( Ω , C k ) is said to be a G-primitivity domain of F. We provide some results about the structure of G-primitivity domains of F at the points of the (suitably defined) G-nonintegrability set of F. A Lusin type theorem for G(D) is also provided. Finally, we give applications to the Maxwell type system and to the multivariate Cauchy-Riemann system.

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“…Then, by the same arguments as in Section 5 of [10] with Theorem 4.1 in place of [10,Theorem 4.1], we obtain the following well known result (cf. [2,14,16]).…”

Section: Alberti's Theoremmentioning

confidence: 99%

Some Results About the Structure of Primitivity Domains for Linear Partial Differential Operators with Constant Coefficients

Delladio

2022

Mediterr. J. Math.

View full text Add to dashboard Cite

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A note on Alberti’s Luzin-type theorem for gradients

Abstract: We give a "soft" proof of Alberti's Luzin-type theorem in [1] (G. Alberti, A Lusintype theorem for gradients, J. Funct. Anal. 100 (1991)), using elementary geometric measure theory and topology. Applications to the C 2 -rectifiability problem are also discussed.

Cited by 1 publication

References 14 publications

Some Results About the Structure of Primitivity Domains for Linear Partial Differential Operators with Constant Coefficients

Some Results About the Structure of Primitivity Domains for Linear Partial Differential Operators with Constant Coefficients

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