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

2020 ◽  
pp. 1-27
Author(s):  
Silvano Delladio
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Type Theorem ◽  
Type System ◽  
Partial Differential Operator ◽  
Integrable Case ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
Linear Partial Differential Operators ◽  
Lower Dimensional

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.

scholarly journals A Construction of asymptotic solutions and the existence of smooth null-solutions for a class of non-Fuchsian partial differential operators

1997 ◽  
Vol 145 ◽  
pp. 125-142
Author(s):  
Takeshi Mandai
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Partial Differential Operator ◽  
Negative Integer ◽  
Asymptotic Solutions ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
Real Analytic

Consider a partial differential operator(1.1) where K is a non-negative integer and aj,a are real-analytic in a neighborhood of (0, 0)

scholarly journals Newton polygons and gevrey indices for linear partial differential operators

1992 ◽  
Vol 128 ◽  
pp. 15-47 ◽  
Author(s):  
Masatake Miyake ◽  
Yoshiaki Hashimoto
Keyword(s):  
Differential Operator ◽  
Power Series ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Differential Operators ◽  
Newton Polygon ◽  
Convergent Power Series ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
Linear Partial Differential Operators

This paper is a continuation of Miyake [7] by the first named author. We shall study the unique solvability of an integro-differential equation in the category of formal or convergent power series with Gevrey estimate for the coefficients, and our results give some analogue in partial differential equations to Ramis [10, 11] in ordinary differential equations.In the study of analytic ordinary differential equations, the notion of irregularity was first introduced by Malgrange [3] as a difference of indices of a differential operator in the categories of formal power series and convergent power series. After that, Ramis extended his theory to the category of formal or convergent power series with Gevrey estimate for the coefficients. In these studies, Ramis revealed a significant meaning of a Newton polygon associated with a differential operator.

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