scholarly journals On a parametrix in some weak sense of a first order linear partial differential operator with two independent variables

1977 ◽  
Vol 29 (1) ◽  
pp. 1-27 ◽  
Author(s):  
Toyohiro AKAMATSU
Keyword(s):  
Differential Operator ◽  
Partial Differential Operator ◽  
Weak Sense ◽  
Linear Partial Differential Operator ◽  
Partial Differential ◽  
Order Linear ◽  
First Order ◽  
Independent Variables ◽  
Two Independent Variables

scholarly journals Note on Hypoellipticity of a First Order Linear Partial Differential Operator

1968 ◽  
Vol 32 ◽  
pp. 323-330
Author(s):  
Yoshio Kato
Keyword(s):  
Analytic Function ◽  
Differential Operator ◽  
Differentiable Function ◽  
Partial Differential Operator ◽  
Linear Partial Differential Operator ◽  
Partial Differential ◽  
First Order ◽  
Open Set ◽  
Euclidian Space ◽  
Infinitely Differentiable Function

Let Ω be a domain in the (n + 1)-dimensional euclidian space Rn+1. A linear partial differential operator P with coefficients in C∞(Ω) (resp. in Cω(Ω)) will be termed hypoelliptic (resp. analytic-hypoelliptic) in Ω if a distribution u on Ω (i.e. u ∈ D′(Ω)) is an infinitely differentiable function (resp. an analytic function) in every open set of Ω where Pu is an infinitely differentiable function (resp. an analytic function).

scholarly journals An integrable system of partial differential equations on the special linear group

2002 ◽  
Vol 44 (1) ◽  
pp. 83-93
Author(s):  
Peter J. Vassiliou
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Group ◽  
Special Linear Group ◽  
Partial Differential ◽  
Special Linear ◽  
First Order ◽  
Independent Variables ◽  
Two Independent Variables

AbstractWe give an intrinsic construction of a coupled nonlinear system consisting of two first-order partial differential equations in two dependent and two independent variables which is determined by a hyperbolic structure on the complex special linear group regarded as a real Lie groupG. Despite the fact that the system is not Darboux semi-integrable at first order, the construction of a family of solutions depending.upon two arbitrary functions, each of one variable, is reduced to a system of ordinary differential equations on the 1-jets. The ordinary differential equations in question are of Lie type and associated withG.

scholarly journals Some Spaces are not the Domain of a Closed Linear Operator in a Banach Space

1980 ◽  
Vol 23 (4) ◽  
pp. 501-503
Author(s):  
Peter Dierolf ◽  
Susanne Dierolf
Keyword(s):  
Banach Space ◽  
Differential Operator ◽  
Linear Operator ◽  
Partial Differential Operator ◽  
Linear Partial Differential Operator ◽  
Closed Linear Operator ◽  
Partial Differential ◽  
Minimal Operator ◽  
Constant Coefficients

Let be a linear partial differential operator with C∞- coefficients. The study of P(∂) as an operator in L2(ℝn) usually starts with the investigation of the minimal operator P0 which is the closure of P(∂) acting on . In the case of constant coefficients it is known that the domain D(P0) of P0 at least contains the space (cf. Schechter [4, p. 58, Lemma 1.2]).

scholarly journals Memoir on the integration of partial differential equations of the second order in three independent variables, when an intermediary integral does not exist in general

1898 ◽  
Vol 62 (379-387) ◽  
pp. 283-285
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
General Feature ◽  
Partial Differential ◽  
First Order ◽  
Independent Variables ◽  
Dependent Variables ◽  
Two Independent Variables

The general feature of most of the methods of integration of any partial differential equation is the construction of an appropriate subsidiary system and the establishment of the proper relations between integrals of this system and the solution of the original equation. Methods, which in this sense may be called complete, are possessed for partial differential equations of the first order in one dependent variable and any number of independent variables; for certain classes of equations of the first order in two independent variables and a number of dependent variables; and for equations of the second (and higher) orders in one dependent and two independent variables.

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