Necessary and sufficient conditions for propagation of singularities for systems of linear partial differential operators with constant coefficients

1983 ◽  
Vol 8 (2) ◽  
pp. 89-198 ◽  
Author(s):  
O. Liess
Keyword(s):  
Differential Operators ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
Propagation Of Singularities ◽  
Constant Coefficients ◽  
Linear Partial Differential Operators ◽  
Necessary And Sufficient

scholarly journals Generalized Vandermonde determinants for reversing Taylor's formula and application to hypoellipticity

2007 ◽  
Vol 38 (2) ◽  
pp. 183-189
Author(s):  
Giuseppe De Donno
Keyword(s):  
Differential Operators ◽  
Partial Differential ◽  
Taylor’S Formula ◽  
Several Variables ◽  
Partial Differential Operators ◽  
Constant Coefficients ◽  
Linear Partial Differential Operators ◽  
Necessary And Sufficient ◽  
Complex Coefficients ◽  
Taylor's Formula

The problem of the hypoellipticity of the linear partial differential operators with constant coefficients was completely solved by H"{o}r-man-der in [5]. He listed many equivalent algebraic conditions on the polynomial symbol of the operator, each necessary and sufficient for hypoellipticity. In this paper we employ two Mitchell's Theorems (1881) regarding a type of Generalized Vandermonde Determinants, for inverting Taylor's formula of polynomials in several variables with complex coefficients. We obtain then a more direct and easy proof of an equivalence for the mentioned H"{o}r-man-der's hypoellipticity conditions.

scholarly journals Some sufficient conditions for the comparability of two differential operators

Filomat ◽  
2002 ◽  
pp. 57-61 ◽  
Author(s):  
Raid Al-Momani ◽  
Qassem Al-Hassan ◽  
Ali Al-Jarrah ◽  
Ghanim Momani
Keyword(s):  
Partial Differential Equations ◽  
Differential Operators ◽  
Complete Solution ◽  
Sufficient Conditions ◽  
Order Relation ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
The Sixties ◽  
Constant Coefficients ◽  
Linear Partial Differential Operators

The comparison of differential operators is a problem of the theory of partial differential operators with constant coefficients. This problem up to now doesn't have a complete solution. It was formulated in the sixties by Lars Hormander in his monograph "The Analysis of Linear Partial Differential Operators". Many facts of the theory of partial differential equations can be formulated by using the concept of pre-order relation over the set of differential operators, however it is too complicated to check the comparability condition of two differential operators. In this paper we get some sufficient conditions for the comparability of two differential operators.

scholarly journals Hypoelliptic differential operators with generalized constant coefficients

1998 ◽  
Vol 41 (1) ◽  
pp. 47-60 ◽  
Author(s):  
M. Nedeljkov ◽  
S. Pilipović
Keyword(s):  
Small Parameter ◽  
Differential Operators ◽  
Sufficient Conditions ◽  
Generalized Functions ◽  
Necessary And Sufficient Conditions ◽  
Order Estimate ◽  
Colombeau Generalized Functions ◽  
Constant Coefficients ◽  
Necessary And Sufficient

The space of Colombeau generalized functions is used as a frame for the study of hypoellipticity of a family of differential operators whose coefficients depend on a small parameter ε.There are given necessary and sufficient conditions for the hypoellipticity of a family of differential operators with constant coefficients which depend on ε and behave like powers of ε as ε→0. The solutions of such family of equations should also satisfy the power order estimate with respect to ε.

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