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 Infinite-dimensional linear algebra and solvability of partial differential equations

2021 ◽  
Vol 13 ◽  
Author(s):  
Todor D. Todorov
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Linear Algebra ◽  
Differential Operators ◽  
Test Functions ◽  
Partial Differential ◽  
Infinite Dimensional ◽  
Partial Differential Operators ◽  
Linear Partial Differential Operators ◽  
Regular Operators

  We discuss linear algebra of infinite-dimensional vector spaces in terms of algebraic (Hamel) bases. As an application we prove the surjectivity of a large class of linear partial differential operators with smooth ($\mathcal C^\infty$-coefficients) coefficients, called in the article \emph{regular}, acting on the algebraic dual $\mathcal D^*(\Omega)$ of the space of test-functions $\mathcal D(\Omega)$. The surjectivity of the partial differential operators guarantees solvability of the corresponding partial differential equations within $\mathcal D^*(\Omega)$. We discuss our result in contrast to and comparison with similar results about the restrictions of the regular operators on the space of Schwartz distribution $\mathcal D^\prime(\Omega)$, where these operators are often non-surjective. 

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.

Sign in / Sign up

Export Citation Format

Share Document