scholarly journals Characterization of the linear partial differential operators with constant coefficients that admit a continuous linear right inverse

1990 ◽  
Vol 40 (3) ◽  
pp. 619-655 ◽  
Author(s):  
B. A. Taylor ◽  
R. Meise ◽  
Dietmar Vogt
Keyword(s):  
Differential Operators ◽  
Continuous Linear ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
Constant Coefficients ◽  
Linear Partial Differential Operators

Characterization of Eigenfunctions by Boundedness Conditions

1992 ◽  
Vol 35 (2) ◽  
pp. 204-213 ◽  
Author(s):  
Ralph Howard ◽  
Margaret Reese
Keyword(s):  
Growth Condition ◽  
Constant Coefficient ◽  
Polynomial Growth ◽  
Differential Operators ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
Linear Partial Differential Operators ◽  
Sequence Of Functions

AbstractSuppose is a sequence of functions on ℝn with Δfk = fk+1 (where Δ is the Laplacian) that satisfies the growth condition: |fk(x)| ≤ Mk{1 + |x|)a where a ≥ 0 and the constants have sublinear growth Then Δf0 = —f0- This characterizes eigenfunctions f of Δ with polynomial growth in terms of the size of the powers Δkf, —∞ < k < ∞. It also generalizes results of Roe (where a = 0, Mk = M, and n = 1 ) and Strichartz (where a = 0, Mk = M for n). The analogue holds for formally self-adjoint constant coefficient linear partial differential operators on ℝn.

Real roots of polynomials and right inverses for partial differential operators in the space of tempered distributions

1990 ◽  
Vol 114 (3-4) ◽  
pp. 169-179 ◽  
Author(s):  
Michael Langenbruch
Keyword(s):  
Differential Operators ◽  
Partial Differential Operator ◽  
Continuous Linear ◽  
Real Polynomial ◽  
Partial Differential ◽  
Tempered Distributions ◽  
Real Roots ◽  
Partial Differential Operators ◽  
Constant Coefficients ◽  
Roots Of Polynomials

SynopsisLet P(D) be a partial differential operator with constant coefficients. If P(D) has a continuous linear right inverse in the space of tempered distributions, then P is the product of a polynomial without real roots and a real polynomial admitting a right inverse. If the polynomial P is real and irreducible, then P(D) admits a right inverse in the tempered distributions if and only if P(×) has the property of zeros of R. Thorn.

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.

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