A Fundamental Solution for a Nonelliptic Partial Differential Operator

1979 ◽  
Vol 31 (5) ◽  
pp. 1107-1120 ◽  
Author(s):  
Peter C. Greiner
Keyword(s):  
Vector Field ◽  
Differential Operator ◽  
Fundamental Solution ◽  
Half Plane ◽  
Partial Differential Operator ◽  
Holomorphic Vector Field ◽  
Partial Differential ◽  
Upper Half Plane ◽  
Cauchy Riemann ◽  
Image Position

Let(1)and set(2)Here . Z is the “unique” (modulo multiplication by nonzero functions) holomorphic vector-field which is tangent to the boundary of the “degenerate generalized upper half-plane”(3)In our terminology t = Re z1. We note that ℒ is nowhere elliptic. To put it into context, ℒ is of the type □b, i.e. operators like ℒ occur in the study of the boundary Cauchy-Riemann complex. For more information concerning this connection the reader should consult [1] and [2].

A fundamental solution for a nonelliptic partial differential operator, II

2018 ◽  
Vol 16 (03) ◽  
pp. 407-433
Author(s):  
Peter Greiner ◽  
Yutian Li
Keyword(s):  
Vector Field ◽  
Differential Operator ◽  
Fundamental Solution ◽  
Half Plane ◽  
Special Functions ◽  
Partial Differential Operator ◽  
Principal Result ◽  
Tangential Vector ◽  
Tangential Vector Field ◽  
Upper Half Plane

Let [Formula: see text] denote the holomorphic tangential vector field to the generalized upper-half plane [Formula: see text]. In our terminology, [Formula: see text]. Consider the [Formula: see text] operator on the boundary of [Formula: see text], [Formula: see text]; note that [Formula: see text] is nowhere elliptic, but it is subelliptic with step three. The principal result of this paper is the derivation of an explicit fundamental solution [Formula: see text] to [Formula: see text]. Our approach is based on special functions and their properties.

scholarly journals Codomains for the Cauchy-Riemann and Laplace operators inℝ2

2008 ◽  
Vol 6 (1) ◽  
pp. 71-87
Author(s):  
Lloyd Edgar S. Moyo
Keyword(s):  
Differential Operator ◽  
Fundamental Solution ◽  
Laplace Operator ◽  
Constant Coefficient ◽  
Partial Differential Operator ◽  
Linear Partial Differential Operator ◽  
Partial Differential ◽  
Nonzero Constant ◽  
Space Of Distributions ◽  
Cauchy Riemann

A codomain for a nonzero constant-coefficient linear partial differential operatorP(∂)with fundamental solutionEis a space of distributionsTfor which it is possible to define the convolutionE*Tand thus solving the equationP(∂)S=T. We identify codomains for the Cauchy-Riemann operator inℝ2and Laplace operator inℝ2. The convolution is understood in the sense of theS′-convolution.

scholarly journals A Fundamental Solution of N. Zeilon's Operator

2000 ◽  
Vol 86 (2) ◽  
pp. 273 ◽  
Author(s):  
Peter Wagner
Keyword(s):  
Differential Operator ◽  
Fundamental Solution ◽  
Partial Differential Operator ◽  
Elliptic Integrals ◽  
Partial Differential

In this paper, we resume earlier work of N. Zeilon and of J. Fehrman and derive an explicit re- presentation by elliptic integrals of a fundamental solution of the partial differential operator $\partial_1^3+\partial_2^3+\partial_3^3$.

Half-eigenvalues of elliptic operators

2002 ◽  
Vol 132 (6) ◽  
pp. 1439-1451 ◽  
Author(s):  
Bryan P. Rynne
Keyword(s):  
Differential Operator ◽  
Bounded Domain ◽  
Trivial Solution ◽  
Partial Differential Operator ◽  
Elliptic Operators ◽  
Second Order ◽  
Partial Differential ◽  
Carathéodory Function ◽  
Image Position ◽  
Semilinear Problem

Suppose that L is a second-order self-adjoint elliptic partial differential operator on a bounded domain Ω ⊂ Rn, n ≥ 2, and a, b ∈ L∞(Ω). If the equation Lu = au+ − bu− + λu (where λ ∈ R and u±(x) = max{±u(x), 0}) has a non-trivial solution u, then λ is said to be a half-eigenvalue of (L; a, b). In this paper, we obtain some general properties of the half-eigenvalues of (L; a, b) and also show that, generically, the half-eigenvalues are ‘simple’.We also consider the semilinear problem where f : Ω × R → R is a Carathéodory function such that, for a.e. x ∈ Ω, and we relate the solvability properties of this problem to the location of the half-eigenvalues of (L; a, b).

scholarly journals Enclosure Theorems for Eigenvalues of Elliptic Operators

1970 ◽  
Vol 13 (1) ◽  
pp. 1-7 ◽  
Author(s):  
John C. Clements
Keyword(s):  
Differential Operator ◽  
Euclidean Space ◽  
Partial Differential Operator ◽  
Positive Definite ◽  
Dimensional Euclidean Space ◽  
Partial Differential ◽  
Partial Differentiation ◽  
The Matrix ◽  
Image Position ◽  
Uniformly Continuous

Let L be the linear, elliptic, self-adjoint partial differential operator given by where Dj denotes partial differentiation with respect to xj, 1 ≤ j ≤ n, b is a positive, continuous real-valued function of x = (x1,…,xn) in n-dimensional Euclidean space En, the aij are real-valued functions possessing uniformly continuous first partial derivatives in En and the matrix {aij} is everywhere positive definite. A solution u of Lu = 0 is assumed to be of class C1.

Estimates for a beam-like partial differential operator and applications

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Meriem Belahdji ◽  
Setti Ayad ◽  
Mohammed Hichem Mortad
Keyword(s):  
Differential Operator ◽  
A Priori Estimates ◽  
A Priori ◽  
Partial Differential Operator ◽  
Partial Differential ◽  
Priori Estimates

Abstract The aim of this paper is to provide some a priori estimates for a beam-like operator. Some applications and counterexamples are also given.

scholarly journals Hausdorff operators on holomorphic Hardy spaces and applications

2019 ◽  
Vol 150 (3) ◽  
pp. 1095-1112 ◽  
Author(s):  
Ha Duy Hung ◽  
Luong Dang Ky ◽  
Thai Thuan Quang
Keyword(s):  
Half Plane ◽  
Hardy Spaces ◽  
Hausdorff Operator ◽  
Upper Half Plane ◽  
Hausdorff Operators ◽  
Operator Norms ◽  
Image Position

AbstractThe aim of this paper is to characterize the non-negative functions φ defined on (0,∞) for which the Hausdorff operator $${\rm {\cal H}}_\varphi f(z) = \int_0^\infty f \left( {\displaystyle{z \over t}} \right)\displaystyle{{\varphi (t)} \over t}{\rm d}t$$is bounded on the Hardy spaces of the upper half-plane ${\rm {\cal H}}_a^p ({\open C}_ + )$, $p\in [1,\infty ]$. The corresponding operator norms and their applications are also given.

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