scholarly journals A duality theorem for solutions of elliptic equations

1990 ◽  
Vol 13 (1) ◽  
pp. 73-85
Author(s):  
Pierre Blanchet
Keyword(s):  
Differential Operator ◽  
Uniform Convergence ◽  
Linear Space ◽  
Elliptic Equations ◽  
Duality Theorem ◽  
Adjoint Equation ◽  
Partial Differential Operator ◽  
Linear Partial Differential Operator ◽  
Elliptic Type ◽  
All Solutions

LetLbe a second order linear partial differential operator of elliptic type on a domainΩofℝmwith coefficients inC∞(Ω). We consider the linear space of all solutions of the equationLu=0onΩwith the topology of uniform convergence on compact subsets and describe the topological dual of this space. It turns out that this dual may be identified with the space of solutions of an adjoint equation “near the boundary” modulo the solutions of this adjoint equation on the entire domain.

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 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).

Faddeev Equations in the Functional Quantum Theory of the Non-Linear Spinorfield

1980 ◽  
Vol 35 (9) ◽  
pp. 964-972
Author(s):  
U. Ramacher
Keyword(s):  
Quantum Theory ◽  
Differential Operator ◽  
Effective Potential ◽  
Partial Differential Operator ◽  
Functional Space ◽  
Baryon Number ◽  
Linear Partial Differential Operator ◽  
Partial Differential ◽  
Faddeev Equations ◽  
Polarization Cloud

Abstract Starting with a linear partial differential operator for a certain system of transition amplitudes associated to a state |a> with baryon number ρ(a) we derive an equation in the ρ(a)-sector of the Functional space which provides the state |a> with an effective potential caused by the polarization cloud. With regard to the needs of n-body scattering we then undertake a cluster decomposition of the effective potential. In particular, we derive the relativistic analoga of Lippmann-Schwinger-and Faddeev equations.

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.

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