Green’s function

Some general preliminary considerations

Let (Ω,μ) be a bounded measure space and (Ω) be a linear functionMathworldPlanetmath space of bounded functions defined on Ω, i.e. (Ω)(Ω). We would like to note two types of functionals from the dual spaceMathworldPlanetmathPlanetmath ((Ω))*, which will be used here:

  1. 1.

    Each function g(x)1(Ω) defines a functional φ((Ω))* in the following way:


    Such functional we will call regularPlanetmathPlanetmath functional and function g — its generator.

  2. 2.

    For each xΩ, we will consider a functional δx((Ω))* defined as follows:

    δx(f)=f(x). (1)

    Since generally, we can not speak about values at the point for functions from (L), in the following, we assume some regularity for functions from considered spaces, so that (1) is correctly defined.

Necessary notations and motivation

Let (Ωx,μx),(Ωy,μy) be some bounded measure spaces; (Ωx),𝒢(Ωy) be some linear function spaces. Let A:(Ωx)𝒢(Ωy) be a linear operatorMathworldPlanetmath which has a well-defined inverse A-1:𝒢(Ωy)(Ωx).

Consider an operator equation:

Af=g (2)

where f(Ωx) is unknown and g𝒢(Ωy) is given. We are interested to have an integral representation for solution of (2). For this purpose we write:


Definition of Green’s function

If xΩx the functional (A-1)*δx is regular with generator G(,y)1(Ωy), then G is called Green’s function of operator A and solution of (2) admits the following integral representation:

Title Green’s function
Canonical name GreensFunction
Date of creation 2013-03-22 14:43:36
Last modified on 2013-03-22 14:43:36
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 7
Author PrimeFan (13766)
Entry type Definition
Classification msc 35C15
Related topic PoissonsEquation