Green’s function

Some general preliminary considerations

Let $(\mathrm{\Omega },\mu )$ be a bounded measure space and $ℱ(\mathrm{\Omega })$ be a linear function space of bounded functions defined on $\mathrm{\Omega }$, i.e. $ℱ(\mathrm{\Omega })\subset {ℒ}^{\mathrm{\infty }}(\mathrm{\Omega })$. We would like to note two types of functionals from the dual space ${(ℱ(\mathrm{\Omega }))}^{*}$, which will be used here:

1. Each function $g(x)\in {ℒ}^1(\mathrm{\Omega })$ defines a functional $\phi \in {(ℱ(\mathrm{\Omega }))}^{*}$ in the following way:

   $$\phi (f)=\underset{\mathrm{\Omega }}{\int }g(x)f(x)𝑑\mu .$$

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

2. For each $x\in \mathrm{\Omega }$, we will consider a functional ${\delta }_x\in {(ℱ(\mathrm{\Omega }))}^{*}$ defined as follows:

   $${\delta }_x(f)=f(x).$$ (1)

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

Necessary notations and motivation

Let $({\mathrm{\Omega }}_x,{\mu }_x),({\mathrm{\Omega }}_y,{\mu }_y)$ be some bounded measure spaces; $ℱ({\mathrm{\Omega }}_x),𝒢({\mathrm{\Omega }}_y)$ be some linear function spaces. Let $A:ℱ({\mathrm{\Omega }}_x)\to 𝒢({\mathrm{\Omega }}_y)$ be a linear operator which has a well-defined inverse $A^{-1}:𝒢({\mathrm{\Omega }}_y)\to ℱ({\mathrm{\Omega }}_x)$.

Consider an operator equation:

$$Af=g$$ (2)

where $f\in ℱ({\mathrm{\Omega }}_x)$ is unknown and $g\in 𝒢({\mathrm{\Omega }}_y)$ is given. We are interested to have an integral representation for solution of (2). For this purpose we write:

$$f(x)={\delta }_x(f)={\delta }_x(A^{-1}(g))=[{(A^{-1})}^{*}{\delta }_x](g).$$

Definition of Green’s function

If $\forall x\in {\mathrm{\Omega }}_x$ the functional ${(A^{-1})}^{*}{\delta }_x$ is regular with generator $G(\cdot ,y)\in {ℒ}^1({\mathrm{\Omega }}_y)$, then $G$ is called Green’s function of operator $A$ and solution of (2) admits the following integral representation:

$$f(x)=\underset{{\mathrm{\Omega }}_y}{\int }G(x,y)g(y)𝑑{\mu }_y$$

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