# Green’s function

## Some general preliminary considerations

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

1. 1.

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

 $\varphi(f)=\int\limits_{\Omega}g(x)\,f(x)\,d\mu.$

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

2. 2.

For each $x\in\Omega$, we will consider a functional $\delta_{x}\in(\mathcal{F}(\Omega))^{*}$ defined as follows:

 $\delta_{x}(f)=f(x).$ (1)

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

## Necessary notations and motivation

Let $(\Omega_{x},\mu_{x}),\,(\Omega_{y},\mu_{y})$ be some bounded measure spaces; $\mathcal{F}(\Omega_{x}),\mathcal{G}(\Omega_{y})$ be some linear function spaces. Let $A:\mathcal{F}(\Omega_{x})\rightarrow\mathcal{G}(\Omega_{y})$ be a linear operator which has a well-defined inverse $A^{-1}:\mathcal{G}(\Omega_{y})\rightarrow\mathcal{F}(\Omega_{x})$.

Consider an operator equation:

 $Af=g$ (2)

where $f\in\mathcal{F}(\Omega_{x})$ is unknown and $g\in\mathcal{G}(\Omega_{y})$ 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 $\forall x\in\Omega_{x}$ the functional $(A^{-1})^{*}\delta_{x}$ is regular with generator $G(\cdot,y)\in\mathcal{L}^{1}(\Omega_{y})$, then $G$ is called Green’s function of operator $A$ and solution of (2) admits the following integral representation:

 $f(x)=\int\limits_{\Omega_{y}}G(x,y)\,g(y)\,d\mu_{y}$
Title Green’s function GreensFunction 2013-03-22 14:43:36 2013-03-22 14:43:36 PrimeFan (13766) PrimeFan (13766) 7 PrimeFan (13766) Definition msc 35C15 PoissonsEquation