Some general preliminary considerations

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

Each function $g(x)\in\EuScript{L}^1(\Omega)$ defines a functional $\varphi\in(\EuScript{F}(\Omega))^*$ in the following way:
$\displaystyle \varphi(f)=\int\limits_{\Omega} g(x)\,f(x)\,d\mu.$
Such functional we will call regular functional and function -- its generator.
For each $x\in\Omega$ , we will consider a functional $\delta_x\in(\EuScript{F}(\Omega))^*$ defined as follows:

$\displaystyle \delta_x(f)=f(x).$ (1)

Since generally, we can not speak about values at the point for functions from $\EuScript(L)^\infty$ , 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; $\EuScript{F}(\Omega_x),\EuScript{G}(\Omega_y)$ be some linear function spaces. Let $A:\EuScript{F}(\Omega_x)\rightarrow\EuScript{G}(\Omega_y)$ be a linear operator which has a well-defined inverse $A^{-1}:\EuScript{G}(\Omega_y)\rightarrow\EuScript{F}(\Omega_x)$ .

Consider an operator equation:

$\displaystyle Af=g$

(2)

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

$\displaystyle f(x)=\delta_x(f)=\delta_x(A^{-1}(g))=[\, (A^{-1})^*\delta_x \,](g).$

Definition of Green's function

If $\forall x\in\Omega_x$ the functional $(A^{-1})^*\delta_x$ is regular with generator $G(\cdot,y)\in\EuScript{L}^1(\Omega_y)$ , then

is called Green's function of operator

and solution of (2) admits the following integral representation:

$\displaystyle f(x)=\int\limits_{\Omega_y}G(x,y)\,g(y)\,d\mu_y$

"Green's function" is owned by PrimeFan. [ owner history (6) ]

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