Green-kernel expansion in the Dirichlet space
= Green-kernel expansion in the Dirichlet space
{c}
For an <orthonormal basis> $(e_n)$ of $H_0^1(D)$, the zero-Dirichlet <Green function of the Laplacian> is the weak kernel represented by
$$
G_D(x,y)=\sum_{n\geq1}e_n(x)e_n(y).
$$
Consequently, whenever the terms are defined and the right side is finite,
$$
\sum_{n\geq1}\left(\int_De_n\,d\rho\right)^2
=\iint_{D\times D}G_D(x,y)\rho(dx)\rho(dy).
$$