 Green-kernel expansion in the Dirichlet space

ID: green-kernel-expansion-in-the-dirichlet-space

(e_{n})

H_{0}^{1} (D)

, the zero-Dirichlet Green function of the Laplacian is the weak kernel represented by

G_{D} (x, y) = \sum_{n \geq 1} e_{n} (x) e_{n} (y) .

Consequently, whenever the terms are defined and the right side is finite,

\sum_{n \geq 1} (\int_{D} e_{n} d ρ)^{2} = \iint_{D \times D} G_{D} (x, y) ρ (d x) ρ (d y) .

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