 Dirichlet Green function in a quadrant

ID: dirichlet-green-function-in-a-quadrant

For the positive Laplacian convention

Δ G = δ

, the method of images in the first quadrant reflects an interior pole

(a, b)

across each axis with negative sign and across both with positive sign:

G_{Q} (x, y; a, b) = \frac{1}{2 π} [lo g ∣ (x, y) - (a, b) ∣ - lo g ∣ (x, y) - (- a, b) ∣ - lo g ∣ (x, y) - (a, - b) ∣ + lo g ∣ (x, y) - (- a, - b) ∣] .

The Dirichlet Green function vanishes on both axes. Its outward normal derivative on the horizontal axis is

\partial_{n} G_{Q} (t, 0; a, b) = \frac{4 ab t}{π [( t - a ) ^{2} + b ^{2} ] [( t + a ) ^{2} + b ^{2} ]},

and the vertical-axis kernel follows by exchange of coordinates. Green second identity converts these derivatives into the quadrant Poisson integral for boundary data.

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