Dirichlet Poisson integral in a quadrant

The conformal map

w = z^{2}

sends

x, y > 0

to the complex upper half-plane. For bounded continuous boundary values

g_{2}

on the horizontal axis and

g_{1}

on the vertical axis with equal corner values, set

f (s) = g_{2} (s)

for

s > 0

and

f (s) = g_{1} (- s)

for

s < 0

. The bounded harmonic function with those Dirichlet boundary conditions is

u (z) = \frac{1}{π} \int_{R} \frac{Im ( z ^{2} ) f ( s )}{∣ s - z ^{2} ∣ ^{2}} d s .

Equivalently, if

z^{2} = a + i c

with

c > 0

, this is

\frac{2 c}{π} \int_{0}^{\infty} r [g_{2} (r) / ((r^{2} - a)^{2} + c^{2}) + g_{1} (r) / ((r^{2} + a)^{2} + c^{2})] d r

. The Poisson kernel is a positive approximate identity, so the prescribed edge values are attained. Boundedness is essential for uniqueness on this unbounded domain.

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Bounded Dirichlet uniqueness in a quadrant

Two bounded harmonic functions continuous to the quadrant edges with the same Dirichlet boundary conditions are equal. Their difference is bounded and zero on both axes. Odd reflection across each axis gives a bounded harmonic extension away from the corner; the removable singularity for a bounded harmonic function fills in that corner. The harmonic Liouville theorem makes that extension constant, and oddness makes the constant zero. This does not exclude unbounded examples such as the zero-boundary harmonic growth in a quadrant.

Zero-boundary harmonic growth in a quadrant

The harmonic function

v (x, y) = x y

vanishes on both axes but is nonzero in the quadrant, and

v_{z} = - i z /2

. More generally

Im z^{2 m}

, for a positive integer

m

, vanishes on both axes and is harmonic. Adding these unbounded functions leaves all edge Dirichlet boundary conditions unchanged. Thus decay of the boundary data does not imply decay of every solution, and an edge-data-only formula cannot determine an arbitrary solution's Wirtinger derivative. A boundedness or suitable growth condition must select the solution.

Holomorphic derivative of a quadrant Dirichlet solution ( $u_{z}$ )

For the bounded Dirichlet Poisson integral in a quadrant, the Wirtinger derivative is

u_{z} (z) = \frac{2 z}{πi} \int_{0}^{\infty} r [\frac{g _{2} ( r )}{( r ^{2} - z ^{2} ) ^{2}} + \frac{g _{1} ( r )}{( r ^{2} + z ^{2} ) ^{2}}] d r .

This follows by differentiating the Schwarz integral formula composed with

z^{2}

, remembering that for real boundary data

u_{z}

is half the derivative of a holomorphic function whose real part is

u

. For complex boundary data, apply the construction to the real and imaginary parts separately and use linearity. If the data and their derivatives decay sufficiently, integration by parts gives

u_{z} (z) = \frac{z}{πi} [\int_{0}^{\infty} \frac{g _{2}^{'} ( r )}{r ^{2} - z ^{2}} d r + \int_{0}^{\infty} \frac{g _{1}^{'} ( r )}{r ^{2} + z ^{2}} d r] .

The endpoint terms cancel exactly when

g_{1} (0) = g_{2} (0)

. Interior denominators do not vanish because

Im z^{2} > 0

.

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