Poisson Kernel Disk Extension

The Poisson kernel solves the Dirichlet problem for the unit disk. Given a continuous function on the boundary circle, it provides the unique harmonic extension into the interior. The kernel weights boundary values using a geometric formulation, acting as a continuous analogue of a weighted average where closer boundary points receive exponentially greater influence.

Worked Example

Consider extending the boundary function f(φ) = cos(nφ) to the interior point at radius r and angle θ. For n = 2, radius r = 0.5, and angle θ = 30° (which is π/6 radians):

1. Boundary Function: f(φ) = cos(2φ)
2. Evaluation Angle: θ = 30° = π/6 rad
3. Boundary value at θ:
   f(π/6) = cos(2 * π/6) = cos(π/3) = 0.5
4. Harmonic Extension Formula: u(r, θ) = r² cos(2θ)
5. Interior Evaluation:
   u(0.5, π/6) = (0.5)² * cos(2 * π/6)
   u(0.5, π/6) = 0.25 * 0.5 = 0.125
6. Kernel Peak:
   Peak = (1 + r) / (1 - r) = (1 + 0.5) / (1 - 0.5) = 1.5 / 0.5 = 3.0

Experiment

Open the Poisson Kernel Disk Extension explorer to visualize the weighting and extension:

Set Mode n to 2 and Radius fraction to 0.5, keeping Angle degrees at 30. Observe the harmonic value of 0.125, confirming our worked example.
Change the Radius fraction to 0. Notice the kernel peak becomes 1.0, making the weighting uniform, and the harmonic value drops to 0 (unless Mode n=0, where it is 1).
Set the Radius fraction to 1. Observe that the kernel peak becomes "Unavailable" as it approaches the continuous boundary limit, acting like a Dirac delta point mass.
Modify Mode n to 0. The boundary function becomes a constant 1, and the harmonic extension uniformly equals 1 regardless of radius.

Limitations

This explorer demonstrates the theoretical Poisson kernel specifically for harmonic polynomial modes f(φ) = cos(nφ). It uses the analytic solution u(r,θ) = rⁿcos(nθ) to evaluate the interior precisely.

It does not implement a general numerical integration solver for arbitrary piecewise boundary functions.
The visualization caps the rendered kernel curve for visual clarity when r approaches 1, even though the theoretical peak goes to infinity.
Boundary limits (r = 1) evaluate the function directly rather than taking a mathematical division by zero.

References

Axler
Harmonic Function Theory
S. Axler, P. Bourdon, W. Ramey. Comprehensive treatment of the Poisson integral and harmonic extensions.

Poisson Kernel Disk Extension