Laplace on a Circle

PDEs · Axiom Academy

Solving Laplace's equation on a disk using separation of variables in polar coordinates 1. Laplacian in Polar Coordinates Consider the disk of radius a : D = (x,y) : x² + y² . In polar coordinates (r, θ), the Laplacian transforms from ∂²u/∂x² + ∂²u/∂y² to a form that respects the circular geometry. The radial symmetry of the disk makes polar coordinates ideal. The Laplacian now includes terms involving r and θ separately. We seek solutions of the form u(r,θ) = R(r)Θ(θ) , where the solution factors into a product of single-variable functions. Substituting into the Laplacian equation separates the r and θ dependencies. Dividing by R(r)Θ(θ) and rearranging yields two ordinary differential equations, each involving only one variable. The separation constant connects them. Since θ represents an angle, the solution must be periodic: Θ(θ + 2π) = Θ(θ) . This periodicity constraint determines the allowed values of the separation constant. The periodic boundary conditions force λ = n² for non-negative integers n, giving solutions cos(nθ) and sin(nθ). The radial equation is an Euler equation. For the solution to be well-defined at the center r = 0, we must reject singular solutions that blow up at the origin. Since r⁻ⁿ → ∞ as r → 0, we must set B = 0. This leaves only the regular solution R(r) = rⁿ. 5. The Poisson Integral Formula

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