Partial Differential Equations · Calculus

Laplace’s Equation and Harmonic Functions

Laplace’s equation describes steady states. Harmonic functions satisfy the mean value property: the value at any point equals the average over any surrounding sphere.

WHAT STUDENTS WILL LEARN

✓Identify harmonic functions as solutions of Δu = 0

✓Apply the mean value property and maximum principle

✓Solve the Dirichlet problem using Poisson’s formula

✓Connect to analytic functions in 2D

BUILDS ON

Classification and Well-PosednessPartial Differential Equations Complex Numbers and Analytic FunctionsComplex Variables

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