Laplace and Poisson Summary

PDEs · Axiom Academy

SUMMARY Laplace and Poisson Equations Review of elliptic PDEs, harmonic functions, and steady-state solutions Elliptic Classification: PDEs where all second derivatives have the same sign, characteristic of steady-state problems Steady-State Solutions: Time-independent solutions representing equilibrium conditions (∂u/∂t = 0) Laplace Operator: The Laplacian ∇²u measures how a function differs from its average at nearby points Physical Interpretation: Describes diffusion, electrostatics, gravity, fluid flow, and heat distribution at equilibrium Laplace Equation: ∇²u = 0 describes systems with no sources or sinks (homogeneous) Poisson Equation: ∇²u = f describes systems with source term f (non-homogeneous) Relationship: Laplace is the special case of Poisson when f = 0 Coordinate Forms: In 2D: u xx + u yy = f; In 3D: u xx + u yy + u zz = f Harmonic Functions & Properties Definition: Functions u satisfying ∇²u = 0 are called harmonic functions Mean Value Property: u at any point equals its average over any surrounding sphere or circle Smoothness: Harmonic functions are infinitely differentiable in their domain No Local Extrema: Cannot have interior maximum or minimum unless constant Strong Form: A harmonic function on a bounded domain achieves its maximum and minimum on the boundary Weak Form: If ∇²u ≥ 0, then u cannot have an interior maximum Consequence: Solutions are heavily influenced by boundary conditions

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