∇²u = 0. Elliptic PDE for equilibrium/steady-state fields: electrostatic potential in vacuum, incompressible flow, harmonic functions. Solutions are uniquely determined by boundary data.
Laplace equation
Related concepts
- Partial differential equation (PDE)
- Poisson equation
- Helmholtz equation
- Schauder estimates
- De Giorgi-Nash-Moser
- Ordinary differential equation (ODE)
- Maximum principle (elliptic)
- Neumann series: (I − λK)⁻¹ = Σ λⁿ Kⁿ for |λ|·‖K‖ < 1
- Harmonic function (Δu = 0): mean-value property and max principle
- u = x² - y² harmonic: Δu ≡ 0, mean over circle ≡ 0 = u(0)
- Poisson integral normalisation: ∫₀^{2π} P_{1/2}(θ) dθ/(2π) ≡ 1
- Upper-half-plane Green's function G(i, 2i) = log(3)/(2π)