Harmonic function (Δu = 0): mean-value property and max principle

A C² function u: Ω ⊂ R^n → R is harmonic iff Δu = ∂²u/∂x₁² + … + ∂²u/∂x_n² = 0. Three equivalent characterisations: (i) C² with Δu = 0; (ii) mean-value property — u(x) = (1/|S_R(x)|) ∫_{∂B_R(x)} u dσ for every ball B_R(x) ⊂ Ω; (iii)…