Mathematical theory of harmonic functions (Δu = 0), their generalisations, and the associated boundary-value problems. Laplace 1782 introduced the Laplace equation and its fundamental solution (Newtonian potential) in celestial mechanics;…
potential-theory
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;…
Poisson kernel P_r(θ) reconstructs harmonic u from ∂D data
The Poisson kernel of the unit disk P_r(θ) = (1−r²)/(1 − 2r cos θ + r²) solves the Dirichlet problem for Δu = 0 on D with continuous…
Newtonian potential U_ρ = ∫ Φ(x-y)ρ(y)dy solves ΔU = ρ
Fundamental solution Φ of the Laplace operator: Φ(x) = −(1/(2π)) · log|x| in R², Φ(x) = 1/((n-2)·ω_n · |x|^{n-2}) in R^n for n ≥ 3 (where…
u = x² - y² harmonic: Δu ≡ 0, mean over circle ≡ 0 = u(0)
Exact sympy verification of the mean-value property on the harmonic polynomial u(x, y) = x² − y² (the real part of z² = (x+iy)²). Step 1 —…
Poisson integral normalisation: ∫₀^{2π} P_{1/2}(θ) dθ/(2π) ≡ 1
Sympy-verified symbolic integral pin: for r = 1/2 the normalisation identity (1/(2π)) ∫_0^{2π} P_r(θ) dθ = 1 for the unit-disk Poisson…
Upper-half-plane Green's function G(i, 2i) = log(3)/(2π)
Green's function of the upper half-plane H = {Im(z) > 0} for the Dirichlet Laplacian: G(z, w) = (1/(2π)) · log |z − w̄|/|z − w| —…
Harmonic function & mean-value property
Δu = 0 on Ω. Mean-value property: u(x) = (1/V_r) ∫_{B_r(x)} u(y) dy for any ball. Implies maximum-principle. Conformally-invariant in…
Dirichlet problem
Δu = 0 in Ω, u = g on ∂Ω. Existence/uniqueness for sufficiently regular Ω. Perron's method via subharmonic envelope. Modern approach:…
Poisson integral formula
u(z) = ∫ P_r(θ-φ) g(e^{iφ}) dφ/(2π) gives harmonic extension to disk from boundary g. Poisson kernel P_r(θ) = (1-r²)/(1-2r cos θ + r²). …
Subharmonic functions & Perron method
u subharmonic if Δu ≥ 0. Maximum-principle: u attains max on boundary. Perron envelope = sup over subharmonics ≤ g on boundary solves…
Capacity (Newtonian potential)
Capacity of compact K ⊂ ℝ^d defined via energy minimisation over probability measures on K. Wiener criterion: regular boundary point iff…
Riesz decomposition (superharmonic)
Every non-negative superharmonic function u on Ω decomposes as Newtonian-potential of Riesz-measure + harmonic remainder: u = ∫G(x,y) dμ(y)…
Dirichlet problem (Laplace equation)
Dirichlet 1850s + Perron 1923: Laplace equation Delta u = 0 on bounded domain with prescribed boundary values; classical existence via…
Green's function (fundamental solution)
Green 1828: G(x,y) satisfies -Delta G = delta(x-y) with vanishing boundary; convolution G * f gives Poisson-equation solution; basis of…
Newtonian potential (volume integral)
Poisson 1813: u(x) = (1/(4pi)) integral rho(y)/|x-y| dy gives potential from mass-distribution; -Delta u = rho; basis of gravitational +…
Mean-value property (harmonic functions)
Gauss 1840: harmonic function u satisfies u(x_0) = (1/|B|) integral_{B} u; equivalent to harmonicity; basis of maximum-principle and…
Riesz potential (fractional integral)
M Riesz 1938: I_alpha f(x) = c_alpha integral f(y)/|x-y|^(n-alpha) dy; bounded operator L^p -> L^q with 1/q = 1/p - alpha/n…
Capacity (electrostatic + Newtonian)
Wiener 1924 / Brelot-Choquet 1955: capacity C(K) = inf |grad u|^2 over admissible u; vanishes iff K is polar; quantitative regularity at…
Dirichlet (1850)
P Dirichlet 1850 + Riemann 1851 Dirichlet-energy minimization; modern variational-PDE + image-analysis + Ginzburg-Landau.
Perron method (1923)
O Perron 1923 sub-harmonic / super-harmonic Perron-construction; modern PDE-existence Caffarelli + viscosity-solutions.
Wiener criterion (1924)
N Wiener 1924 boundary-regularity capacity-criterion; modern p-Laplacian-regularity + nonlinear-PDE Bjorn-Bjorn.
Kellogg (1929)
O Kellogg 1929 'Foundations of Potential Theory'; modern Newtonian-potential layer-potential boundary-integral methods.
Newton 1/r potential (1687)
I Newton 1687 1/r gravitational-potential; modern multipole-expansion + Laplace-1773 + boundary-integral-Stokes 1854.
Gauss divergence (1813)
C Gauss 1813 Ostrogradsky-divergence-theorem; modern de-Rham + Stokes' theorem + boundary-integral electrostatics.