Laplaceoperator

The Laplace operator, denoted Δ, is the divergence of the gradient, Δ f = div(∇ f). In Euclidean space ℝ^n it is the sum of second partial derivatives: Δ f = ∑_{i=1}^n ∂^2 f/∂x_i^2.

It is linear, elliptic, and invariant under isometries. It is self-adjoint with respect to the L^2 inner product on a domain with appropriate boundary conditions and is a scalar differential operator of second order.

Harmonic functions satisfy Δ f = 0 and have the mean value property; the maximum principle says a non-constant harmonic function attains its maximum and minimum on the boundary of a domain.

It figures prominently in Poisson's equation Δu = f, the heat equation ∂u/∂t = κ Δ u, and the wave equation. It also underlies potential theory and electrostatics.

The fundamental solution (Green's function) in ℝ^n is proportional to |x|^{2-n} for n ≥ 3 and to log|x| in n = 2. Solutions are analyzed via Green's identities and spectral theory on bounded domains.

A discrete analogue exists on graphs, where the Laplacian is defined by Δ f(x) = ∑_{y∼x} (f(y) - f(x)).

In differential geometry, the Laplace–Beltrami operator generalizes to functions on manifolds. The operator is usually denoted Δ or ∆.