Laplacian

The Laplacian, denoted Δ, is a linear differential operator that measures how a function diverges from its average in a neighborhood. For a twice-differentiable scalar field f on n-dimensional Euclidean space, Δf equals the sum of second partial derivatives: Δf = ∑_{i=1}^n ∂^2 f/∂x_i^2. Equivalently, it is the divergence of the gradient, Δf = div(grad f). For a vector field, the Laplacian is applied componentwise.

Generalizing to curved spaces, the Laplace–Beltrami operator extends Δ to Riemannian manifolds, capturing intrinsic geometry. In flat coordinates it reduces to the familiar Δ; its definition does not depend on the chosen coordinates.

Properties and interpretation: Δ is linear and, under suitable boundary conditions, self-adjoint and negative semidefinite; accordingly, −Δ is positive semidefinite. A function satisfying Δu = 0 is harmonic and enjoys the mean value property, which underpins many physical and geometric phenomena. The Laplacian appears in the heat equation, ∂u/∂t = κΔu, describing diffusion, and in Poisson’s equation, Δu = f, modeling potential fields.

Discrete counterpart: on graphs, the graph Laplacian L = D − A (with D the degree matrix and A the adjacency matrix) governs diffusion on networks and underlies spectral clustering and graph signal processing.

Spectral and analytic perspectives: under Fourier transform, Δ corresponds to multiplication by -|ξ|^2, so its eigenfunctions form a basis in appropriate domains; eigenvalues reflect domain geometry and boundary conditions.

Applications span physics, engineering, geometry, and image processing, including edge detection and Laplacian-of-Gaussian filtering.