Laplacians

The Laplacian, denoted Δ, is a differential operator that maps twice-differentiable functions to a scalar. In Euclidean n-space, Δf equals the sum of second partial derivatives, Δf = ∂²f/∂x₁² + ... + ∂²f/∂x_n², equivalently Δf = div(grad f). In many contexts, −Δ is taken to be a positive semidefinite operator.

In partial differential equations, the Laplacian is central: Laplace's equation Δu = 0 characterizes harmonic functions; Poisson's equation Δu = f appears with source terms. Time-dependent problems include the heat equation ∂u/∂t = κΔu and the wave equation ∂²u/∂t² = c²Δu, each with appropriate boundary and initial conditions.

On graphs, the combinatorial Laplacian L = D − A, where D is the degree matrix and A the adjacency matrix, is symmetric and positive semidefinite. A normalized version, L_norm = I − D⁻¹/²AD⁻¹/², provides a scale-invariant operator. L acts on functions on the vertex set, and its eigenvalues encode connectivity. The kernel consists of constant functions on each connected component.

In differential geometry, the Laplace–Beltrami operator Δ_g generalizes the Laplacian to functions on a Riemannian manifold, defined as div_g(grad_g f). It depends on the metric and reduces to the ordinary Laplacian in Euclidean space, linking analysis with the geometry of the space.

Key properties include self-adjointness and Green’s identities. A function is harmonic when Δf = 0 and harmonic functions minimize Dirichlet energy. Laplacians have broad applications in physics, geometry, and data analysis, including potential theory, image processing, spectral clustering, and diffusion and wave simulations.