In mathematics, the Dirichlet energy is a measure of how variable a function is. More abstractly, it is a quadratic functional on the Sobolev space H1 . The Dirichlet energy is intimately connected to Laplace's equation and is named after the German mathematician Peter Gustav Lejeune Dirichlet.

Definition

Given an open set Ω ⊆ Rn and a function u : Ω → R the Dirichlet energy of the function u is the real number where ∇u : Ω → Rn denotes the gradient vector field of the function u .

Properties and applications

Since it is the integral of a non-negative quantity, the Dirichlet energy is itself non-negative, i.e. E[u] ≥ 0 for every function u . Solving Laplace's equation for all, subject to appropriate boundary conditions, is equivalent to solving the variational problem of finding a function u that satisfies the boundary conditions and has minimal Dirichlet energy. Such a solution is called a harmonic function and such solutions are the topic of study in potential theory. In a more general setting, where Ω ⊆ Rn is replaced by any Riemannian manifold M , and u : Ω → R is replaced by u : M → Φ for another (different) Riemannian manifold Φ , the Dirichlet energy is given by the sigma model. The solutions to the Lagrange equations for the sigma model Lagrangian are those functions u that minimize/maximize the Dirichlet energy. Restricting this general case back to the specific case of u : Ω → R just shows that the Lagrange equations (or, equivalently, the Hamilton–Jacobi equations) provide the basic tools for obtaining extremal solutions.