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.
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.