Dirichlet energy

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

$E[u]={\frac {1}{2}}\int _{\Omega }\|\nabla u(x)\|^{2}\,dx,$ 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 $-\Delta u(x)=0$ for all $x\in \Omega$ , 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.