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 $H^{1}$. 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 $Ω ⊆ R^{n}$ 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 $∇u : Ω → R^{n}$ 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.

In a more general setting, where $Ω ⊆ R^{n}$ 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.