Dirichlet's energy

In mathematics, the Dirichlet's 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 function u : Ω → R the Dirichlet's energy of the function u is the real number

$E[u] = \frac1{2} \int_{\Omega} \| \nabla u (x) \|^{2} \, \mathrm{d} V,$

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's energy is itself non-negative, i.e. E[u] ≥ 0 for every function u.

Solving Laplace's equation

$- \Delta u (x) = 0 \text{ for all } x \in \Omega$

(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.

See also

• Dirichlet's principle
• Total variation
• Oscillation
• Harmonic map