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.
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Definition
Given an open set Ω ⊆ Rn and function u : Ω → R the Dirichlet's 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's energy is itself non-negative, i.e. E[u] ≥ 0 for every function u.
Solving Laplace's equation
(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.