Dirichlet energy
Template:Short description 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 Template:Math. 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 Template:Math and a function Template:Math the Dirichlet energy of the function Template:Math is the real number
where Template:Math denotes the gradient vector field of the function Template:Math.
Properties and applications
Since it is the integral of a non-negative quantity, the Dirichlet energy is itself non-negative, i.e. Template:Math for every function Template:Math.
Solving Laplace's equation for all , subject to appropriate boundary conditions, is equivalent to solving the variational problem of finding a function Template:Math 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 Template:Math is replaced by any Riemannian manifold Template:Math, and Template:Math is replaced by Template:Math for another (different) Riemannian manifold Template:Math, the Dirichlet energy is given by the sigma model. The solutions to the Lagrange equations for the sigma model Lagrangian are those functions Template:Math that minimize/maximize the Dirichlet energy. Restricting this general case back to the specific case of Template:Math just shows that the Lagrange equations (or, equivalently, the Hamilton–Jacobi equations) provide the basic tools for obtaining extremal solutions.
See also
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