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
- [math]\displaystyle{ E[u] = \frac 1 2 \int_\Omega \| \nabla u(x) \|^2 \, dx, }[/math]
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 [math]\displaystyle{ -\Delta u(x) = 0 }[/math] for all [math]\displaystyle{ x \in \Omega }[/math], 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 ⊆ Rn 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.
See also
References
- Lawrence C. Evans (1998). Partial Differential Equations. American Mathematical Society. ISBN 978-0821807729.
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