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

${\displaystyle E[u]={\frac {1}{2}}\int _{\Omega }\|\nabla u(x)\|^{2}\,dx,}$

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

• Dirichlet's principle – Concept in potential theory
• Dirichlet eigenvalue – fundamental modes of vibration of an idealized drum with a given shapePages displaying wikidata descriptions as a fallback
• Total variation – Measure of local oscillation behavior
• Bounded mean oscillation – real-valued function whose mean oscillation is boundedPages displaying wikidata descriptions as a fallback
• Harmonic map – smooth map that is a critical point of the Dirichlet energy functionalPages displaying wikidata descriptions as a fallback
• Capacity of a set – In Euclidean space, a measure of that set's "size"

References

• Lawrence C. Evans (1998). Partial Differential Equations. American Mathematical Society. ISBN 978-0821807729.