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
[edit]Given an open set
where ∇u :
Properties and applications
[edit]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 for all , 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
See also
[edit]- Dirichlet's principle
- Dirichlet eigenvalue – fundamental modes of vibration of an idealized drum with a given shape
- Total variation – Measure of local oscillation behavior
- Bounded mean oscillation – real-valued function whose mean oscillation is bounded
- Harmonic map – smooth map that is a critical point of the Dirichlet energy functional
- Capacity of a set – in Euclidean space, a measure of that set's "size"
References
[edit]- Lawrence C. Evans (1998). Partial Differential Equations. American Mathematical Society. ISBN 978-0821807729.