Dirichlet's energy

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 $H 1$ . 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 $Ω \subseteq R n$ and function $u : Ω \to R$ the Dirichlet's energy of the function $u$ is the real number

E[u] = \frac1{2} \int_{\Omega} \| \nabla u (x) \|^{2} \, \mathrm{d} V,

where $\nabla u : Ω \to R n$ 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

- \Delta u (x) = 0 \text{ for all } 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.

References

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

Dirichlet's energy

Definition

Properties and applications

See also

References