Dirichlet energy

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In mathematics, the Dirichlet energy is a numerical measure of how "non-constant" 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 Lejeune Dirichlet.

[edit] Definition

Given an open set Ω ⊆ Rⁿ and function u : Ω → R the Dirichlet energy of u is the real number

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

where ∇u : Ω → Rⁿ denotes the gradient vector field of u.

[edit] Properties and applications

Since it is the integral of a non-negative quantity, it is clear that the Dirichlet energy is itself non-negative, i.e. E[u] ≥ 0 for every function u.

Solving Laplace's equation

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

(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.

[edit] References

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