Dirichlet's energy

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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 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 Ω ⊆ Rn and function u : Ω → 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 u : Ω → Rn denotes the gradient vector field of the function u.

Properties and applications[edit]

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.

See also[edit]

References[edit]

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