In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Dirichlet energy is a measure of how ''variable'' a
function is. More abstractly, it is a
quadratic
In mathematics, the term quadratic describes something that pertains to squares, to the operation of squaring, to terms of the second degree, or equations or formulas that involve such terms. ''Quadratus'' is Latin for ''square''.
Mathematics ...
functional
Functional may refer to:
* Movements in architecture:
** Functionalism (architecture)
** Form follows function
* Functional group, combination of atoms within molecules
* Medical conditions without currently visible organic basis:
** Functional sy ...
on the
Sobolev space . The Dirichlet energy is intimately connected to
Laplace's equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as
\nabla^2\! f = 0 or \Delta f = 0,
where \Delta = \nab ...
and is named after the German mathematician
Peter Gustav Lejeune Dirichlet.
Definition
Given an
open set and a function the Dirichlet energy of the function is the
real number
:
where denotes the
gradient vector field of the function .
Properties and applications
Since it is the integral of a non-negative quantity, the Dirichlet energy is itself non-negative, i.e. for every function .
Solving Laplace's equation
for all
, subject to appropriate
boundary conditions
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...
, is equivalent to solving the
variational problem
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
of finding a function 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 is replaced by any
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
, and is replaced by for another (different) Riemannian manifold , the Dirichlet energy is given by the
sigma model. The solutions to the
Lagrange equations for the sigma model
Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
are those functions that minimize/maximize the Dirichlet energy. Restricting this general case back to the specific case of 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
*
Dirichlet eigenvalue
*
Total variation
*
Oscillation
*
Harmonic map
In the mathematical field of differential geometry, a smooth map between Riemannian manifolds is called harmonic if its coordinate representatives satisfy a certain nonlinear partial differential equation. This partial differential equation for a ...
References
*{{cite book , author=Lawrence C. Evans , title=Partial Differential Equations , publisher=American Mathematical Society , year=1998 , isbn=978-0821807729
Calculus of variations
Partial differential equations