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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 :E = \frac 1 2 \int_\Omega \, \nabla u(x) \, ^2 \, dx, 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 -\Delta u(x) = 0 for all x \in \Omega, 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