In the branch of mathematics known as potential theory, a Dirichlet form is a generalization of the Laplacian that can be defined on any measure space, without the need for mentioning partial derivatives. This allows mathematicians to study the Laplace equation and heat equation on spaces that are not manifolds: for example, fractals. To accomplish this generalization, one focuses not on the Laplacian itself but on the quantity
that is minimized when the Laplacian vanishes.
Technically, a Dirichlet form is a Markovian closed symmetric form on an L2-space. Such objects are studied in abstract potential theory, based on the classical Dirichlet's principle. The theory of Dirichlet forms originated in the work of Beurling and Deny (1958, 1959) on Dirichlet spaces.
A Dirichlet form on a measure space
such that
1) The domain
2)
3)
4) The set
5) For any
In other words, a Dirichlet form is nothing but a positive symmetric bilinear form defined on a dense subset of
The best known Dirichlet form is the Dirichlet energy of functions on
which gives rise to the space
where
If the kernel
where