In the branch of mathematics known as potential theory, a Dirichlet form is a generalization of the Laplacian that can be defined on every 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. The benefit on these spaces is that one can do this without needing a gradient operator, and in particular, one can even weakly define a "Laplacian" in this manner if starting with the Dirichlet form. The classical Dirichlet form on is given by: where one often discusses which is often referred to as the "energy" of the function. Functions that minimize the energy given certain boundary conditions are called harmonic, and the associated Laplacian will be zero on the interior, as expected. As an alternative example, the standard graph Dirichlet form is given by: where means they are connected by an edge. Let a subset of the vertex set be chosen, and call it the boundary of the graph. Assign a Dirichlet boundary condition. One can find a function that minimizes the graph energy, and it will be harmonic. In particular, it will satisfy the averaging property, which is embodied by the graph Laplacian, that is, if is a graph harmonic then which can of course be rearranged to showing the averaging property. Technically, a Dirichlet form is a Markovianclosedsymmetric 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 on Dirichlet spaces. A Dirichlet form on a measure space is a bilinear function such that 1) is a dense subset of 2) is symmetric, that is for every. 3) for every. 4) The set equipped with the inner product defined by is a real Hilbert space. 5) For every we have that and In other words, a Dirichlet form is nothing but a non negative symmetric bilinear form defined on a dense subset of such that 4) and 5) hold. Alternatively, the quadratic form itself is known as the Dirichlet form and it is still denoted by, so. The best-known Dirichlet form is the Dirichlet energy of functions on which gives rise to the Sobolev space. Another example of a Dirichlet form is given by where is some non-negative symmetric integral kernel. If the kernel satisfies the bound, then the quadratic form is bounded in. If moreover,, then the form is comparable to the norm in squared and in that case the set defined above is given by . Thus Dirichlet forms are natural generalizations of the Dirichlet integrals where is a positive symmetric matrix. The Euler-Lagrange equation of a Dirichlet form is a non-local analogue of an elliptic equations in divergence form. Equations of this type are studied using variational methods and they are expected to satisfy similar properties.
