Dirichlet form

{{Short description|Mathematical form}}

In [[potential theory]] (the study of [[harmonic functions]]) and [[functional analysis]], '''Dirichlet forms''' generalize the [[Laplacian]] (the [[mathematical operator]] on scalar fields). Dirichlet forms 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]]. The benefit on these spaces is that one can do this without needing a [[Gradient|gradient operator]], and in particular, one can even weakly define a "Laplacian" in this manner if starting with the Dirichlet form.

In [[potential theory]] (the study of [[harmonic functions]]) and [[functional analysis]], '''Dirichlet forms''' generalize the [[Laplacian]] (the [[mathematical operator]] on scalar fields). Dirichlet forms 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]]. The benefit on these spaces is that one can do this without needing a [[Gradient|gradient operator]], and in particular, one can even weakly define a "Laplacian" in this manner if starting with the Dirichlet form.