Introduction to the Theory of Dirichlet Forms

Abstract

A Dirichlet form is a generalization of the energy form $f\mapsto \int _\Omega |\nabla f|^2 d\lambda$ introduced in the 1840s especially by William Thomson (Lord Kelvin) (cf. Temple, 100 Years of Mathematics, 1981, [351], Chap. 15) in order to solve by minimization the problem without second member $\Delta f=0$ in the open set $\Omega$ (Dirichlet principle). Riemann adopted the expression Dirichlet form (Riemann Grundlagen für eine allgemeine Theorie der Funktionen einer veränderlischen komplexen Grösse, 1851, [314]). The generalization now known as a Dirichlet form keeps the notion in the same relationship with the semigroup as the energy form holds with the heat semigroup.