On the closability of classical Dirichlet forms in the plane. (Q1432652)

The author exhibits a measure \(\mu\) on the plane such that the Dirichlet form \(E(f,g)=\int(\nabla f,\nabla g)\,d\mu\) is closable, whereas the form \(E_x(f,g)=\int \partial_xf\partial_xg\,d\mu\) is not. This gives a positive answer to a question of \textit{S. Albeverio} and \textit{M. Röckner} [J. Funct. Anal. 88, No. 2, 395--436 (1990; Zbl 0737.46036)]. The measure \(\mu\) is restriction of Lebesgue measure to an open subset of the unit square and the construction is based on a Cantor set of positive measure.