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Zbl 1085.46023
Pugachev, O.V.
On the closability of classical Dirichlet forms in the plane.
(English. Russian original)
[J] Dokl. Math. 64, No. 2, 197-200 (2001); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 380, No. 3, 315-318 (2001). ISSN 1064-5624; ISSN 1531-8362/e

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 {\it S. Albeverio} and {\it 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.
[Alan J. Pryde (Clayton)]
MSC 2000:
*46E35 Sobolev spaces and generalizations
31C25 Dirichlet spaces
46N20 Appl. of functional analysis to differential and integral equations
47A07 Forms on topological linear spaces
47B37 Operators on sequence spaces, etc.

Citations: Zbl 0737.46036

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