On closability of classical Dirichlet forms.
type: Item
instance of: scholarly article
title: On closability of classical Dirichlet forms.
zbMATH Open document ID: 1039.47002
author: O. V. Pugachev
published in: Journal of Functional Analysis
publication date: 2004-03-14T00:00:00Z
review text: Let \(H\) be a Hilbert space, \(D({\mathcal E})\) a linear subspace of \(H\) and \({\mathcal E}:D({\mathcal E})\times D({\mathcal E})\to \mathbb R\) a nonnegative symmetric bilinear form. The form is closable if \(u_n\in D({\mathcal E})\), with \(u_n\to0\) in \(H\) and \({\mathcal E}(u_n-u_m,u_n-u_m)\to0\) implies that \({\mathcal E}(u_n,u_n)\to0\). Consider the space \(L^2(\mu)\), where \(\mu\) is a probability measure on the Borel sets of \(\mathbb R^n\). It is known that if the partial Dirichlet forms \({\mathcal E}_i(f,g)=\int {\partial f\over \partial x_i} {\partial g\over \partial x_i}\,d\mu(x)\) are closable, then so is the classical gradient Dirichlet form \({\mathcal E}(f,g)=\int (\nabla f,\nabla g)\,d\mu(x)\). The converse was conjectured some years ago by M. Röckner to be, in general, false. The paper under review establishes this conjecture by constructing (via a Cantor set type procedure) a bounded open set \(F\) in \(\mathbb R^2\) such that the gradient Dirichlet form is closed on \(L^2(\lambda^2| _F)\) but the partial form \({\mathcal E}_1\) is not (here \(\lambda^2\) is Lebesgue measure in \(\mathbb R^2\)). The construction extends to a measure with support \(\mathbb R^2\).
reviewed by: Guillermo P. Curbera
Mathematics Subject Classification ID: 47A07
Mathematics Subject Classification ID: 31C25
zbMATH DE Number: 2055846
zbMATH Keywords: partial Dirichlet forms
zbMATH Keywords: Hilbert space
zbMATH Keywords: gradient Dirichlet form
zbMATH Keywords: closability
MaRDI profile type: Publication
full work available at URL: j.jfa.2001.11.001
OpenAlex ID: W2033792713
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DOI: 10.1016/J.JFA.2001.11.001
description: scientific article; zbMATH DE number 2055846