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A haracterization of closure of the set of compactly supported functions in Dirichlet generalized integral metric and its applications

Abstract

We obtain conditions under which a function $u(x)$ with finite Dirichlet ge e alized integral over a domain $G$ ($u(x)\in H(G))$ belongs to the closure of the set $C_0^\infty(G)$ in the metrics of this Dirichlet integral (i.e., to the space $H_0(G)$). In the case where $G=R^n \;(n \geq 2)$ using these conditions we construct examples of Dirichlet integrals such that $H(R^n) \neq H_0(R^n)$. For $n=2$ these examples show that in the known Mazia theorem uniform positivity of the Dirichlet integral matrix cannot be replaced with its pointwise positivity. The characterization of the space $H_0(G)$ is also applied to the problem of relative equivalence of the spaces $H(G)$ and $H_0(G)$ concerning the part of the boundary $\Gamma (\Gamma\subseteq \partial G)$. This problem in fact coincides with the problem of possibility to set boundary conditions of corresponding boundary-value problems.

Article Information

 Title A haracterization of closure of the set of compactly supported functions in Dirichlet generalized integral metric and its applications Source Methods Funct. Anal. Topology, Vol. 15 (2009), no. 3, 237-250 MathSciNet MR2567308 Copyright The Author(s) 2009 (CC BY-SA)

Authors Information

A. G. Brusentsev
Belgorod Shukhov State Technological University, 46 Kostyukova, Belgorod, 308012, Russia

Citation Example

A. G. Brusentsev, A haracterization of closure of the set of compactly supported functions in Dirichlet generalized integral metric and its applications, Methods Funct. Anal. Topology 15 (2009), no. 3, 237-250.

BibTex

@article {MFAT527,
AUTHOR = {Brusentsev, A. G.},
TITLE = {A haracterization of closure of the set of compactly supported functions in Dirichlet generalized integral metric and its applications},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {15},
YEAR = {2009},
NUMBER = {3},
PAGES = {237-250},
ISSN = {1029-3531},
URL = {http://mfat.imath.kiev.ua/article/?id=527},
}