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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.


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Article Information

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

Authors Information

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


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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},
}


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