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Nonzero capacity sets and dense subspaces in scales of Sobolev spaces

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Abstract

We show that for a compact set $K\subset{\mathbb R}^n$ of nonzero $\alpha$-capacity, $C_\alpha(K)>0$, $\alpha\geq 1$, the subspace $\overset{\circ}{W}{^{\alpha,2}}(\Omega)$, $\Omega={\mathbb R}^n\setminus K$ in ${W}{^{\alpha,2}}({\mathbb R}^n)$ is dense in $W^{m,2}({\mathbb R}^n)$, $m\leq\alpha-1$, iff the $m$-capacity of $K$ is zero, $C_{m}(K)=0$.

Key words: Singular perturbations, rigged Hilbert spaces, capacity, Berezansky canonical isomorphisms, Sobolev spaces, dense subspaces.


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Article Information

TitleNonzero capacity sets and dense subspaces in scales of Sobolev spaces
SourceMethods Funct. Anal. Topology, Vol. 20 (2014), no. 3, 213-218
MathSciNet MR3242705
zbMATH 1324.46045
MilestonesReceived 18/02/2014
CopyrightThe Author(s) 2014 (CC BY-SA)

Authors Information

Mykola E. Dudkin
National Technical University of Ukraine (KPI), 37 Peremogy Av., Kyiv, 03056, Ukraine

Volodymyr D. Koshmanenko
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, Ukraine 18/02/2014


Citation Example

Mykola E. Dudkin and Volodymyr D. Koshmanenko, Nonzero capacity sets and dense subspaces in scales of Sobolev spaces, Methods Funct. Anal. Topology 20 (2014), no. 3, 213-218.


BibTex

@article {MFAT740,
    AUTHOR = {Dudkin, Mykola E. and Koshmanenko, Volodymyr D.},
     TITLE = {Nonzero capacity  sets and dense subspaces in scales of Sobolev spaces},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {20},
      YEAR = {2014},
    NUMBER = {3},
     PAGES = {213-218},
      ISSN = {1029-3531},
  MRNUMBER = {MR3242705},
 ZBLNUMBER = {1324.46045},
       URL = {http://mfat.imath.kiev.ua/article/?id=740},
}


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