Nonzero capacity sets and dense subspaces in scales of Sobolev spaces

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.

Full Text

Article Information

Title	Nonzero capacity sets and dense subspaces in scales of Sobolev spaces
Source	Methods Funct. Anal. Topology, Vol. 20 (2014), no. 3, 213-218
MathSciNet	MR3242705
zbMATH	1324.46045
Milestones	Received 18/02/2014
Copyright	The 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},
}