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# 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.

### 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},
}