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 Mat. Zametki: Year: Volume: Issue: Page: Find

 Mat. Zametki, 2004, Volume 75, Issue 6, Pages 803–817 (Mi mz82)

Estimates of $C^m$-Capacity of Compact Sets in $\mathbb{R}^N$

A. M. Voroncov

M. V. Lomonosov Moscow State University

Abstract: For a given homogeneous elliptic partial differential operator $L$ with constant complex coefficients, the Banach space $V$ of distributions in $\mathbb{R}^N$ and a compact set $X$ in $\mathbb{R}^N$, we study the quantity $\lambda_{V,L}(X)$ equal to the distance in $V$ from the class of functions $f_0$ satisfying the equation $Lf_0 = 1$ in a neighborhood of $X$ (depending on $f_0$) to the solution space of the equation $Lf= 0$ in the neighborhoods of $X$. For $V=BC^m$, we obtain upper and lower bounds for $\lambda_{V,L}(X)$ in terms of the metric properties of the set $X$, which allows us to obtain estimates for $\lambda_{V,L}(X)$ for a wide class of spaces $V$.

DOI: https://doi.org/10.4213/mzm82

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English version:
Mathematical Notes, 2004, 75:6, 751–764

Bibliographic databases:

UDC: 517.538.5+517.956.2

Citation: A. M. Voroncov, “Estimates of $C^m$-Capacity of Compact Sets in $\mathbb{R}^N$”, Mat. Zametki, 75:6 (2004), 803–817; Math. Notes, 75:6 (2004), 751–764

Citation in format AMSBIB
\Bibitem{Vor04} \by A.~M.~Voroncov \paper Estimates of $C^m$-Capacity of Compact Sets in $\mathbb{R}^N$ \jour Mat. Zametki \yr 2004 \vol 75 \issue 6 \pages 803--817 \mathnet{http://mi.mathnet.ru/mz82} \crossref{https://doi.org/10.4213/mzm82} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2085808} \zmath{https://zbmath.org/?q=an:1064.31004} \elib{http://elibrary.ru/item.asp?id=6618285} \transl \jour Math. Notes \yr 2004 \vol 75 \issue 6 \pages 751--764 \crossref{https://doi.org/10.1023/B:MATN.0000030985.99917.0a} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000222492400019}