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Mat. Sb. (N.S.), 1976, Volume 99(141), Number 2, Pages 282–294 (Mi msb2745)  

This article is cited in 8 scientific papers (total in 8 papers)

On an estimate of the Dirichlet integral in unbounded domains

A. K. Gushchin


Abstract: For an arbitrary unbounded region $\Omega$ satisfying a certain condition ($\operatorname{meas}\Omega=\infty$, and $\Omega$ can be such that
$$ \lim_{R\to\infty}\frac1R\operatorname{meas}(\Omega\cap\{|x|<R\})=0) $$
a lower bound for the Dirichlet integral $\int_\Omega|\nabla f(x)|^2 dx$ is established for all functions $f(x)$ in $W_2^1(\Omega)\cap L_r(\Omega)$ which have finite moment $\mu_l=\int_\Omega|x| |f(x)|^l dx$, $0<l<2<r$. The bound of the Dirichlet integral is a positive function of the variables $\mu_l$, $\|f\|_{L_r(\Omega)}$, $\|f\|_{L_2(\Omega)}$ and $\|f\|_{L_q(\Omega)}$, $q\geqslant1$, $l\leqslant q<2$, and is determined by certain geometric characteristics of $\Omega$.
Bibliography: 4 titles.

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English version:
Mathematics of the USSR-Sbornik, 1976, 28:2, 249–261

Bibliographic databases:

UDC: 517.5
MSC: Primary 26A86; Secondary 35K20, 35A15
Received: 26.06.1975

Citation: A. K. Gushchin, “On an estimate of the Dirichlet integral in unbounded domains”, Mat. Sb. (N.S.), 99(141):2 (1976), 282–294; Math. USSR-Sb., 28:2 (1976), 249–261

Citation in format AMSBIB
\Bibitem{Gus76}
\by A.~K.~Gushchin
\paper On an estimate of the Dirichlet integral in unbounded domains
\jour Mat. Sb. (N.S.)
\yr 1976
\vol 99(141)
\issue 2
\pages 282--294
\mathnet{http://mi.mathnet.ru/msb2745}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=626999}
\zmath{https://zbmath.org/?q=an:0338.35009}
\transl
\jour Math. USSR-Sb.
\yr 1976
\vol 28
\issue 2
\pages 249--261
\crossref{https://doi.org/10.1070/SM1976v028n02ABEH001650}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1976EM69100008}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. K. Gushchin, “Stabilization of the solutions of the second boundary value problem for a second order parabolic equation”, Math. USSR-Sb., 30:4 (1976), 403–440  mathnet  crossref  mathscinet  zmath  isi
    2. A. K. Guščin, “On the behaviour ast→∞ of solutions of the second mixed problem for a second-order parabolic equation”, Appl Math Optim, 6:1 (1980), 169  crossref  mathscinet  isi
    3. V. I. Ushakov, “Stabilization of solutions of the third mixed problem for a second order parabolic equation in a noncylindrical domain”, Math. USSR-Sb., 39:1 (1981), 87–105  mathnet  crossref  mathscinet  zmath  isi
    4. A. K. Gushchin, “On the uniform stabilization of solutions of the second mixed problem for a parabolic equation”, Math. USSR-Sb., 47:2 (1984), 439–498  mathnet  crossref  mathscinet  zmath
    5. A. I. Ibragimov, “Some qualitative properties of solutions of the mixed problem for equations of elliptic type”, Math. USSR-Sb., 50:1 (1985), 163–176  mathnet  crossref  mathscinet  zmath
    6. A. K. Gushchin, V. P. Mikhailov, Yu. A. Mikhailov, “On uniform stabilization of the solution of the second mixed problem for a second order parabolic equation”, Math. USSR-Sb., 56:1 (1987), 141–162  mathnet  crossref  mathscinet  zmath
    7. Lezhnev A., “Bounds of Green-Function and Solutions of a 2nd Mixed Problem for a Parabolic Equation”, Differ. Equ., 25:4 (1989), 478–486  mathnet  mathscinet  zmath  isi
    8. Andreucci D., Tedeev A., “Optimal Bounds and Blow Up Phenomena for Parabolic Problems in Narrowing Domains”, Proc. R. Soc. Edinb. Sect. A-Math., 128:Part 6 (1998), 1163–1180  crossref  mathscinet  zmath  isi
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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