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

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

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
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
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
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
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
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
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
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
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