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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
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\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
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=626999}
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\transl
\jour Math. USSR-Sb.
\yr 1976
\vol 28
\issue 2
\pages 249--261
\crossref{https://doi.org/10.1070/SM1976v028n02ABEH001650}
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Linking options:
http://mi.mathnet.ru/eng/msb2745 http://mi.mathnet.ru/eng/msb/v141/i2/p282
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This publication is cited in the following articles:
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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
-
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
-
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
-
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
-
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
-
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
-
Lezhnev A., “Bounds of Green-Function and Solutions of a 2nd Mixed Problem for a Parabolic Equation”, Differ. Equ., 25:4 (1989), 478–486
-
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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