Geodesic
On an estimate of the Dirichlet integral in unbounded domains
Sbornik. Mathematics, Tome 28 (1976) no. 2, pp. 249-261 
A. K. Gushchin. On an estimate of the Dirichlet integral in unbounded domains. Sbornik. Mathematics, Tome 28 (1976) no. 2, pp. 249-261. http://geodesic.mathdoc.fr/item/SM_1976_28_2_a7/
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     author = {A. K. Gushchin},
     title = {On an estimate of the {Dirichlet} integral in unbounded domains},
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     year = {1976},
     volume = {28},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1976_28_2_a7/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

For an arbitrary unbounded region $\Omega$ satisfying a certain condition ($\operatorname{meas}\Omega=\infty$, and $\Omega$ can be such that $$ \lim_{R\to\infty}\frac1R\operatorname{meas}\bigl(\Omega\cap\{|x|<R\}\bigr)=0\bigr) $$ 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. 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.

[1] A. K. Guschin, “Nekotorye svoistva obobschennogo resheniya vtoroi kraevoi zadachi dlya parabolicheskogo uravneniya”, Matem. sb., 97 (139) (1975), 242–261 | Zbl

[2] J. Nash, “Continuity of solutions of parabolic and elliptic equations”, Amer. J. Math., 80 (1958), 931–953 ; Matematika, 4:1 (1960), 31–52 | DOI | MR

[3] A. K. Guschin, “Ob otsenkakh reshenii kraevykh zadach dlya parabolicheskogo uravneniya vtorogo poryadka”, Trudy matem. in-ta im. V. A. Steklova, CXXVI (1973), 5–45

[4] E. De Giorgi, “Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari”, Mem. Accad. Sci. Torino, ser. 3, 3 (1957), 25–43 ; Matematika, 4:6 (1960), 23–38 | MR | Zbl