On an estimate of the Dirichlet integral in unbounded domains

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 $\mu_l=\int_\Omega|x|\,|f(x)|^l\,dx$, $0<l<2<r$. 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 $\Omega$.
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