On $\varepsilon$-Friedrichs inequalities and its application

Let $n\in \mathbb{N}$ be a fixed number, $\Omega$ be a bounded domain in $\mathbb{R}^n$, $L^2(\Omega),\ L^{\infty}(\Omega)$ be the Lebesgue spaces, $H^1(\Omega)$ and $H^1_0(\Omega)$ be the Sobolev spaces, $$ \Pi_{\ell}(\alpha):= \mathop{\otimes}\limits_{k=1}^{n}(\alpha_k;\alpha_k+\ell), \;\; \alpha=(\alpha_1,\ldots,\alpha_n)\in \mathbb{R}^n, \;\; \ell>0. $$ There are proved the following assertions about $\varepsilon$-Friedrichs inequality (Theorem 1) in the space $H^1(\Omega)$ (Theorem 1) and $H^1_0(\Omega)$ (Theorem 2) : for every $\varepsilon>0$ there exist $N_{\varepsilon}\in \mathbb{N}$ and $\omega_{1},\ldots,\omega_{N_{\varepsilon}}\in L^{\infty}(\Omega)$ such that the inequality $\displaystyle \int\nolimits_{\Omega} |v(x)|^2 \; dx \le \varepsilon \int\nolimits_{\Omega} |\nabla v(x)|^2 \; dx +\sum\limits_{j=1}^{N_{\varepsilon}} \Bigl( \int\nolimits_{\Omega} v(x)\omega_j(x) \; dx \Bigr)^2 $ holds for every $v\in H^1(\Omega)$ (Theorem 1) and $v\in H^1_0(\Omega)$ (Theorem 2), where $\Omega$ is a bounded domain in $\mathbb{R}^n$ for which there exist numbers $\ell>0$, $m\in \mathbb{N}$, and $\alpha^1,\ldots,\alpha^m\in \mathbb{R}^n$ satisfying the following conditions 1) $\overline{\Omega}=\overline{\Pi_{\ell}(\alpha^1)}\cup \ldots \cup \overline{\Pi_{\ell}(\alpha^m)}$; 2) for every $i,j\in \{1,\ldots,m\}$ with $i\not=j$ we obtain: $\Pi_{\ell}(\alpha^i)\cap \Pi_{\ell}(\alpha^j)=\varnothing$.