$\square$

[*Proof of Theorem* ]{} By using the inequality $$\big(1+\theta^{-1}\big)^{\theta}\le e, \qquad \forall\theta>0,$$ with $\theta=8\Vert V_n\Vert_{\infty}/{\lambda_1(\Omega_n)}$, we obtain by , $$\Phi(\Omega_n,V_n)\le e2^{2(3m+4)\eta}(1+8\eta)\Phi(\Omega_n),$$ and by , $$\Phi(\Omega_n,V_n)\ge e^{-1}2^{-2(3m+4)\eta}(1+8\eta)^{-1}\Phi(\Omega_n).$$ This proves Theorem \[the2\]. $\square$

[*Proof of Theorem .*]{} To verify we have by Lemma 4 in [@MvdBS], $$\int_{\Omega}dy\, p_{\Omega}(x,y;t)\ge 1-2^{(m+2)/2}e^{-d_{\Omega}(x)^2/(8t)}.$$ Hence we find by that $$\label{e53}
\int_{\Omega}dx\,\int_{\Omega}dy\, p_{\Omega}(x,y;t)\ge |\Omega|-2^{(m+2)/2}\int_{\Omega}dx\,e^{-d_{\Omega}(x)^2/(8t)}.$$ Multiplying both sides of by $e^{-ct}$ and integrating with respect to $t$ gives that $$\begin{aligned}
\label{e54}
\| v_{\Omega,c} \|_1&\ge \frac{|\Omega|}{c}-2^{(m+2)/2}\int_{\Omega}dx\,\int_{{\mathbb{R}}^+}dt\, e^{-ct-d_{\Omega}(x)^2/(8t)}\nonumber \\ &
\ge \frac{|\Omega|}{c}-2^{(m+2)/2}\int_{\Omega}dx\,\int_{{\mathbb{R}}^+}dt\, e^{-ct/2}\sup\{e^{-ct/2-d_{\Omega}(x)^2/(8t)}:t>0\}\nonumber \\ &
=\frac{|\Omega|}{c}-\frac{2^{(m+4)/2}}{c}\int_{\Omega}dx\,e^{-c^{1/2}d_{\Omega}(x)/2}.\end{aligned}$$ On the other hand using , $$\begin{aligned}
\label{e55}
\| v_{\Omega,c} \|_\infty &=\sup_{x\in\Omega}\int_{{\mathbb{R}}^+}dt\, e^{-ct}\int_{\Omega}dy\, p_{\Omega}(x,y;t)\nonumber \\ &
\le\sup_{x\in\Omega}\int_{{\mathbb{R}}^+}dt\, e^{-ct}\int_{{\mathbb{R}}^m}dy\, p_{{\mathbb{R}}^m}(x,y;t)\nonumber \\ &
=\frac{1}{c}.\end{aligned}$$ The second inequality in follows from and . Finally, follows from , and Lebesgue’s dominated convergence theorem, while follows from of Theorem \[the1\]. $\square$

[*Proof of Theorem* ]{} Let $\frac12<\kappa<1$ be arbitrary, and let $(A_{n})\in\mathfrak{A}((\Omega)_n)$ be such that for all $n$ sufficiently large, $\Vert v_{\Omega_n,V_n}\Vert_1\le (2\kappa-1)^{-1}\int_{A_{n}} v_{\Omega_n,V_n}$, and $|A_n|/|\Omega_n|<1-\kappa$. Then, for all such $n$, $$\begin{aligned}
\Vert v_{\Omega_n,V_n}\Vert_1&\le (2\kappa-1)^{-1}\int_{A_{n}}v_{\Omega_n,V_n}\nonumber \\ &\le (2\kappa-1)^{-1}\Vert v_{\Omega_n,V_n}\Vert_{\infty}|A_{n}|\nonumber \\ &\le \frac{1-\kappa}{2\kappa-1}\Vert v_{\Omega_n,V_n}\Vert_{\infty}|\Omega_n|.\end{aligned}$$ Hence for all $n$ sufficiently large, $\Phi(\Omega_n,V_n)\le \frac{1-\kappa}{2\kappa-1}$. Since $\frac12<\kappa<1$ was arbitrary, $\lim_{n\rightarrow\infty}\Phi(\Omega_n,V_n)=0.$ $\square$

Theorem \[the5\] implies that if $(\Omega_n,V_n)$ satisfies the $\eta$ condition of Theorem 