in a straightforward way from since $$\frac{\int_{A_\nu} v_\nu}{\| v_\nu \|_1} = 1 - \frac{\int_{1/\nu}^1 v_\nu}
{\int_{0}^{1/\nu} v_\nu + \int_{1/\nu}^1 v_\nu}\ge 1-\frac{\int_{1/\nu}^1 v_\nu}{\int_{0}^{1/\nu} v_\nu}.$$

 Example \[exa2\] {#sec4}
=================

In this section we analyse localisation for a sequence of open, bounded sets $(\Omega_{\alpha,n}), n\in{\mathbb{N}}$ in ${\mathbb{R}}^2$, where $\Omega_{\alpha,n}$ is the disjoint union of $n$ rectangles of size $n^{-1}\times 1$, and $1$ rectangle of size $cn^{-\alpha}\times 1$. The constant $c$ is strictly positive, and $0<\alpha<1$.

\[exa2\] Let $$N_{\alpha,c}=\min\{n\in{\mathbb{N}}:n^{\alpha}>c,\, n^{1-\alpha}>c^{-1}\}.$$

-   If $n\ge N_{\alpha,c}$, then $$\label{e16}
    \frac{8}{45c^2(1+c)}\big(n^{2\alpha-2}+c^3n^{-\alpha}\big)\le\Phi(\Omega_{\alpha,n})\le \frac{4}{3c^2}\big(n^{2\alpha-2}+c^3n^{-\alpha}\big).$$

-   If $\frac23<\alpha<1$, then $(v_{\Omega_{\alpha,n}})$ is not localising, $\kappa=0.$

-   If $0<\alpha<\frac23$, then $(v_{\Omega_{\alpha,n}})$ is localising, $\kappa=1.$

-   If $\alpha=\frac23$, then $(v_{\Omega_{\alpha,n}})$ is $\kappa_c$-localising with $$\label{e13}
    \kappa_c=\frac{c^3}{1+c^3},$$ and the supremum defining $\kappa_c$ in is a maximum.

First we establish the following estimate for the torsion function of a rectangle in ${\mathbb{R}}^2$.

Let $R_{a,b}=(-a/2,a/2)\times(-b/2,b/2)$ be a rectangle with side lengths $a$ and $b$ respectively, and let $x=(x_1,x_2)\in R_{a,b}$. $$\label{e49a}
v_{R_{a,b}}(x_1,x_2)\le v_{(-a/2,a/2)}(x_1)=\frac{a^2-4x_1^2}{8},\qquad \forall (x_1,x_2)\in R_{a,b}.$$

By for $V=0$, separation of variables, and monotonicity and non-negativity of Dirichlet heat kernels we have $$\begin{aligned}
\label{e49b}
v_{R_{a,b}}(x_1,x_2)&=\int_0^{\infty}dt\,\int_{(-a/2,a/2)}dy_1\,p_{(-a/2,a/2)}(x_1,y_1;t)\nonumber \\ &
\hspace{20mm}\times\int_{(-b/2,b/2)}dy_2\,p_{(-b/2,b/2)}(x_2,y_2;t)\nonumber \\ &
\le\int_0^{\infty}dt\,\int_{(-a/2,a/2)}dy_1\,p_{(-a/2,a/2)}(x_1,y_1;t)\int_{{\mathbb{R}}}dy_2\,p_{{\mathbb{R}}}(x_2,y_2;t)\nonumber \\ &
=\int_0^{\infty}dt\,\int_{(-a/2,a/2)}dy_1\,p_{(-a/2,a/2)}(x_1,y_1;t)\nonumber \\ &
=v_{(-a/2,a/2)}(x_1)\nonumber \\&
=\frac{a^2-4x_1^2}{8},\end{aligned}$$ where we have used formula , $p_{{\mathbb{R}}}(x_2,y_2;t)=(4\pi t)^{-1/2}e^{-|x_2-y_2|^2/(4t)}$, identity , 