Finite samples outside a uniform BV bound

The square $[7\mathrm{/}16,9\mathrm{/}16{]}^2$ has side $1\mathrm{/}8$ and lies in ${\mathrm{\Omega }}^{\mathrm{\prime }}$ because its corner radius from $(1\mathrm{/}2,1\mathrm{/}2)$ is $\sqrt{2}\mathrm{/}16<1\mathrm{/}8$. The grid spacing is $s_N=1\mathrm{/}(8q)$ and $r_N=s_N\mathrm{/}4$. Distinct cell centers are separated by at least $s_N>2r_N$, and each disk lies inside its cell; hence the disks are disjoint and contained in ${\mathrm{\Omega }}^{\mathrm{\prime }}$. Thus $1\le {\gamma }_N\le 1+a=5\mathrm{/}4<3\mathrm{/}2$ and ${\gamma }_N=1$ on $\mathrm{\Omega }\setminus {\mathrm{\Omega }}^{\mathrm{\prime }}$.

For a disk $B(c,r)$ and a smooth compactly supported vector field $\phi$, polar coordinates give ${\int }_{B(c,r)}\mathrm{div}\phi dx=r{\int }_0^{2\pi }\phi (c+r(\cos \theta ,\sin \theta ))\cdot (\cos \theta ,\sin \theta )d\theta$. Indeed, in polar variables $\mathrm{div}\phi ={\rho }^{-1}{\mathrm{\partial }}_{\rho }(\rho {\phi }_r)+{\rho }^{-1}{\mathrm{\partial }}_{\theta }{\phi }_{\theta }$; integrating against $\rho d\rho d\theta$ makes the angular derivative integrate to zero and the inner radial term vanish. The distributional definition therefore gives $D{\mathbf{1}}_{B(c,r)}=-\nu {\mathcal{H}}^1{\mathrm{\mid }}_{\mathrm{\partial }B(c,r)}$, whose variation has mass $2\pi r$. Summing these measures over the disjoint disks shows $D{\mathbf{1}}_{E_N}$ is finite with variation mass $2\pi r_{N}N$, so [F3] gives ${\mathbf{1}}_{E_N}\in BV(\mathrm{\Omega })$ and [F2] gives the stated upper bound.

The annuli of radii $(r_N\mathrm{/}2,3r_N\mathrm{/}2)$ about the chosen centers are pairwise disjoint: their outer radii sum to $3r_N=3s_N\mathrm{/}4<s_N$, while distinct centers are at least $s_N$ apart. They lie in $\mathrm{\Omega }$, since each center is within $\sqrt{2}\mathrm{/}16$ of $(1\mathrm{/}2,1\mathrm{/}2)$ and $\sqrt{2}\mathrm{/}16+3r_N\mathrm{/}2\le \sqrt{2}\mathrm{/}16+3\mathrm{/}64<1\mathrm{/}4$. On each annulus choose a smooth radial cutoff equal to one at radius $r_N$, vanishing outside the annulus and between zero and one. The sum of the corresponding radial unit-vector fields is smooth, compactly supported in $\mathrm{\Omega }$, has norm at most one, and equals the outward normal on every circle. Its flux is $2\pi r_{N}N$, so [F2] gives the matching lower bound. Since $D{\gamma }_N=aD{\mathbf{1}}_{E_N}$ by [F3] and $a>0$, step 1.2 gives $TV({\mathbf{1}}_{E_N};\mathrm{\Omega })=2\pi r_{N}N$ and $TV({\gamma }_N;\mathrm{\Omega })=2\pi a{r}_{N}N=\pi aN\mathrm{/}(16q)$.

Since $q=\lceil \sqrt{N}\rceil \le 2\sqrt{N}$, step 2.1 gives $TV({\gamma }_N;\mathrm{\Omega })\ge (\pi a\mathrm{/}32)\sqrt{N}$. The value and collar bounds were checked in step 1.1, and every finite union of disks has the finite perimeter computed in step 2.1. Thus each conductivity is individually finite-BV while the total variations diverge, so no fixed $R$ contains the whole family in $X_2(R)$.