Coarea for finite-variation L² functions

By [F1], choose compactly supported $v_R\in BV({\mathbb{R}}^2)\cap L^1$ with $v_R\to u$ in $L^2$ and ${\mathrm{lim\hspace{0.17em}sup}}_{R}TV(v_R)\le T{V}_2(u;{\mathbb{R}}^2)$. For each fixed $t\mathrm{\ne }0$, ${\chi }_{E_{R,t}}\to {\chi }_{E_t}$ in $L^1$: the symmetric difference is outside $B(0,R)$ for the cutoffs constructed in [F1] and contained in $\{\mathrm{\mid }u\mathrm{\mid }>\mathrm{\mid }t\mathrm{\mid }\}$, whose tail measure is at most $\mathrm{\mid }t{\mathrm{\mid }}^{-2}{\int }_{{\mathbb{R}}^2\setminus B(0,R)}\mathrm{\mid }u{\mathrm{\mid }}^2$. Here $E_{R,t}$ and $E_t$ are the corresponding signed superlevel or sublevel sets.

For $t>0$, $\mathrm{\mid }{E}_t^{+}\mathrm{\mid }\le t^{-2}\mathrm{\parallel }u{\mathrm{\parallel }}_2^2$, and for $t<0$, $\mathrm{\mid }{E}_t^{-}\mathrm{\mid }\le \mathrm{\mid }t{\mathrm{\mid }}^{-2}\mathrm{\parallel }u{\mathrm{\parallel }}_2^2$. The indicator-valued maps are Borel on each half-line: on positive levels they are right-continuous in $L^1$ when $t_j\downarrow t>0$, by continuity from below of measure; on negative levels they are left-continuous when $t_j\uparrow t<0$. Composing with the lower semicontinuous extended variation in [F3] makes their perimeter functions measurable. Lower semicontinuity in [F3] and step 1.1 give $P(E_t^{\pm };{\mathbb{R}}^2)\le {\mathrm{lim\hspace{0.17em}inf}}_{R\to \mathrm{\infty }}P(E_{R,t}^{\pm };{\mathbb{R}}^2)$ for every applicable nonzero $t$. Fatou's lemma [F5], the BV coarea identity [F2], and the variation bound in step 1.1 imply, for every $\epsilon >0$, ${\int }_{\epsilon }^{\mathrm{\infty }}P(E_t^{+};{\mathbb{R}}^2)dt+{\int }_{-\mathrm{\infty }}^{-\epsilon }P(E_t^{-};{\mathbb{R}}^2)dt\le T{V}_2(u;{\mathbb{R}}^2)\mathrm{.}$ Letting $\epsilon \downarrow 0$ and using monotone convergence [F5] proves that the full perimeter sum is at most $T{V}_2(u;{\mathbb{R}}^2)$; its finiteness and [F3] give finite perimeter for almost every applicable level.

Let $\phi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^2;{\mathbb{R}}^2)$ with $\mathrm{\parallel }\phi {\mathrm{\parallel }}_{\mathrm{\infty }}\le 1$. The signed layer-cake identity is $u(x)={\int }_0^{\mathrm{\infty }}{\chi }_{E_t^{+}}(x)dt-{\int }_{-\mathrm{\infty }}^0{\chi }_{E_t^{-}}(x)dt$. On the compact support of $\mathrm{div}\phi$, $\int \mathrm{\mid }u\mathrm{\mid }\mathrm{\mid }\mathrm{div}\phi \mathrm{\mid }<\mathrm{\infty }$, so Tonelli and Fubini [F4] permit integration of this identity against $\mathrm{div}\phi$. At almost every level the absolute value of the resulting indicator test integral is at most its perimeter by [F3]. Therefore $\int u\mathrm{div}\phi$ is at most the perimeter sum in step 2.1. Taking the supremum over admissible $\phi$ gives $T{V}_2(u;{\mathbb{R}}^2)$ at most that sum, which together with step 2.1 proves equality.