Coarea formula for spatial BV

For any real $f\in L^1(\mathrm{\Omega })$, define its extended test variation $V(f)$ as the supremum of $\int f\mathrm{div}\phi$ over smooth compactly supported vector fields with $\mathrm{\mid }\phi \mathrm{\mid }\le 1$. It is a lower semicontinuous extended nonnegative function on $L^1$: each test integral is a continuous linear functional and $\{f:V(f)>a\}$ is the union of its open superlevel sets. By [F2], $V(f)<\mathrm{\infty }$ is equivalent to $f\in BV$, in which case $V(f)=TV(f;\mathrm{\Omega })$. For $t>0$ the indicator ${\chi }_{\{f>t\}}$ lies in $L^1$, since its norm is at most $\mathrm{\parallel }f{\mathrm{\parallel }}_1\mathrm{/}t$, and it is right continuous as an $L^1$-valued function of $t$: if $t_j\downarrow t>0$, its indicators increase to ${\chi }_{\{f>t\}}$ and their errors tend to zero by monotone convergence. Dyadic right-step approximations therefore make this map Borel measurable. For $t<0$, $t\mapsto {\chi }_{\{f<t\}}$ is similarly left continuous and Borel. Composition with the lower semicontinuous $V$ shows that both extended level-variation functions are measurable.

By [F1], take smooth strict approximants $u_j$ with $\mathrm{\parallel }{u}_j-u{\mathrm{\parallel }}_1<2^{-j}$ and $\int \mathrm{\mid }\mathrm{\nabla }{u}_j\mathrm{\mid }\to TV(u;\mathrm{\Omega })$, passing to a subsequence by least indices if needed. For real numbers $a,b$, the total length of the positive and negative levels at which their signed superlevel or sublevel indicators differ is $\mathrm{\mid }a-b\mathrm{\mid }$. Hence [F3] gives ${\int }_0^{\mathrm{\infty }}\mathrm{\parallel }{\chi }_{\{u_j>t\}}-{\chi }_{\{u>t\}}{\mathrm{\parallel }}_{1}dt+{\int }_{-\mathrm{\infty }}^0\mathrm{\parallel }{\chi }_{\{u_j<t\}}-{\chi }_{\{u<t\}}{\mathrm{\parallel }}_{1}dt=\mathrm{\parallel }{u}_j-u{\mathrm{\parallel }}_1\mathrm{.}$ Sum over $j$ and apply [F3]: for almost every applicable nonzero level, the indicator errors are summable, and in particular the level indicators converge in $L^1$. The lower semicontinuity from step 1.1 gives $V({\chi }_{E_t^{\pm }})\le {\mathrm{lim\hspace{0.17em}inf}}_{j}V({\chi }_{\{u_j\gtrless t\}})$ at those levels. Fatou and the smooth coarea identity [F1] consequently give ${\int }_0^{\mathrm{\infty }}V({\chi }_{E_t^{+}})dt+{\int }_{-\mathrm{\infty }}^{0}V({\chi }_{E_t^{-}})dt\le {\mathrm{lim\hspace{0.17em}inf}}_j{\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\nabla }{u}_j\mathrm{\mid }dx=TV(u;\mathrm{\Omega })\mathrm{.}$ The integral is finite, so its integrands are finite almost everywhere; [F2] then proves the stated almost-everywhere finite-perimeter assertion.

For any admissible smooth test field $\phi$, signed layer cake gives $u(x)={\int }_0^{\mathrm{\infty }}{\chi }_{E_t^{+}}(x)dt-{\int }_{-\mathrm{\infty }}^0{\chi }_{E_t^{-}}(x)dt$ almost everywhere. The absolute integral against $\mathrm{div}\phi$ is bounded by $\mathrm{\parallel }\mathrm{div}\phi {\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }u{\mathrm{\parallel }}_1$, so iterated integration is valid. Each level test integral in absolute value is at most its extended variation, using also $-\phi$ in step 1.1. Therefore $\int u\mathrm{div}\phi$ is at most the sum of the two perimeter integrals. Take the supremum over $\phi$ and apply [F2] to $u$ to obtain the reverse of the inequality in step 2.1. Equality follows. Changing the representative of $u$ changes each level set only by a spatial null set, so all conclusions concern the given $L^1$ class.