I am working on the proof of the co-area formula for $BV$ functions. Suppose $u\in BV(\Omega)$ then the co-area formula states that $$ \|Du\|(\Omega)=\int_{-\infty}^\infty \|\partial E_t(u)\|(\Omega)\,dt $$ where $E_t(u):=\{x\in\Omega,\, u(x)>t\}$ and $\|\partial E_t(u)\|$ denote the perimeter measure.

The proof of $$ \|Du\|(\Omega)\leq\int_{-\infty}^\infty \|\partial E_t(u)\|(\Omega)\,dt $$ is fine, but for converse Evans & Gariepy use a not very intuitive proof, instead, the proof given by Ambrosio & Fusco & Palla [AFP] is more straight but I have a concern in it. Basically, they also use the idea that $BV$ function can be approx by smooth function in some sense. Indeed, we can obtain a sequence $(u_n)\subset C^\infty(\Omega)\cap W^{1,1}(\Omega)$ such that $u_n\to u$ in $L^1(\Omega)$ and $\|Du_n\|(\Omega)\to \|Du\|(\Omega)$. Then, on AFP they write $$\|Du\|(\Omega)=\lim_{n\to\infty}\|Du_n\|(\Omega)=\\\lim_{n\to\infty} \int_{-\infty}^\infty \|\partial E_t(u_n)\|(\Omega)\,dt\geq\int_{-\infty}^\infty \lim_{n\to\infty} \|\partial E_t(u_n)\|(\Omega)\,dt\geq \int_{-\infty}^\infty \|\partial E_t(u)\|(\Omega)\,dt$$

My concern is on second equality. They are using the fact that for $C^\infty\cap W^{1,1}$ function we have $$ \int_\Omega |Du_n|dx=\|Du_n\|(\Omega)= \int_{\infty}^\infty \|\partial E_t(u_n)\|(\Omega)\,dt \tag 1$$

The only theorem close to statement $(1)$ is the Co-area formula for Lipschitz function which states that if $f$ is Lipschitz, we have $$ \int_\Omega|Df|dx= \int_{-\infty}^\infty H^{N-1}(f=t)\,dt \tag 2$$

So I am trying to use $(2)$ to justify $(1)$.

To do so, we need to prove that $u_n$ is actually Lip, but even if the $L^1$ norm of $\nabla u_n$ is finite, could it implies that $u_n$ is actually Lipschitz? I don't think so.

Secondly, we should prove that a.e. $t\in \mathbb R$, we have $$ |\partial E_t(u_n)|(\Omega)=H^{N-1}(\{u_n=t\}) \tag 3$$ This part looks fine to me, because if $u_n$ is $C^\infty$, then by Sard theorem, a.e. level set of $u_n$ has smooth boundary and hence $(3)$ hold. Am I right?

  • 1
    $\begingroup$ A couple of ideas that may or may not help: 1. Try to see if the coarea formula generalizes in a sensible way to locally Lipschitz functions (then since a $C^1$ function is locally Lipschitz we would be done, assuming this generalization works how it should). 2. Since $C^\infty \cap BV \subset W^{1,1}$, maybe we could try to prove first a coarea formula for this functions (it's a somewhat more natural first candidate because of the factor $\int_\Omega |\nabla u|$, and you have better convergence properties by the Meyer-Serrin theorem). Hopefully this is useful... $\endgroup$ – Jose27 Dec 28 '14 at 6:50
  • $\begingroup$ What is the problem with Evans proof? Claim 2 in Evans proof is just what you need here. Where is the this proof in [AFP] book? $\endgroup$ – Tomás Jan 11 '15 at 11:42
  • $\begingroup$ @Tomás this are no problems with evans proof, but I just like AFP better. The AFP prove you could find it in page 146 $\endgroup$ – spatially Jan 11 '15 at 16:50
  • $\begingroup$ I see @wisher. Have you seen in my last comment the answer for your problem? $\endgroup$ – Tomás Jan 11 '15 at 17:29
  • $\begingroup$ @Tomás of course! Thank you so much for your comments! $\endgroup$ – spatially Jan 11 '15 at 17:32

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