# Total variation of (weakly) differentiable functions

the total variation of a function $u\in L^1(\Omega)$, $\Omega\subset \mathbb{R}^n$, can be defined as

$$\sup \{ \int_\Omega u \; \mathrm{div} g \; dx:\; g \in C_c^1(\Omega,\mathbb{R}^n), \; \lvert g(x) \rvert \leq 1,\; x \in \Omega \}$$

for (weakly) differentiable functions $u$, this supremum equals the $L^1$ norm of the (weak) gradient $\int_\Omega \lvert \nabla u \rvert\; dx$. however, i can't seem to find the rigorous argument to show this. can someone help me?

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What do you assume on $\Omega$? –  Davide Giraudo Mar 11 '12 at 18:48
$\Omega$ is an open set, no further assumptions. –  begeistzwerst Mar 11 '12 at 18:52

Let $\epsilon > 0$. Pick $f$ in $C^\infty(\Omega) \cap BV(\Omega)$ so $\|u-f\|_{L^1(\Omega)} < \epsilon$ and $\Big|\int_\Omega |Du|- \int_\Omega |Df|\Big| < \epsilon$. By integration by parts,

\begin{aligned} \int_\Omega |Df|&= \sup\{ \int_\Omega f\text{ div}g \mid g\in C_0^1(\Omega, \mathbb{R}^n)|g| \leq 1\}\\ &=\sup\{ \int_\Omega -\nabla f \cdot g \mid g\in C_0^1(\Omega, \mathbb{R}^n),\ |g| \leq 1\}\\ &=\sup\{ \int_\Omega \nabla f \cdot g \mid g\in C_0^1(\Omega, \mathbb{R}^n)\, |g| \leq 1\} = \|\nabla f\|_{L^1({\Omega})}\\ \end{aligned} % which can be confirmed by letting $g$ close to $\frac{1}{\lvert \nabla f \rvert} \nabla f$. Now let $\epsilon \to 0$

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thank you for your answer. why do we have to make the detour of picking a smooth $f$? i mean, why can't we use this argument directly for $u$? –  begeistzwerst Mar 11 '12 at 23:48
I was thinking of BV functions when I wrote it, but I guess the integration by parts works for $u \in W^{1,1}(\Omega)$ without introducing the smooth $f$. In the last line, you have to be careful about what happens if $\nabla u$ vanishes anywhere. I think you can let $\varphi_\epsilon(x) = 0$ if $|\nabla(u)| < \epsilon$ and $\varphi_\epsilon(x) = 1$ if $|\nabla(u)| \geq 2\epsilon$, with $0 \leq \varphi \leq 1$, and take the limit as $\epsilon \to 0$. –  Stefan Smith Mar 13 '12 at 10:50
what do you intend to do with these $\varphi_\epsilon$? use them as a sequence of $g$'s in the supremum? this is a good idea, but they must be of class $C_0^1$ (apart from the fact that you did not define them for $\epsilon\leq|\nabla u| < 2\epsilon$). –  begeistzwerst Mar 13 '12 at 22:23
thank you for your suggestions. I discussed this question with one of my colleagues. I am still not sure about fine details and technicalities, but I think we worked out its main arguments. I posted it as an answer below. I originally thought this whole thing would be easier. Or am I still missing the simplest argument? –  begeistzwerst Mar 14 '12 at 22:09

Let $\Omega\subset\mathbb{R}^n$ be an open set, $u\in W^{1,1}(\Omega)$, $\nabla u$ its weak gradient and $\int_\Omega |Du|$ its total variation as defined above. We want to show that $$\int_\Omega |Du| = \int_\Omega |\nabla u|\;dx.$$

"$\leq$" clearly holds, since

$$\int_\Omega u \; \text{div}g \; dx = - \int_\Omega \nabla u \cdot g \; dx \leq \int_\Omega |\nabla u|\; dx$$

due to $|g|\leq 1$. It remains to show that equality is achieved or, equivalently, that "$\geq$" holds. This can be done by constructing an appropriate sequence of functions $(g_\epsilon)$ living in $C_c^1$ that, in some sense, approximates $\frac{\nabla u}{|\nabla u|}$. Then, given we found a sequence that does the trick, we would be done, due to the following argument.

\begin{align} \int_\Omega |Du| &= \sup { \int_\Omega \nabla u \cdot g \; dx: g \in C_c^1(\Omega, \mathbb{R}^n), \; |g| \leq 1 } \newline &\geq \lim_{\epsilon \rightarrow 0} \int_\Omega \nabla u \cdot g_\epsilon \; dx \newline &=\int_\Omega|\nabla u|\;dx \end{align}

Thus, the crucial point is to find such $g_\epsilon$'s. Let $$\tilde{g}:= \begin{cases} \frac{\nabla u}{|\nabla u|}, & \text{if } \nabla u \neq 0 \newline 0, & \text{otherwise}\end{cases}$$ and let $\tilde{g}_{\epsilon_1}$ be this function multiplied with the characteristic function of the set $\Omega_{\epsilon_1} \cap B_{\epsilon^{-1}_1}$, where $\Omega_{\epsilon_1} = \{x\in\Omega: \text{dist}(x,\partial\Omega) \geq \epsilon_1\}$ and $B_{\epsilon^{-1}_1}$ is the closed ball centered at the origin with radius $\frac{1}{\epsilon_1}$. For all positive $\epsilon_1$, $\tilde{g}_{\epsilon_1}$ is a compactly supported vector field satisfying $|\tilde{g}_{\epsilon_1}| \leq 1$ and $\tilde{g}_{\epsilon_1}\overset{\epsilon_1\rightarrow 0}{\rightarrow}\tilde{g}$. By convolution with a suitable mollifier $\eta_{\epsilon_2}$ (see e.g. Giusti 1984, pp. 10-11) the resulting functions $g_\epsilon := \eta_{\epsilon_2} \ast \tilde{g}_{\epsilon_1}$, $\epsilon := (\epsilon_1, \epsilon_2)$, are compactly supported, smooth and $g_\epsilon \overset{\epsilon_2\rightarrow 0}{\rightarrow}\tilde{g}_{\epsilon_1}$ in $L_1$. This sequence of functions (I believe) fulfills the above equality, which should conclude the proof.

Is this the right way? I have to emphasize that I am not very sure about the double limit, and might need some input to make it fully rigorous. Any ideas?

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for the sake of completeness: i just stumbled over this wikipedia article, which contains the proof for differentiable $u$. unless i am missing something, it seems to directly carry over to weakly differentiable functions as it is essentially the same as this answer. –  begeistzwerst May 7 '12 at 14:48
Your proof looks like it is probably correct, but there is no need for 2 $\epsilon$'s. You proved $\int_\Omega |Du| \leq \int_\Omega |\nabla u|\,dx$. To prove the opposite direction, let $\epsilon > 0$. Let $f_\epsilon \in C^\infty(\Omega) \cap W^{1,1}(\Omega)$ with $\|u - f_\epsilon\|_{W^{1,1}} < \epsilon$ (this is necessary because later we want $f_\epsilon$ to be $C^1$). For $t > 0$ define $S_t = \{x \in \Omega \mid d(x,\partial\Omega) > t \text{ and }|\nabla u(x)| > t\}$. Define $$g_\epsilon(x) = \begin{cases} \frac{\nabla f_\epsilon(x)}{|\nabla f_\epsilon(x)|}, &\text{x \in S_{2\epsilon}}\newline 0, &\text{x \in \Omega \setminus S_\epsilon} \end{cases}$$ with $g_\epsilon$ smooth and $|g_\epsilon(x)| \leq 1$ for all $x \in \Omega$. Then by the dominated convergence theorem, \begin{aligned} \int_\Omega |Df_\epsilon| = &\sup\{\int_\Omega f_\epsilon \text{ div}g \mid g\in C_0^1(\Omega, \mathbb{R}^n), |g| \leq 1\}\\ \end{aligned} $$= \sup\{\int_\Omega -\nabla f_\epsilon \cdot g \mid g\in C_0^1(\Omega, \mathbb{R}^n), |g| \leq 1\}$$
$$= \sup\{\int_\Omega \nabla f_\epsilon \cdot g \mid g\in C_0^1(\Omega, \mathbb{R}^n), |g| \leq 1\}$$
$$\geq \int_\Omega \nabla f_\epsilon \cdot g_\epsilon$$ $$\geq \int_{S_{2\epsilon}} |\nabla f_{\epsilon}|\,dx - \int_{{S_\epsilon}\setminus{S_{2\epsilon}}}1\,dx$$
So $$\int_\Omega|Du| =\lim_{\epsilon\to 0} \int_\Omega |Df_\epsilon| \geq \lim_{\epsilon \to 0} \int_\Omega |\nabla f_{\epsilon}|\,dx = \int_\Omega |\nabla u|\,dx.$$