A function of bounded variation in a regular set has bounded variation in $\mathbb R^N$ and a formula for its variation

Question

Let $\Omega \subset \mathbb R^N$ be an open set. The total variation of a function $u \in L^1(\Omega)$ in $\Omega$ is given by $$ |Du|(\Omega) := \sup\left\{\int_\Omega u \ \text{div}\varphi \ dx \ : \ \varphi \in C_c^1(\Omega; \mathbb R^N), \|\varphi\|_\infty \leq 1 \right\}, $$ and $u$ is said to have bounded variation in $\Omega$ if $|Du|(\Omega) < \infty$. In these notes on the Cheeger problem, Parini states that if $\Omega$ has Lipschitz boundary and $u$ has bounded variation in $\Omega$, then $u$ has bounded variation in $\mathbb R^N$ and $$ |Du|(\mathbb R^N) = |Du|(\Omega) + \int_{\partial \Omega}|u| d \mathcal H^{N - 1}, $$ where $\mathcal H^{N - 1}$ denotes the Hausdorff measure.

How to prove this identity?

First I thought about approximating by smooth subsets (see Proposition 3.4 in the notes), and later I tried to work directly with the definition, but didn't accomplish much with neither approaches.

Thanks in advance and kind regards.

Answer 1

Let $\Omega$ be a bounded domain with smooth boundary.

If $u$ is smooth, then $|Du|(\Omega) = \lVert \nabla u\rVert_{L^1(\Omega)}$. In fact, $\int_{\Omega} u\,\text{div}\varphi = -\int_{\Omega} \nabla u\cdot\varphi \le \lVert \nabla u\rVert_{L^1(\Omega)}$, so $|Du(\Omega)|\le \lVert \nabla u\rVert_{L^1(\Omega)}$. The other inequality follows taking $\varphi = -\zeta\,\nabla u/(|\nabla u|^2+\varepsilon)^{1/2}$, where $\zeta \in C_c^\infty(\Omega)$, $|\zeta|\le 1$ and $\zeta = 1$ in a compact subset $K\subset\Omega$; let $K$ approach $\Omega$.

Now, in $\mathbb{R}^N$, notice that $\int_{\mathbb{R}^N} u\,\text{div}\varphi = \int_{\Omega} u\,\text{div}\varphi = -\int_{\Omega} \nabla u\cdot \varphi + \int_{\partial\Omega}u\nu\cdot\varphi\,d\mathcal{H}^{N-1}$, where $\nu$ is the outward normal vector. Hence, $|Du(\mathbb{R}^N)| \le |Du(\Omega)| + \int_{\partial\Omega}|u|\,d\mathcal{H}^{N-1}$. In the other direction, take $\varphi = \varphi_1 + \varphi_2$ with $\varphi_1$ supported in $K\subset \Omega$ and such that $|Du(\Omega)| - \varepsilon < \int_{\mathbb{R}^N} u\,\text{div}\varphi_1$. Take $\varphi_2$ with support outside $K$ and in a neighborhood of $\partial\Omega$ such that $\int_{\partial\Omega}u\nu\cdot\varphi_2\,d\mathcal{H}^{N-1} = \int_{\partial\Omega}|u|\,d\mathcal{H}^{N-1}$.

When neither $\Omega$ nor $u$ satisfy the above properties of regularity, then I suppose that you can approach them by a sequence of smooth objects.