Today, we will discuss the coarea formula for BV functions, established by Fleming and Rishel in 1960 (W. Fleming, R. Rishel, An integral formula for total gradient variation, Arch. Math. 11 (1960), 218-222.)
Let us first recall the lower semicontinuity property of the total variation with respect to the strong convergence in L1(Ω):
Let {un}n∈Ω be a sequence in BV(Ω) strongly converging to some u in L1(Ω) and satisfying |un|BV<∞. Then u∈BV(Ω) and |u|BV(Ω)≤lim The coarea formula is as stated as follows:
Coarea formula: Let u be a given function in BV(\Omega). Then for almost every t in \mathbb{R}, the level set E_t=\{ x\in \Omega \subset \mathbb{R}^N: u(x) > t\} of u is a set of finite perimeter in \Omega, and
\begin{align} &Du=\int_{-\infty}^{\infty} D\chi_{E_t}\, dt,\\ &|u|_{BV(\Omega)}=|Du|(\Omega)=\int_{-\infty}^{\infty} \int_{\Omega}|D\chi_{E_t}|\, dt. \end{align}
The second assertion above is often written in terms of the perimeter of level sets as follows:
\begin{align} |u|_{BV(\Omega)}=\int_{-\infty}^{\infty} Per(E_t, \Omega) dt, \end{align}
where Per(E_t, \Omega)=\mathcal{H}^{N-1}(\Omega \cap \partial E_t) is the perimeter of the level set E_t.
Proof of the coarea formula: We follow the treatment of Variational Analysis in Sobolev and BV spaces by Attouch et al.
Part I: Proof of |Du|(\Omega) \leq \int_{-\infty}^{\infty}\int_{\Omega}|D\chi_{E_t}|\, dt:
Let us assume that D\chi_{E_t} belongs to \mathbf{M}(\Omega, \mathbb{R}^N), the set of all \mathbb{R}^N- valued Borel measures which is the set of all the \sigma- additive set functions \mu:\mathcal{B}(\Omega)\rightarrow \mathbb{R}^N, with \mu(\emptyset)=0. For all t in \mathbb{R} set
\begin{equation} f_t=\left\{ \begin{array}{ll} \chi_{E_t} & \mbox{if } t \geq 0 \\ -\chi_{\Omega \backslash E_t} & \mbox{if } t < 0 \end{array} \right. \end{equation} It is easy to see that u(x)=\int_{-\infty}^{\infty}f_t(x)\,dt for all x\in \Omega. For all \bar{\phi} \in C_c^1(\Omega, \mathbb{R}^N) we have \begin{align*} &\langle Du, \bar{\phi} \rangle \\ &= -\int_{\Omega} u \, \mbox{div} \,\bar{\phi} \, dx \\ & = - \int_{\Omega} \int_{-\infty}^{\infty} f_t \, \mbox{div} \,\bar{\phi} \, dt\,dx\\ &= - \int_{\Omega} \int_{-\infty}^{0} f_t \, \mbox{div} \,\bar{\phi} \, dt\,dx -\int_{\Omega} \int_{0}^{\infty} f_t \, \mbox{div} \,\bar{\phi} \, dt\,dx \\ &= \int_{\Omega} \int_{-\infty}^{0} \chi_{\Omega \backslash E_t} \, \mbox{div} \,\bar{\phi} \, dt\,dx -\int_{\Omega} \int_{0}^{\infty} \chi_{E_t} \, \mbox{div} \,\bar{\phi} \, dt\,dx \\ &=\int_{\Omega} \int_{-\infty}^{0} (1-\chi_{E_t}) \, \mbox{div} \,\bar{\phi} \, dt\,dx -\int_{\Omega} \int_{0}^{\infty} \chi_{E_t} \, \mbox{div} \,\bar{\phi} \, dt\,dx \\ &=\int_{\Omega} \int_{-\infty}^{0} \mbox{div} \,\bar{\phi} \, dt\,dx - \int_{\Omega} \int_{-\infty}^{0} \chi_{E_t} \, \mbox{div} \,\bar{\phi} \, dt\,dx-\int_{\Omega} \int_{0}^{\infty} \chi_{E_t} \, \mbox{div} \,\bar{\phi} \, dt\,dx \\ &= \int_{\Omega} \int_{-\infty}^{0} \mbox{div} \,\bar{\phi} \, dt\,dx-\int_{\Omega} \int_{-\infty}^{\infty} \chi_{E_t} \, \mbox{div} \,\bar{\phi} \, dt\,dx \\ &= 0-\int_{\Omega} \int_{-\infty}^{\infty} \chi_{E_t} \, \mbox{div} \,\bar{\phi} \, dt\,dx \\ &= \int_{-\infty}^{\infty} \langle D\chi_{E_t}, \bar{\phi}\rangle \,dt \end{align*} Hence, Du=\int_{-\infty}^{\infty} D\chi_{E_t}\, dt, and |Du|(\Omega) \leq \int_{-\infty}^{\infty}\int_{\Omega}|D\chi_{E_t}|\, dx dt.
Part II: (coverse) Proof of \int_{-\infty}^{\infty}\int_{\Omega}|D\chi_{E_t}|\, dxdt\leq |Du|(\Omega) :
Step i. We first assume that u belongs to the space \mathcal{A}(\Omega) of piecewise linear and continuous functions in \Omega. By linearity, one can assume that u=a\cdot x+b with a\in \mathbb{R}^N and b\in \mathbb{R}, so that
\begin{align*} &\int_{\Omega} |D\chi_{E_t}|\\ &=\mathcal{H}(\Omega \cap \partial E_t)\\ &=\mathcal{H}(\Omega \cap \{x| a\cdot x +b=t\})\\ &=\int_{\Omega \cap \{x| a\cdot x +b=t\}} d\mathcal{H}^{N-1}(x)\\ &=\int_{\{x| a\cdot x +b=t\}}\chi_{\Omega}(x)\, d\mathcal{H}^{N-1}(x) \end{align*}
Thus we get,
\begin{align*} &\int_{-\infty}^{\infty} \int_{\Omega} |D\chi_{E_t}|\, dx dt\\ &=|a|\mathcal{L}(\Omega)\\ &=\int_{\{x| a\cdot x +b=t\}}\chi_{\Omega}(x)\, d\mathcal{H}^{N-1}(x)\\ &=\int_{\Omega} |Du| \end{align*}
Hence we have,
\begin{align*} \int_{-\infty}^{\infty} \int_{\Omega} |D\chi_{E_t}|\, dx dt=|Du|(\Omega) \dots \mbox{ if } u=a\cdot x+b. \end{align*}
Step ii. Now we prove that \int_{-\infty}^{\infty} \int_{\Omega} |D\chi_{E_t}|\, dx dt\leq|Du|(\Omega) for all u\in BV(\Omega). We know that the space of piecewise linear and continuous functions \mathcal{A}(\Omega) is dense in W^{1, 1}(\Omega) when equipped with the strong topology. We also know that the space C^{\infty}\cap W^{1, 1}(\Omega) is dense in BV when equipped with the intermediate convergence. Thus, there exists a sequence \{u_n\}_{n\in \mathbb{N}} \in \mathcal{A}(\Omega) such that u_n\rightharpoonup u for the intermediate convergence. For each of the functions u_n we set E_{n, t}:=\{x\in \Omega: u_n(x) > t\}. Due to the intermediate convergence, we have
\begin{align*} &\int_{\Omega}|Du|=\lim_{n\rightarrow \infty}\int_{\Omega}|Du_n| \dots \mbox{the intermediate convergence}\\ &=\lim_{n\rightarrow \infty} \int_{-\infty}^{\infty} \int_{\Omega} |D\chi_{E_{n,t}}|\, dx dt \dots \mbox{from step i}\\ &\geq \int_{-\infty}^{\infty} \lim\inf_{n\rightarrow \infty} \int_{\Omega} |D\chi_{E_{n,t}}|\, dx dt \dots \mbox{Fatou's lemma}\\ \end{align*}
From the intermediate convergence we also have u_n\rightarrow u in L^1(\Omega). i.e.
\begin{align*} \int_{\Omega} |u_n-u|=\int_{\Omega} \int_{-\infty}^{\infty} |\chi_{E_{n,t}}-\chi_{E_t}|\ dt dx = \int_{-\infty}^{\infty} \Big(\int_{\Omega} |\chi_{E_{n,t}}-\chi_{E_t}|\ dx \Big) dt=0. \end{align*}
Thus, for a subsequence E_{n_m, t}, and for almost all t \in \mathbb{R}, \chi_{E_{n_m, t}}\rightarrow \chi_{E_{n, t}} strongly in L^1(\Omega). The lower semicontinuity of the total variation with respect to the strong convergence in L^1(\Omega) we get \lim\inf_{n\rightarrow \infty} \int_{\Omega} |D\chi_{E_{n_m,t}}|\, dx \geq \int_{\Omega} |D\chi_{E_{t}}|\, dx. Thus, we get,
\int_{\Omega}|Du| \geq \int_{-\infty}^{\infty} \int_{\Omega} |D\chi_{E_{t}}|\, dx dt.
This completes the proof!
(By the way, step ii also proves that D\chi_{E_t} \in \mathbf{M}(\Omega, \mathbb{R}^N) for a.e. t\in \mathbb{R}, something that we used in Part I of the proof.)
Let us first recall the lower semicontinuity property of the total variation with respect to the strong convergence in L1(Ω):
Let {un}n∈Ω be a sequence in BV(Ω) strongly converging to some u in L1(Ω) and satisfying |un|BV<∞. Then u∈BV(Ω) and |u|BV(Ω)≤lim The coarea formula is as stated as follows:
Coarea formula: Let u be a given function in BV(\Omega). Then for almost every t in \mathbb{R}, the level set E_t=\{ x\in \Omega \subset \mathbb{R}^N: u(x) > t\} of u is a set of finite perimeter in \Omega, and
\begin{align} &Du=\int_{-\infty}^{\infty} D\chi_{E_t}\, dt,\\ &|u|_{BV(\Omega)}=|Du|(\Omega)=\int_{-\infty}^{\infty} \int_{\Omega}|D\chi_{E_t}|\, dt. \end{align}
The second assertion above is often written in terms of the perimeter of level sets as follows:
\begin{align} |u|_{BV(\Omega)}=\int_{-\infty}^{\infty} Per(E_t, \Omega) dt, \end{align}
where Per(E_t, \Omega)=\mathcal{H}^{N-1}(\Omega \cap \partial E_t) is the perimeter of the level set E_t.
Proof of the coarea formula: We follow the treatment of Variational Analysis in Sobolev and BV spaces by Attouch et al.
Part I: Proof of |Du|(\Omega) \leq \int_{-\infty}^{\infty}\int_{\Omega}|D\chi_{E_t}|\, dt:
Let us assume that D\chi_{E_t} belongs to \mathbf{M}(\Omega, \mathbb{R}^N), the set of all \mathbb{R}^N- valued Borel measures which is the set of all the \sigma- additive set functions \mu:\mathcal{B}(\Omega)\rightarrow \mathbb{R}^N, with \mu(\emptyset)=0. For all t in \mathbb{R} set
\begin{equation} f_t=\left\{ \begin{array}{ll} \chi_{E_t} & \mbox{if } t \geq 0 \\ -\chi_{\Omega \backslash E_t} & \mbox{if } t < 0 \end{array} \right. \end{equation} It is easy to see that u(x)=\int_{-\infty}^{\infty}f_t(x)\,dt for all x\in \Omega. For all \bar{\phi} \in C_c^1(\Omega, \mathbb{R}^N) we have \begin{align*} &\langle Du, \bar{\phi} \rangle \\ &= -\int_{\Omega} u \, \mbox{div} \,\bar{\phi} \, dx \\ & = - \int_{\Omega} \int_{-\infty}^{\infty} f_t \, \mbox{div} \,\bar{\phi} \, dt\,dx\\ &= - \int_{\Omega} \int_{-\infty}^{0} f_t \, \mbox{div} \,\bar{\phi} \, dt\,dx -\int_{\Omega} \int_{0}^{\infty} f_t \, \mbox{div} \,\bar{\phi} \, dt\,dx \\ &= \int_{\Omega} \int_{-\infty}^{0} \chi_{\Omega \backslash E_t} \, \mbox{div} \,\bar{\phi} \, dt\,dx -\int_{\Omega} \int_{0}^{\infty} \chi_{E_t} \, \mbox{div} \,\bar{\phi} \, dt\,dx \\ &=\int_{\Omega} \int_{-\infty}^{0} (1-\chi_{E_t}) \, \mbox{div} \,\bar{\phi} \, dt\,dx -\int_{\Omega} \int_{0}^{\infty} \chi_{E_t} \, \mbox{div} \,\bar{\phi} \, dt\,dx \\ &=\int_{\Omega} \int_{-\infty}^{0} \mbox{div} \,\bar{\phi} \, dt\,dx - \int_{\Omega} \int_{-\infty}^{0} \chi_{E_t} \, \mbox{div} \,\bar{\phi} \, dt\,dx-\int_{\Omega} \int_{0}^{\infty} \chi_{E_t} \, \mbox{div} \,\bar{\phi} \, dt\,dx \\ &= \int_{\Omega} \int_{-\infty}^{0} \mbox{div} \,\bar{\phi} \, dt\,dx-\int_{\Omega} \int_{-\infty}^{\infty} \chi_{E_t} \, \mbox{div} \,\bar{\phi} \, dt\,dx \\ &= 0-\int_{\Omega} \int_{-\infty}^{\infty} \chi_{E_t} \, \mbox{div} \,\bar{\phi} \, dt\,dx \\ &= \int_{-\infty}^{\infty} \langle D\chi_{E_t}, \bar{\phi}\rangle \,dt \end{align*} Hence, Du=\int_{-\infty}^{\infty} D\chi_{E_t}\, dt, and |Du|(\Omega) \leq \int_{-\infty}^{\infty}\int_{\Omega}|D\chi_{E_t}|\, dx dt.
Part II: (coverse) Proof of \int_{-\infty}^{\infty}\int_{\Omega}|D\chi_{E_t}|\, dxdt\leq |Du|(\Omega) :
Step i. We first assume that u belongs to the space \mathcal{A}(\Omega) of piecewise linear and continuous functions in \Omega. By linearity, one can assume that u=a\cdot x+b with a\in \mathbb{R}^N and b\in \mathbb{R}, so that
\begin{align*} &\int_{\Omega} |D\chi_{E_t}|\\ &=\mathcal{H}(\Omega \cap \partial E_t)\\ &=\mathcal{H}(\Omega \cap \{x| a\cdot x +b=t\})\\ &=\int_{\Omega \cap \{x| a\cdot x +b=t\}} d\mathcal{H}^{N-1}(x)\\ &=\int_{\{x| a\cdot x +b=t\}}\chi_{\Omega}(x)\, d\mathcal{H}^{N-1}(x) \end{align*}
Thus we get,
\begin{align*} &\int_{-\infty}^{\infty} \int_{\Omega} |D\chi_{E_t}|\, dx dt\\ &=|a|\mathcal{L}(\Omega)\\ &=\int_{\{x| a\cdot x +b=t\}}\chi_{\Omega}(x)\, d\mathcal{H}^{N-1}(x)\\ &=\int_{\Omega} |Du| \end{align*}
Hence we have,
\begin{align*} \int_{-\infty}^{\infty} \int_{\Omega} |D\chi_{E_t}|\, dx dt=|Du|(\Omega) \dots \mbox{ if } u=a\cdot x+b. \end{align*}
Step ii. Now we prove that \int_{-\infty}^{\infty} \int_{\Omega} |D\chi_{E_t}|\, dx dt\leq|Du|(\Omega) for all u\in BV(\Omega). We know that the space of piecewise linear and continuous functions \mathcal{A}(\Omega) is dense in W^{1, 1}(\Omega) when equipped with the strong topology. We also know that the space C^{\infty}\cap W^{1, 1}(\Omega) is dense in BV when equipped with the intermediate convergence. Thus, there exists a sequence \{u_n\}_{n\in \mathbb{N}} \in \mathcal{A}(\Omega) such that u_n\rightharpoonup u for the intermediate convergence. For each of the functions u_n we set E_{n, t}:=\{x\in \Omega: u_n(x) > t\}. Due to the intermediate convergence, we have
\begin{align*} &\int_{\Omega}|Du|=\lim_{n\rightarrow \infty}\int_{\Omega}|Du_n| \dots \mbox{the intermediate convergence}\\ &=\lim_{n\rightarrow \infty} \int_{-\infty}^{\infty} \int_{\Omega} |D\chi_{E_{n,t}}|\, dx dt \dots \mbox{from step i}\\ &\geq \int_{-\infty}^{\infty} \lim\inf_{n\rightarrow \infty} \int_{\Omega} |D\chi_{E_{n,t}}|\, dx dt \dots \mbox{Fatou's lemma}\\ \end{align*}
From the intermediate convergence we also have u_n\rightarrow u in L^1(\Omega). i.e.
\begin{align*} \int_{\Omega} |u_n-u|=\int_{\Omega} \int_{-\infty}^{\infty} |\chi_{E_{n,t}}-\chi_{E_t}|\ dt dx = \int_{-\infty}^{\infty} \Big(\int_{\Omega} |\chi_{E_{n,t}}-\chi_{E_t}|\ dx \Big) dt=0. \end{align*}
Thus, for a subsequence E_{n_m, t}, and for almost all t \in \mathbb{R}, \chi_{E_{n_m, t}}\rightarrow \chi_{E_{n, t}} strongly in L^1(\Omega). The lower semicontinuity of the total variation with respect to the strong convergence in L^1(\Omega) we get \lim\inf_{n\rightarrow \infty} \int_{\Omega} |D\chi_{E_{n_m,t}}|\, dx \geq \int_{\Omega} |D\chi_{E_{t}}|\, dx. Thus, we get,
\int_{\Omega}|Du| \geq \int_{-\infty}^{\infty} \int_{\Omega} |D\chi_{E_{t}}|\, dx dt.
This completes the proof!
(By the way, step ii also proves that D\chi_{E_t} \in \mathbf{M}(\Omega, \mathbb{R}^N) for a.e. t\in \mathbb{R}, something that we used in Part I of the proof.)