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)≤liminfn→∞|un|BV(Ω). The coarea formula is as stated as follows:
Coarea formula: Let u be a given function in BV(Ω). Then for almost every t in R, the level set Et={x∈Ω⊂RN:u(x)>t} of u is a set of finite perimeter in Ω, and
Du=∫∞−∞DχEtdt,|u|BV(Ω)=|Du|(Ω)=∫∞−∞∫Ω|DχEt|dt.
The second assertion above is often written in terms of the perimeter of level sets as follows:
|u|BV(Ω)=∫∞−∞Per(Et,Ω)dt,
where Per(Et,Ω)=HN−1(Ω∩∂Et) is the perimeter of the level set Et.
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|(Ω)≤∫∞−∞∫Ω|DχEt|dt:
Let us assume that DχEt belongs to M(Ω,RN), the set of all RN− valued Borel measures which is the set of all the σ− additive set functions μ:B(Ω)→RN, with μ(∅)=0. For all t in R set
ft={χEtif t≥0−χΩ∖Etif t<0 It is easy to see that u(x)=∫∞−∞ft(x)dt for all x∈Ω. For all ˉϕ∈C1c(Ω,RN) we have ⟨Du,ˉϕ⟩=−∫Ωudivˉϕdx=−∫Ω∫∞−∞ftdivˉϕdtdx=−∫Ω∫0−∞ftdivˉϕdtdx−∫Ω∫∞0ftdivˉϕdtdx=∫Ω∫0−∞χΩ∖Etdivˉϕdtdx−∫Ω∫∞0χEtdivˉϕdtdx=∫Ω∫0−∞(1−χEt)divˉϕdtdx−∫Ω∫∞0χEtdivˉϕdtdx=∫Ω∫0−∞divˉϕdtdx−∫Ω∫0−∞χEtdivˉϕdtdx−∫Ω∫∞0χEtdivˉϕdtdx=∫Ω∫0−∞divˉϕdtdx−∫Ω∫∞−∞χEtdivˉϕdtdx=0−∫Ω∫∞−∞χEtdivˉϕdtdx=∫∞−∞⟨DχEt,ˉϕ⟩dt Hence, Du=∫∞−∞DχEtdt, and |Du|(Ω)≤∫∞−∞∫Ω|DχEt|dxdt.
Part II: (coverse) Proof of ∫∞−∞∫Ω|DχEt|dxdt≤|Du|(Ω):
Step i. We first assume that u belongs to the space A(Ω) of piecewise linear and continuous functions in Ω. By linearity, one can assume that u=a⋅x+b with a∈RN and b∈R, so that
∫Ω|DχEt|=H(Ω∩∂Et)=H(Ω∩{x|a⋅x+b=t})=∫Ω∩{x|a⋅x+b=t}dHN−1(x)=∫{x|a⋅x+b=t}χΩ(x)dHN−1(x)
Thus we get,
∫∞−∞∫Ω|DχEt|dxdt=|a|L(Ω)=∫{x|a⋅x+b=t}χΩ(x)dHN−1(x)=∫Ω|Du|
Hence we have,
∫∞−∞∫Ω|DχEt|dxdt=|Du|(Ω)… if u=a⋅x+b.
Step ii. Now we prove that ∫∞−∞∫Ω|DχEt|dxdt≤|Du|(Ω) for all u∈BV(Ω). We know that the space of piecewise linear and continuous functions A(Ω) is dense in W1,1(Ω) when equipped with the strong topology. We also know that the space C∞∩W1,1(Ω) is dense in BV when equipped with the intermediate convergence. Thus, there exists a sequence {un}n∈N∈A(Ω) such that un⇀u for the intermediate convergence. For each of the functions un we set En,t:={x∈Ω:un(x)>t}. Due to the intermediate convergence, we have
∫Ω|Du|=limn→∞∫Ω|Dun|…the intermediate convergence=limn→∞∫∞−∞∫Ω|DχEn,t|dxdt…from step i≥∫∞−∞liminfn→∞∫Ω|DχEn,t|dxdt…Fatou's lemma
From the intermediate convergence we also have un→u in L1(Ω). i.e.
∫Ω|un−u|=∫Ω∫∞−∞|χEn,t−χEt| dtdx=∫∞−∞(∫Ω|χEn,t−χEt| dx)dt=0.
Thus, for a subsequence Enm,t, and for almost all t∈R, χEnm,t→χEn,t strongly in L1(Ω). The lower semicontinuity of the total variation with respect to the strong convergence in L1(Ω) we get liminfn→∞∫Ω|DχEnm,t|dx≥∫Ω|DχEt|dx. Thus, we get,
∫Ω|Du|≥∫∞−∞∫Ω|DχEt|dxdt.
This completes the proof!
(By the way, step ii also proves that DχEt∈M(Ω,RN) for a.e. t∈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)≤liminfn→∞|un|BV(Ω). The coarea formula is as stated as follows:
Coarea formula: Let u be a given function in BV(Ω). Then for almost every t in R, the level set Et={x∈Ω⊂RN:u(x)>t} of u is a set of finite perimeter in Ω, and
Du=∫∞−∞DχEtdt,|u|BV(Ω)=|Du|(Ω)=∫∞−∞∫Ω|DχEt|dt.
The second assertion above is often written in terms of the perimeter of level sets as follows:
|u|BV(Ω)=∫∞−∞Per(Et,Ω)dt,
where Per(Et,Ω)=HN−1(Ω∩∂Et) is the perimeter of the level set Et.
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|(Ω)≤∫∞−∞∫Ω|DχEt|dt:
Let us assume that DχEt belongs to M(Ω,RN), the set of all RN− valued Borel measures which is the set of all the σ− additive set functions μ:B(Ω)→RN, with μ(∅)=0. For all t in R set
ft={χEtif t≥0−χΩ∖Etif t<0 It is easy to see that u(x)=∫∞−∞ft(x)dt for all x∈Ω. For all ˉϕ∈C1c(Ω,RN) we have ⟨Du,ˉϕ⟩=−∫Ωudivˉϕdx=−∫Ω∫∞−∞ftdivˉϕdtdx=−∫Ω∫0−∞ftdivˉϕdtdx−∫Ω∫∞0ftdivˉϕdtdx=∫Ω∫0−∞χΩ∖Etdivˉϕdtdx−∫Ω∫∞0χEtdivˉϕdtdx=∫Ω∫0−∞(1−χEt)divˉϕdtdx−∫Ω∫∞0χEtdivˉϕdtdx=∫Ω∫0−∞divˉϕdtdx−∫Ω∫0−∞χEtdivˉϕdtdx−∫Ω∫∞0χEtdivˉϕdtdx=∫Ω∫0−∞divˉϕdtdx−∫Ω∫∞−∞χEtdivˉϕdtdx=0−∫Ω∫∞−∞χEtdivˉϕdtdx=∫∞−∞⟨DχEt,ˉϕ⟩dt Hence, Du=∫∞−∞DχEtdt, and |Du|(Ω)≤∫∞−∞∫Ω|DχEt|dxdt.
Part II: (coverse) Proof of ∫∞−∞∫Ω|DχEt|dxdt≤|Du|(Ω):
Step i. We first assume that u belongs to the space A(Ω) of piecewise linear and continuous functions in Ω. By linearity, one can assume that u=a⋅x+b with a∈RN and b∈R, so that
∫Ω|DχEt|=H(Ω∩∂Et)=H(Ω∩{x|a⋅x+b=t})=∫Ω∩{x|a⋅x+b=t}dHN−1(x)=∫{x|a⋅x+b=t}χΩ(x)dHN−1(x)
Thus we get,
∫∞−∞∫Ω|DχEt|dxdt=|a|L(Ω)=∫{x|a⋅x+b=t}χΩ(x)dHN−1(x)=∫Ω|Du|
Hence we have,
∫∞−∞∫Ω|DχEt|dxdt=|Du|(Ω)… if u=a⋅x+b.
Step ii. Now we prove that ∫∞−∞∫Ω|DχEt|dxdt≤|Du|(Ω) for all u∈BV(Ω). We know that the space of piecewise linear and continuous functions A(Ω) is dense in W1,1(Ω) when equipped with the strong topology. We also know that the space C∞∩W1,1(Ω) is dense in BV when equipped with the intermediate convergence. Thus, there exists a sequence {un}n∈N∈A(Ω) such that un⇀u for the intermediate convergence. For each of the functions un we set En,t:={x∈Ω:un(x)>t}. Due to the intermediate convergence, we have
∫Ω|Du|=limn→∞∫Ω|Dun|…the intermediate convergence=limn→∞∫∞−∞∫Ω|DχEn,t|dxdt…from step i≥∫∞−∞liminfn→∞∫Ω|DχEn,t|dxdt…Fatou's lemma
From the intermediate convergence we also have un→u in L1(Ω). i.e.
∫Ω|un−u|=∫Ω∫∞−∞|χEn,t−χEt| dtdx=∫∞−∞(∫Ω|χEn,t−χEt| dx)dt=0.
Thus, for a subsequence Enm,t, and for almost all t∈R, χEnm,t→χEn,t strongly in L1(Ω). The lower semicontinuity of the total variation with respect to the strong convergence in L1(Ω) we get liminfn→∞∫Ω|DχEnm,t|dx≥∫Ω|DχEt|dx. Thus, we get,
∫Ω|Du|≥∫∞−∞∫Ω|DχEt|dxdt.
This completes the proof!
(By the way, step ii also proves that DχEt∈M(Ω,RN) for a.e. t∈R, something that we used in Part I of the proof.)