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In the book Elliptic partial differential equations of second order written by Gilbarg and Trudinger, I saw the following sentence on page 17 in section 2.4 Green’s Representation:

As a prelude to existence considerations we derive now some further consequences of the divergence theorem, namely, Green identities. Let
$Omega$ be a domain for which the divergence theorem holds and let $u$ and $v$ be $C^2(barOmega)$ functions.

It is well known that the divergence theorem holds when $Omega$ is a bounded domain with $C^1$ boundary.

Are there any other domain than a bounded one with $C^1$ boundary for which the theorem holds?

I would be grateful if you could give any comment for this question.

As suggested by fourierwho, perhaps the most the natural domains for which the divergence (also called Gauss-Green) theorem holds are the sets of finite perimeter, i.e. Caccioppoli sets, so lets precisely see why.

Definition 1 ([1], §3.3 p. 143). Let $Omega$ a Lebesgue measurable set in $mathbb{R}^n$. For any open subset $Gsubseteqmathbb{R}^n$ the perimeter of $Omega$ in $G$, denoted as $P(Omega,G)$, is the variation of $chi_Omega$ in $Omega$ i.e.
$$
begin{split}
P(Omega,G)&=supleft{int_Omega nablacdotvarphi,mathrm{d}x,:,varphiin [C_c^1(G)]^n, |varphi|_inftyleq1right}\
& =| nabla chi_{Omegacap G}|=TV(Omega,G)
end{split}tag{1}label{1}
$$
where $[C_c^1(G)]^n$ is the set of compact support continuously differentiable vector functions in $G$ and $TV$ is the total variation of the set function $nabla chi_{Omegacap G}$.

The set $Omega$ is a set of finite perimeter (a Caccioppoli set) in $Gsubseteqmathbb{R}^n$ if $P(Omega,G)<infty$.

• If $G=mathbb{R}^n$, then we can speak of perimeter of $Omega$ tout court, and denote it as $P(Omega)$.


• If $P(Omega,G^prime)<infty$ for every bounded open set $G^primeSubsetmathbb{R}^n$, $Omega$ is a set of locally finite perimeter.

Why definition eqref{1} implies a natural extension of the classical divergence (Gauss-Green) theorem? For simplicity lets consider sets of finite perimeter: $P(Omega)<infty$ implies that the distributional derivative of the characteristic function of $Omega$ is a vector Radon measure whose total variation is the perimeter defined by eqref{1}, i.e.
$$
nablachi_Omega(varphi)=int_Omeganablacdotvarphi,mathrm{d}x=int_Omega varphi,mathrm{d}nablachi_Omegaquad
varphiin [C_c^1(mathbb{R}^n)]^ntag{2}label{2}
$$
Now the support in the sense of distributions of $nablachi_Omega$ is $subseteqpartialOmega$ ([2], §1.8 pp. 6-7): to see this note that if $xnotinpartialOmega$, it should belong to an open set $ASubsetmathbb{R}^n$ such that either $ASubsetOmega$ or $ASubsetmathbb{R}^nsetminusOmega$:

1. if $ASubsetOmega$, then $chi_Omega=1$ on $A$ and hence eqref{2} is equal to zero for each $varphiin [C_c^1(A)]^n$
   


2. if $ASubsetmathbb{R}^nsetminusOmega$, then $chi_Omega=0$ on $A$ and hence eqref{2} is again equal to zero for each $varphiin [C_c^1(A)]^n$
   

Also, as a general corollary of (one of the versions of) Radon-Nikodym theorem ([1], §1.1 p. 14) we can apply a polar decomposition to $nablachi_Omega$ and obtain
$$
nablachi_Omega=nu_Omega|nablachi_Omega|_{TV}tag{3}label{3}
$$
where $nu_Omega$ is a $L^1$ function taking values on the unit sphere $mathbf{S}^{n-1}Subsetmathbb{R}^n$, and rewriting eqref{2} by using eqref{3} we obtain the sought for general divergence (Gauss-Green) theorem
$$
int_Omega!nablacdot varphi, mathrm{d}x =int_{partialOmega} !varphi,cdotnu_Omega, mathrm{d}|nablachi_Omega|quadforallvarphiin [C_c^1(mathbb{R}^n)]^ntag{4}label{4}
$$
Note that this result is an almost direct consequence of definition 1 above, with minimal differentiability requirement imposed on the data $varphi$: it seems to follow directly from the given definition of perimeter eqref{2} through the application of general (apparently unrelated) theorems on the structure of measures and distributions, and in this sense it is the most "natural form" of the divergence/Gauss-Green theorem.

Further notes

• When $Omega$ is a smooth bounded domain, eqref{4} "reduces" the standard divergence (Gauss-Green) theorem.


• There are more general statement of the theorem, relaxing further both the conditions on $Omega$ and on $varphi$: however they require further, more technical, assumptions and therefore are in some sense "less natural".


• The notion of perimeter eqref{1} was introduced by Ennio De Giorgi by using a gaussian kernel in order to "mollify" the set $Omega$. By using De Giorgi's ideas, Calogero Vinti and Emilio Bajada further generalized the notion of perimeter: however I am not aware of a corresponding generalization of the divergence theorem.

[1] Ambrosio, Luigi; Fusco, Nicola; Pallara, Diego (2000), Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs, New York and Oxford: The Clarendon Press/Oxford University Press, New York, pp. xviii+434, ISBN 0-19-850245-1, MR1857292, Zbl 0957.49001.

[2] Giusti, Enrico (1984), Minimal surfaces and functions of bounded variations, Monographs in Mathematics, 80, Basel–Boston–Stuttgart: Birkhäuser Verlag, pp. XII+240, ISBN 978-0-8176-3153-6, MR 0775682, Zbl 0545.49018