# Domains for which the divergence theorem holds

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 $C^2(\bar\Omega)$ 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.

• My (minimal) experience with such questions is you have to dive into geometric measure theory a bit. Francesco Maggi touches on exactly this question in his book on geometric measure theory, titled Sets of Finite Perimeter and Geometric Variational Problems. – fourierwho Jun 29 '18 at 2:31
• Thanks for your reply. I will check the book. – 04170706 Jun 29 '18 at 2:33
• Now that I have the book in front of me you will want to see the synopsis of Part 2 in Maggi's book. Also see: mathoverflow.net/questions/253488/… – fourierwho Jun 29 '18 at 5:50

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 let's precisely see why.

Definition 1 ([1], §3.3 p. 143). Let $$\Omega$$ a Lebesgue measurable set in $$\mathbb{R}^n$$. For any open subset $$G\subseteq\mathbb{R}^n$$ the perimeter of $$\Omega$$ in $$G$$, denoted as $$P(\Omega,G)$$, is the variation of $$\chi_\Omega$$ in $$\Omega$$ i.e. $$\begin{split} P(\Omega,G)&=\sup\left\{\int_\Omega \nabla\cdot\varphi\,\mathrm{d}x\,:\,\varphi\in [C_c^1(G)]^n, \|\varphi\|_\infty\leq1\right\}\\ & =| \nabla \chi_{\Omega\cap G}|=TV(\Omega,G) \end{split}\tag{1}\label{1}$$ 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 $$\nabla \chi_{\Omega\cap G}$$.

The set $$\Omega$$ is a set of finite perimeter (a Caccioppoli set) in $$G\subseteq\mathbb{R}^n$$ 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 $$G^\prime\Subset\mathbb{R}^n$$, $$\Omega$$ 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. $$\nabla\chi_\Omega(\varphi)=\int_\Omega\nabla\cdot\varphi\,\mathrm{d}x=\int_\Omega \varphi\,\mathrm{d}\nabla\chi_\Omega\quad \varphi\in [C_c^1(\mathbb{R}^n)]^n\tag{2}\label{2}$$ Now the support in the sense of distributions of $$\nabla\chi_\Omega$$ is $$\subseteq\partial\Omega$$ ([2], §1.8 pp. 6-7): to see this note that if $$x\notin\partial\Omega$$, it should belong to an open set $$A\Subset\mathbb{R}^n$$ such that either $$A\Subset\Omega$$ or $$A\Subset\mathbb{R}^n\setminus\Omega$$:

1. if $$A\Subset\Omega$$, then $$\chi_\Omega=1$$ on $$A$$ and hence \eqref{2} is equal to zero for each $$\varphi\in [C_c^1(A)]^n$$
2. if $$A\Subset\mathbb{R}^n\setminus\Omega$$, then $$\chi_\Omega=0$$ on $$A$$ and hence \eqref{2} is again equal to zero for each $$\varphi\in [C_c^1(A)]^n$$

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 $$\nabla\chi_\Omega$$ and obtain $$\nabla\chi_\Omega=\nu_\Omega|\nabla\chi_\Omega|_{TV}\equiv\nu_\Omega|\nabla\chi_\Omega|\tag{3}\label{3}$$ where $$\nu_\Omega$$ is a $$L^1$$ function taking values on the unit sphere $$\mathbf{S}^{n-1}\Subset\mathbb{R}^n$$, and rewriting \eqref{2} by using \eqref{3} we obtain the sought for general divergence (Gauss-Green) theorem $$\int_\Omega\!\nabla\cdot \varphi\, \mathrm{d}x =\int_{\partial\Omega} \!\varphi\,\cdot\nu_\Omega\, \mathrm{d}|\nabla\chi_\Omega|\quad\forall\varphi\in [C_c^1(\mathbb{R}^n)]^n\tag{4}\label{4}$$ 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