Domains for which the divergence theorem holds

Question

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.

Answer 1

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