In order to use the integration by parts formula(or more generally the divergence theorem) for functions of several variables $$\int_{\Omega} \nabla u\cdot v d \Omega = \int_{\partial \Omega}(u(v \cdot \nu))d \Omega - \int_{\Omega}u\nabla \cdot v d\Omega$$

is it required that $\Omega$ is compact or at least just bounded subset of $\mathbb{R}^{n}$? Is it required that $\partial \Omega$ is at least Lipschitz continuous? If not, are there any restrictions on $\Omega$ and $\partial \Omega$?

This question was inspired by the wiki entry 'integration by parts' and 'divergence theorem'. Thanks for any assistance.


1 Answer 1


Usually, $\Omega$ is an open subset of $\mathbb R^n$. It does not have to be bounded, but maybe we want it to be. To define the normal vector, we may want the boundary to be $C^1$ smooth. Or at least Lipschitz smooth. But not necessarily, another condition may be imposed. In general, $\Omega$ need not be open. And the gradients -- maybe they have to be defined in $\Omega$ and be continuous. Or not. Depends on whether Sobolev spaces were developed.

My point is: "the divergence theorem" is a generic name for results that share some spirit but differ in details. There may not be "the most general version" of the theorem because when allowing worse sets of integration, one may need better behavior of functions, and vice versa. Opening three different books on real analysis, you'll likely find three different versions of the theorem. Here is one, from Evans and Gariepy; it's quite permissive as far as the set is concerned.

Theorem. Let $E\subset \mathbb R^n$ have locally finite perimeter. Then for $\mathcal H^{n-1}$-a.e. $x\in\partial_*E$ (the measure-theoretic boundary of $E$) there is a unique measure theoretic unit outer normal $\nu(x)$ such that $$\int_E \operatorname{div} \varphi \,dx = \int_{\partial_* E} \varphi\cdot \nu \,d\mathcal H^{n-1}$$ for all $\varphi\in C^1_c(\mathbb R^n;\mathbb R^n)$.


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