In PDE books, it is well known fact that, for any vector field $w$ in $C^1(\overline \Omega),$ $$ \int_\Omega div~w~ dx=\int_{\partial \Omega} w\cdot \nu~ ds, $$ wnere $ds$ is the $(n-1)$-dimensional area element in $\partial \Omega.$

Or if $u \in C^2(\overline \Omega)$, $$ \int_\Omega \Delta u~ dx=\int_{\partial \Omega} \frac{\partial u}{\partial \nu}~ ds. $$ In this case, the functions under consideration are good enough so that both Lebesgue and Riemann integral exist. Am I right? When we consider weak solutions to some differential equation, it is not smooth enough. Thus I think we should use Lebesgue integral.

(1) I was wondering if the surface integrals above are Lebesgue integral or not.

(2) In general, when we see the integrals, how can we know if it is Riemann or Lebesgue integral?

Pleas let me know if you have any comment for this question. Thanks in advance!

  • $\begingroup$ Maybe I'm not right, but I almost sure that almost everytime books talk about Lebesgue integrals. $\endgroup$ – Gonzalo Benavides Jul 2 '18 at 1:08

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