Ufyukyu

6 2 $begingroup$ I am looking for a "counter-example" to integration by parts of the following type: $Omega subseteq mathbb R^n$ is an open, bounded, connected domain with smooth boundary. $u,v:bar Omega to mathbb R$ are real-valued functions, where $u$ is smooth and compactly supported in $Omega$ , and $v$ is smooth on an open subset of $bar Omega$ whose complement is a closed subset of measure zero. I want $int_{Omega}(partial_iu)v neq -int_{Omega}u(partial_iv)$ , i.e. to demonstrate failure of integration by parts. Edit: Does the answer change if we assume in addition that $v$ is continuous everywhere on $bar Omega$ ?. BigbearZzz gave here an example with a non-continuous $v$ . If $v$ was smooth on all $bar Omega$ , then integration by parts would work. The point is that I am limiting ...