Integration by parts for smooth, compactly supported functions

On the Wiki article for the Sobolev Space, it says that for any u∈Ck(Ω)u\in C^k(\Omega) and for any compactly supported φ∈C∞c(Ω)\varphi\in C^{\infty}_c(\Omega), integration by parts yields
∫ΩuDαφdx=(−1)|α|∫ΩφDαudx.\int_{\Omega}u\mathrm{D}^{\alpha}\varphi\mathrm{d}x = (-1)^{|\alpha|}\int_{\Omega}\varphi\mathrm{D}^{\alpha}u\mathrm{d}x.

For example, if α=1\alpha=1, we get

∫Ωu∂φ∂xdx=uφ−∫Ωφ∂u∂xdx,\int_{\Omega}u\frac{\partial \varphi}{\partial x} \mathrm{d}x = u\varphi -\int_{\Omega}\varphi\frac{\partial u}{\partial x}\mathrm{d}x,

which has an extra term compared with the formula in the Wiki article.

How does their formula follow from integration by parts?




Since φ∈Ckc(Ω)\varphi \in C_c^k(\Omega), the boundary terms from the integration by parts all vanish.
– Daniel Fischer♦
2 days ago



Got it, thank you!
– man_in_green_shirt
2 days ago