Compact support of derivatives of $u$ in weak sense

Question

I would like to know if let $u:\mathbb{R}^n\to \mathbb{R}$ be a function in $W^{k,p}(\mathbb{R}^n)$ such that $u$ has compact support in $\mathbb{R}^n$ then, for each $|\alpha|\le k,$ $D^\alpha u$ has compact support in $\mathbb{R}^n$?

Because since $D^\alpha u\in L^1_{loc}(\mathbb{R}^n)$ by definition of $W^{k,p}(\mathbb{R}^n)$, then I conclude that $D^\alpha u\in L^1(\mathbb{R}^n)$, if $p=2$ by Hölder Inequality.

Can someone help me? Thanks.

Answer 1

Let $K$ be the support of $u$. For any test function $\phi$ with support in $\mathbb{R}^n\setminus K$ $$ \int_{\mathbb{R}^n}D^\alpha u\,\phi\,dx=(-1)^{|\alpha|}\int_{\mathbb{R}^n}u\,D^\alpha\phi\,dx=0 $$ because the support of $u$ and $\phi$ are disjoint. This means that $D^\alpha u$ is supported no $K$.