Let $W^{1,p}(U)$ be the Sobolev space, where $U$ is a connected   bounded domain in $\mathbb{R}^n$ and $u\in W^{1,p}(U)$ satisfying   $Du=0$ a.e. in $U$. Then $u$ is constant a.e. in $U$.   I don't know how to prove this. Especially, I don't know how to use "connected".  Please guide me.
Let $W^{1,p}(U)$ be the Sobolev space. Suppose that $U$ is connected bounded domain in $\mathbb{R}^n$ and $u \in W^{1,p}(U)$ satisfies $Du=0$ a.e. in $U$. How can I prove that $u$ is constant a.e. in $U$?
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