Let $W^{1,p}(U)$ be the Sobolev space. Suppose that U is connected bounded domain in Rn and $u∈W^{1,p}(U)$ satisfies $Du=0$ a.e. in U. Prove that u is constant a.e. in U.
I have worked out the method which use $u_ε=u∗ρ_ε, Du_ε=0$ and convergence.
Could you help me to come up with another way to prove it?
The statement that $u$ has weak derivative zero is equivalent to the statement that for all smooth functions $\varphi:U \to \Bbb R$, we have $$ \int_U u\,\nabla \varphi = 0. $$ Now, it suffices to note that for any characteristic function $\Bbb 1_S$ associated with a closed $S \subset U$, a (bounded) sequence $\varphi_k$ can be selected such that the first component of $\nabla \varphi_k$ approaches $\Bbb 1_S$.