Evans PDE Chapter 5 Problem 11: Does $Du=0$ a.e. implie $u=c$ a.e.?

Solution 1:

For a Dirac-family $(\varphi_\varepsilon)_\varepsilon$ with $\mathrm{supp}\, \varphi_\varepsilon \subseteq \overline{B_\varepsilon(0)}$ you have $D(u*\varphi_\varepsilon) = (Du)*\varphi_\varepsilon = 0$ on all open sets $O \subseteq U$ with $\mathrm{dist}(O,\partial U) > \delta$, with $\delta > \varepsilon$, so $u * \varphi_\varepsilon$ is constant on all connected ones. Also, $u*\varphi_\varepsilon \to u$ in $L^p$ for $p \geq 1$, in particular $u * \varphi_\varepsilon \to u$ almost everywhere for some subsequence. But this means $u$ is constant almost everywhere on all $O$ with $\mathrm{dist}(O,\partial U) > 0$, so it is constant almost everywhere on $U$.