The following is about exercise number 11 of section 5.10 of the text Lawrence C. Evans - Partial Differential Equations_ Second Edition. This is an exercise that has been posed by other people on this page, but I have noticed that the demos that are given lack some details, rather they are about sketches.Although I still have some concerns, I will try to demonstrate this exercise in detail, while asking for your cooperation to clarify my concerns in the lines highlighted in bold below:
Exercise. Suppose $U\subseteq\mathbb{R}^n$ is open and connected, and $u\in W^{1,p}(U)$ satisfies $$Du=0\quad\mbox{a.e in }U. $$ Prove $U$ is constant a.e in $U$.
Proof: Let $\epsilon>0$. Consider the function $$u^\epsilon=\eta_\epsilon *u,$$ by Theorem 7 of appendix C.5 of the book Lawrence C. Evans - Partial Differential Equations_ Second Edition (properties of mollifiers), $u^\epsilon$ is $C^\infty$ in the set $$U_\epsilon=\{\mathbf{x}\in U:\mbox{d}{(\mathbf{x},\partial U)}>\epsilon\}.$$ Note that $U_\epsilon$ is open, since if $y\in U_\epsilon$, that is, $$y\in U\quad\mbox{and}\quad\mbox{d}(\mathbf{y}, \partial U)> \epsilon,$$ By doing $r=\mbox{d}(\mathbf{y},\partial U)-\epsilon>0$, it's easy to see that $B(\mathbf{y},r)\subset U_\epsilon$. In fact, let $\mathbf{w}\in B(\mathbf{y},r) $, then \begin{align*} \mbox{d}(\mathbf{y},\mathbf{w}) <r & \Longleftrightarrow \mbox{d}(\mathbf{y},\mathbf{w})<\mbox{d}(\mathbf{y},\partial U)-\epsilon\\ & \Longleftrightarrow \mbox{d}(\mathbf{y},\mathbf{w}) <\mbox{d}(\mathbf{y},\mathbf{w})+\mbox{d}(\mathbf{w},\partial U)-\epsilon \\ & \Longleftrightarrow \mbox{d}(\mathbf{w},\partial U)>\epsilon.\end{align*} So $\mathbf{w}\in U_\epsilon$, and $ U_\epsilon$ is open. On the other hand, by the proof of Theorem 1 from section 5.3 of the book Lawrence C. Evans - Partial Differential Equations_ Second Edition (local approximation by smooth functions), we have that \begin{align*} Du^\epsilon=\eta_\epsilon * Du\quad\mbox{on } U_\epsilon, \end{align*} where $\eta_\epsilon $ is the standard mollifier. Since $Du=0$ a.e in $U $, it follows from the previous line that $Du^\epsilon=0 $ a.e in $U_\epsilon $, O well, $u$ is constant in every connected component of $U_\epsilon $ (is it correct to state this? if yes, why?). Now let $\mathbf{x},\mathbf{y}\in U $, since $U$ is open and connected, so this is connected by paths, there is a path $\Gamma\subset U$ that connects to $\mathbf{x} $ with $\mathbf{y}$. Let $\delta=\min_{\mathbf{z}\in\Gamma} \mbox{d}(\mathbf{z},\partial U) $, that is, $\delta$ is the minimum distance between the points of $\Gamma $ and the boundary of $U$. For the choice of $\delta$, for all $\epsilon<\delta $, the path $\Gamma$ is contained in $ U_\epsilon $. Therefore, given any two points of $U$, for all $\epsilon<\delta $, these can be connected by a path contained in $ U_\epsilon $, in other words $U_\epsilon $ is connected by paths, thus $U_\epsilon $ is connected. It follows that $\mathbf{x},\mathbf{y} $ are in the same connected component $U_\epsilon $, so $ u^\epsilon(\mathbf{x})=u^\epsilon(\mathbf{y})$ is constant (it is correct to state that if $Du^\epsilon=0$ a.e, then $u^\epsilon$ is constant in each open connected $U_\epsilon$? why?). Finally, by hypothesis $u\in W^{1,p} (U) $, therefore, by Theorem 7 of appendix C.5 of the book Lawrence C. Evans - Partial Differential Equations_ Second Edition (properties of mollifiers) implies that $u^\epsilon\longrightarrow u $ a.e in $U$ as $\epsilon \to 0 $, therefore $ u $ is constant a.e in $ U $ (Why if $u^\epsilon$ is constant a.e in every connected component $U_\epsilon$ and $u^\epsilon\longrightarrow u$ a.e in $U$, then $u$ is constant a.e in $U$?).
Observations: This demostration uses some results and notation from the text Lawrence C. Evans - Partial Differential Equations_ Second Edition.
Definitions: (i) Define $\eta\in C^\infty(\mathbb{R}^n)$ by $$\eta(x):=\left\{\begin{array}{ll}C\exp\left(\frac{1}{|x|^2-1}\right), & \mbox{if }|x|<1\\0, & \mbox{if } |x|\geq 1\end{array}\right.$$ the constant $C>0$ selected so that $\displaystyle{\int_{\mathbb{R}^n}}\eta\,dx=1$.
(ii) For each $\epsilon>0$, set $$\eta_\epsilon(x):=\frac{1}{\epsilon^n}\eta\left(\frac{x}{\epsilon}\right).$$
We call $\eta$ standard mollifier. The functions $\eta_\epsilon$ are $C^\infty$ and satisfy $$\int_{\mathbb{R}^n}\eta_\epsilon\,dx=1, \mbox{spt}(\eta_\epsilon)\subset B(0,\epsilon).$$ spt denotes support.
(iii) $d$ denotes the distance function associated with the metric of the space in question.
(iv) $*$ denotes convolution.
(v) $W^{1,p}$ denotes Sobolev space.
The reason for the different questions raised above is that I have researched and found different solutions for this exercise, more like demonstration sketches, but these are not very detailed. In the end, after understanding and solving most of the above demo on my own, I was left with the concerns raised.