Evans PDE: $Du =0$ a.e. implies that $u$ is a constant a.e.

Question

I am aware that this question has been asked on Math SE. I'm curious why my proposed solution was marked incorrect by the grader in my partial differential equations course.

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Original question in Chapter $5$ of Evans (Exercise $5.10.11$)

Let $U$ be an open subset of $\mathbb R^n$ with a smooth boundary $\partial U$. Suppose $U$ is connected and $u\in W^{1,p}(U)$ satisfies $$Du=0 \quad \text{a.e. in $U$}.$$ Prove $u$ is constant a.e. in $U$.

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Proposed solution:

Let $\varepsilon > 0$. Then consider the smooth functions $$u_{\varepsilon}= \eta_{\varepsilon}\ast u \in C^{\infty}(U_{\varepsilon}),$$ where $U_{\varepsilon}=\{x\in U: d(x,\partial U) > \varepsilon\}$. By Theorem 5.3.1 in Evans, we have $$D_{x_i}(u_{\varepsilon})=\eta_{\varepsilon}\ast D_{x_i}u.$$ Therefore, by assumption, $D_{x_i}u_{\varepsilon}=0$ a.e. in $U_{\varepsilon}$. So $u_{\varepsilon}$ is constant on each connected subset of $U_{\varepsilon}$.

Next, let $x,y\in U$. Since $U$ is connected, there exists a polygonal path $\Gamma\subseteq U$ which connects $x$ and $y$.

Figure $1$: If $U$ is connected then its subdomain $U_{\varepsilon}$ may not be connected. However, any two points $x,y\in U$ can be connected by a polygonal path $\Gamma$ remaining inside $U$. So, if $\varepsilon >0$ is sufficiently small, then $x$ and $y$ belong to the same connected component of $U_{\varepsilon}$.

Let $\delta=\underset{z\in\Gamma}{\min} ~d(z,\partial U)$ be the minimum distance of points in $\Gamma$ to the boundary of $U$. Then for every $\varepsilon < \delta$ the whole polygonal curve $\Gamma$ is in $U_{\varepsilon}$. So $x,y$ lie in the same connected component of $U_{\varepsilon}$. Hence $u_{\varepsilon}(x)=u_{\varepsilon}(y)\equiv \text{const}.$

As $u\in W^{1,p}(U)$, Theorem 7 in Appendix C of Evans tells us that $$u_{\varepsilon}\overset{\varepsilon \to0}{\longrightarrow } u ~\text{a.e. in $U$}.$$

Thus, $u$ is a constant a.e. in $U$.

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The grader marked my answer as incorrect because connected does not imply path connected. While this is certainly true, I don't think that it destroys the analysis in this context.

Is this analysis incorrect? Could you fix the analysis by a similar argument?