Zero weak gradient gives componentwise constants

Apply the cutoff lemma to $K=\{0\}$ inside $B(0,1)$ to obtain $\chi \in C_c^{\mathrm{\infty }}(B(0,1))$ with $0\le \chi \le 1$ and $\chi =1$ on a neighborhood of $0$. The ball-measure bound gives $0<a:=\int \chi <\mathrm{\infty }$; set $\rho =\chi \mathrm{/}a$. Then $\rho$ is real, has mass one, and is supported in $B(0,1)$. Under AC, Countable Choice is available for the cited approximation and integration interfaces.

Fix $q\in {\mathbb{R}}^n$ and $r>0$ with $\overline{B(q,2r)}\subset \mathrm{\Omega }$, and write $B=B(q,r)$ and $B_0=B(q,2r)$. The closure of $B_0$ is compact by Heine–Borel, so [F2] places $u{\mathrm{\mid }}_{B_0}$ in $W^{1,p}(B_0;\mathbb{K})$. The ball $B_0$ has finite measure by [F7]; Hölder in [F5] shows that each real component of $u$ has an $L^1(B_0)$ representative, including $p=1$ and $p=\mathrm{\infty }$. Extend that representative by zero to $F\in L^1({\mathbb{R}}^n;\mathbb{R})$. The weak derivative of each component on $B_0$ is zero: restrict the global identity using [F3], and replace its a.e.-zero value by the zero representative.

For $0<\epsilon <r$ define $F_{\epsilon }=F\ast {\rho }_{\epsilon }$. By [F11] this convolution is smooth. If $x\in B$, then the test $y\mapsto {\rho }_{\epsilon }(x-y)$ is supported in $B_0$. Differentiating under the integral and using ${\mathrm{\partial }}_{x_i}{\rho }_{\epsilon }(x-y)=-{\mathrm{\partial }}_{y_i}{\rho }_{\epsilon }(x-y)$, the weak identity on $B_0$ gives ${\mathrm{\partial }}_i{F}_{\epsilon }(x)=-{\int }_{B_0}F(y){\mathrm{\partial }}_{y_i}{\rho }_{\epsilon }(x-y)dy={\int }_{B_0}{D}_{i}F(y){\rho }_{\epsilon }(x-y)dy=0.$ This applies separately to the real and imaginary components.

The first partials of each smooth real component of $F_{\epsilon }{\mathrm{\mid }}_B$ are continuous and zero. By [F12] it is totally differentiable on $B$ with zero derivative, and therefore constant there because $B$ is connected and open. Hence each real mollified component is a constant on $B$.

For either real component $f$ of $u$, its zero extension $F$ lies in $L^1({\mathbb{R}}^n)$. By [F9]–[F10], $\mathrm{\parallel }F\ast {\rho }_{\epsilon }-F{\mathrm{\parallel }}_{L^1({\mathbb{R}}^n)}\to 0$ as $\epsilon \downarrow 0$. On $B$, the convolution uses only values in $B_0$, so this is convergence to $f$ in $L^1(B)$. Take ${\epsilon }_m=r\mathrm{/}(m+1)$; write the constant value on $B$ as $a_m$ and set $a=\lambda (B{)}^{-1}{\int }_{B}f$. Since $0<\lambda (B)<\mathrm{\infty }$, [F7, F13] and linearity give $\lambda (B)\mathrm{\mid }{a}_m-a\mathrm{\mid }=\mid {\int }_B(F\ast {\rho }_{{\epsilon }_m}-f)\mid \le {\int }_B\mathrm{\mid }F\ast {\rho }_{{\epsilon }_m}-f\mathrm{\mid }\mathrm{.}$ Therefore ${\int }_B\mathrm{\mid }f-a\mathrm{\mid }\le 2{\int }_B\mathrm{\mid }F\ast {\rho }_{{\epsilon }_m}-f\mathrm{\mid }⟶0,$ so [F10] gives $f=a$ almost everywhere on $B$. Applying this to both components proves that $u$ has one constant value almost everywhere on every such inner ball $B$.

For each $x\in \mathrm{\Omega }$, choose an inner ball $B(q,r)$ containing $x$ with $\overline{B(q,2r)}\subset \mathrm{\Omega }$; openness supplies one. Let $\kappa (x)$ be the a.e.-constant value of $u$ on that ball, which is unique because the ball has positive measure. If two admissible inner balls contain $x$, their intersection is a nonempty open set and contains a positive-measure ball. On that intersection both constants equal $u$ almost everywhere. A finite union of null sets is null, so the positive-measure intersection forces the constants to agree. Thus $\kappa$ is well defined, and it is constant on each admissible inner ball. In particular $\kappa$ is locally constant on $\mathrm{\Omega }$.

Let $C$ be a connected component of $\mathrm{\Omega }$. By [F15], $C$ is open and connected. For a fixed $x_0\in C$, the set $\{x\in C:\kappa (x)=\kappa (x_0)\}$ and its complement are both open in $C$ because $\kappa$ is locally constant. The first set is nonempty, so connectedness gives $\kappa (x)=\kappa (x_0)$ for every $x\in C$; call this value $c_C$.

Consider the countable family of all balls $B(q,1\mathrm{/}m)$ with $q\in {\mathbb{Q}}^n$, $m\ge 1$, and $\overline{B(q,2\mathrm{/}m)}\subset C$. It covers $C$: around any $x\in C$ take $B(x,\delta )\subset C$. By [F17], choose $m\ge 1$ with $3\mathrm{/}m<\delta$ and then $q\in {\mathbb{Q}}^n$ with $\mathrm{\mid }q-x\mathrm{\mid }<1\mathrm{/}m$. Thus $x\in B(q,1\mathrm{/}m)$ and $\overline{B(q,2\mathrm{/}m)}\subset B(x,\delta )\subset C$. By steps 4.1 and 6.1, $u=c_C$ almost everywhere on each such inner ball. The exceptional sets are measurable and null; by [F19] their countable union is null. Hence $u=c_C$ almost everywhere on all of $C$. If $\mathrm{\Omega }=\mathrm{\varnothing }$, there is no component to consider. [F4, F15, F17, F19, step 4.1, step 6.1] \square