= Solution
Subtract the two <weak solution> identities and test with their difference $w\in H^1(U)$. Then
$$
\int_U|\nabla w|^2=0.
$$
A <Sobolev function with zero weak gradient> is constant on each connected component. One justification is to mollify locally: each mollification has zero gradient and is constant on its ball, and overlaps identify the constants; taking limits gives the original assertion. Since $U$ is connected, $w$ is one constant on $U$.
Conversely, adding a constant changes neither the <weak derivative> nor the weak identity. \b[The solution is unique up to an additive constant.]
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