 Solution

Subtract the two weak solution identities and test with their difference $w\in H^1(U)$. ThenA 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. The solution is unique up to an additive constant.