= Sobolev function with zero weak gradient
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{title2=$\nabla u=0\Longrightarrow u\text{ is constant}$}
On a connected open set, a <Sobolev space> function with zero <weak gradient> is constant almost everywhere. Local <mollification> gives constant smooth representatives on interior balls; overlaps and connectedness identify their constants. Without connectedness, the constant may differ between components. This explains the constant ambiguity in a <Neumann Poisson problem>.
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