= Solution
{c}
\b[False.] The <constant function> $u=1$ is a <harmonic function> and has zero <normal derivative> on every smooth <boundary>, but it is nonzero in $V$.
In fact, the first of <Green's identities> gives
$$
\iiint_V|\nabla u|^2\,dV=\iint_{\partial V}u\partial_nu\,dS-\iiint_Vu\Delta u\,dV=0.
$$
Hence $\nabla u=0$; <path connectedness> makes $u$ constant throughout $V$. Thus this homogeneous <Neumann boundary condition> determines a <harmonic function> only up to an additive constant.
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