Solution
= Solution
{c}
\b[True.] The first of <Green's identities> and the <Robin boundary condition> $\partial_nu=-u$ give
$$
\iiint_V|\nabla u|^2\,dV=-\iint_{\partial V}u^2\,dS,
\qquad
\boxed{\iiint_V|\nabla u|^2\,dV+\iint_{\partial V}u^2\,dS=0.}
$$
Both <integrands> are nonnegative, so $\nabla u=0$ in $V$ and $u=0$ on the <boundary>. By <path connectedness>, $u$ is a <constant function>, and its <boundary value> makes that constant zero. Therefore $\boxed{u=0}$.