= Solution
Let $E(t)=\int_V|\nabla w|^2\,dV$, using real-valued smooth $w$ and a fixed bounded region. <Differentiation under the integral sign> and <Green's first identity> yield
$$
\begin{aligned}
E'(t)&=2\int_V\nabla w\cdot\nabla w_t\,dV\\
&=2\int_Sw_t\partial_nw\,dS-2\int_Vw_t\Delta w\,dV.
\end{aligned}
$$
The boundary term vanishes because $w_t=0$ on $S$, and the <heat equation> gives $w_t=\Delta w$ in $V$. Consequently
$$
\boxed{E'(t)=-2\int_V(\Delta w)^2\,dV\leq0.}
$$
The integrand is continuous and nonnegative. Thus at any fixed time, equality is equivalent to $\Delta w=0$ everywhere in $V$, also giving $w_t=0$ there at that time. This proves <Dirichlet energy dissipation for the heat equation>.
The boundary condition fixes the boundary temperature in time; it is not a zero normal derivative condition. No sign condition on $\partial_nw$ is used. The stated smoothness licenses the time derivative, spatial <integration by parts> and boundary trace; without that regularity an appropriate weak energy argument would be required.
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