Let $S$ be a piecewise smooth closed surface in $\mathbb{R}^{3}$ which is the boundary of a volume $V$.

(a) The smooth functions $\phi$ and $\phi_{1}$ defined on $\mathbb{R}^{3}$ satisfy

$$\nabla^{2} \phi=\nabla^{2} \phi_{1}=0$$

in $V$ and $\phi(\mathbf{x})=\phi_{1}(\mathbf{x})=f(\mathbf{x})$ on $S$. By considering an integral of $\boldsymbol{\nabla} \psi \cdot \boldsymbol{\nabla} \psi$, where $\psi=\phi-\phi_{1}$, show that $\phi_{1}=\phi$.

(b) The smooth function $u$ defined on $\mathbb{R}^{3}$ satisfies $u(\mathbf{x})=f(\mathbf{x})+C$ on $S$, where $f$ is the function in part (a) and $C$ is constant. Show that

$$\int_{V} \nabla u \cdot \nabla u d V \geqslant \int_{V} \nabla \phi \cdot \nabla \phi d V$$

where $\phi$ is the function in part (a). When does equality hold?

(c) The smooth function $w(\mathbf{x}, t)$ satisfies

$$\nabla^{2} w=\frac{\partial w}{\partial t}$$

in $V$ and $\frac{\partial w}{\partial t}=0$ on $S$ for all $t$. Show that

$$\frac{d}{d t} \int_{V} \nabla w \cdot \nabla w d V \leqslant 0$$

with equality only if $\nabla^{2} w=0$ in $V$.