Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics V / 2

Let $\Omega \subset {\mathbb{R}}^3$ be a bounded region with a piecewise smooth closed boundary, and let $\mathbf{F}$ be a continuously differentiable vector field on an open neighbourhood of its closure. With $\mathbf{n}$ the outward unit normal vector, the divergence theorem statesThe boundary surface integral is the total outward flux. All boundary components are included; on the boundary of a cavity, the outward unit normal vector points into the cavity.

The specific heat capacity $c$ is the energy needed per unit mass per unit increase in temperature. With $c$ constant, a small element of volume $dV$ has mass $\rho dV$ and thermal energy $\rho cTdV$, measured relative to the reference at which $T=0$. Summing over $\Omega$ givesChoosing a different reference temperature only adds a time-independent constant to this energy, so it does not affect the heat balance with volumetric heat generation.

For any fixed subregion $\Omega$, with no internal heat generation, conservation of energy says that the rate of stored thermal energy equals the incoming heat flux:The minus sign is needed because $\mathbf{q}\cdot \mathbf{n}>0$ denotes outgoing heat flux. Substitute Fourier's law $\mathbf{q}=-k\nabla T$ and apply the divergence theorem:Here differentiation under the integral uses a fixed region and sufficient regularity of $T$. Since the identity holds for every $\Omega$, continuity of the integrands gives the local heat equationFor spatially varying thermal conductivity, the product rule gives $\nabla \cdot (k\nabla T)=k{\nabla }^{2}T+\nabla k\cdot \nabla T$; the second term cannot generally be discarded.

Suppose $T^{(1)}$ and $T^{(2)}$ are two sufficiently regular steady state solutions with the same prescribed temperature on the boundary. Their difference $w=T^{(1)}-T^{(2)}$ satisfiesThe identical PDE source terms cancel. Apply the product rule and the divergence theorem to $wk\nabla w$; this is the weighted form of Green's first identity:Since $k>0$, the nonnegative integrand must vanish, so $\nabla w=0$. Hence $w$ is constant on each connected component of $R$, and its zero Dirichlet boundary condition makes each constant zero. Therefore $\boxed{T^{(1)}=T^{(2)}}$: the solution, if it exists, is unique. This proves Dirichlet uniqueness for variable-conductivity heat conduction; it does not require a constant thermal conductivity.

Integrate the steady state equation over $R$ and use the divergence theorem:The zero heat flux condition makes the boundary term vanish. But $\rho >0$ and $S>0$ throughout a region of positive volume, soa contradiction. Thus no steady-state solution exists. This is the insulated steady heat balance: positive internal heat generation continually increases the stored thermal energy when no heat can escape.