= Dirichlet uniqueness for variable-conductivity heat conduction
{c}
{title2=$\nabla\cdot(k\nabla T)+H=0,\quad T|_{\partial R}=h$}
Suppose $k>0$ and two sufficiently regular <steady state> <temperature> profiles solve the same <heat equation> with the same <PDE source term> $H$ and the same <Dirichlet boundary condition> $h$. Their difference $w$ satisfies $\nabla\cdot(k\nabla w)=0$ and $w=0$ on the boundary. The <divergence theorem> applied to $wk\nabla w$ gives
$$
0=\int_Rw\nabla\cdot(k\nabla w)\,dV=-\int_Rk\|\nabla w\|^2\,dV.
$$
Thus $\nabla w=0$, and the zero boundary values force $w=0$ on each connected component. The argument proves uniqueness without asserting existence and without replacing the spatially varying <thermal conductivity> by a constant. For <weak solutions>, the same calculation is the energy argument obtained by choosing $w$ as the <test function>, under the usual boundedness and positive lower bound assumptions on $k$.
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