= Uniqueness for a harmonic transmission problem
{title2=$k_1\int_{\Omega_1}|\nabla w|^2+k_2\int_{\Omega_2}|\nabla w|^2=0$}
Let a bounded connected body consist of two regions with positive constant <thermal conductivities> $k_1,k_2$. A temperature is <harmonic> in each region, continuous across their interface, has continuous normal heat flux there, and has prescribed outer <Dirichlet boundary conditions>. There is at most one sufficiently regular solution.
For the difference $w$ of two solutions, apply <Green's first identity> in each region and multiply by its <thermal conductivity>. The interface outward normals are opposite, so continuity of $w$ and $k\partial_n w$ cancels their boundary terms. The outer term vanishes because $w=0$. Hence the displayed sum of nonnegative <Dirichlet energies> is zero. The <gradient> vanishes in both regions; continuity and the outer data then force $w=0$. The proof also applies to finitely many connected subregions, provided the union is connected and the same transmission conditions hold.
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