Dirichlet uniqueness for the Poisson equation ($\Delta {\varphi }_1=\Delta {\varphi }_2,{\varphi }_1{|}_{\partial V}={\varphi }_2{|}_{\partial V}⟹{\varphi }_1={\varphi }_2$)

The difference of two classical Poisson equation solutions with the same Dirichlet boundary conditions is a harmonic function vanishing on the boundary. Under sufficient boundary smoothness, Green's first identity gives ${\int }_V|\nabla ({\varphi }_1-{\varphi }_2){|}^2=0$. More generally the maximum principle for harmonic functions proves the same uniqueness for functions continuous up to the boundary and twice continuously differentiable in the bounded interior.

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.