= Dirichlet uniqueness for the Poisson equation
{title2=$\Delta\phi_1=\Delta\phi_2,\quad\phi_1|_{\partial V}=\phi_2|_{\partial V}\Longrightarrow\phi_1=\phi_2$}
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 $\int_V|\nabla(\phi_1-\phi_2)|^2=0$. 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.
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