# Harnack theorem

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Harnack's first theorem: If a sequence of functions which are harmonic in a bounded domain $G$ and continuous on $\overline G$ converges uniformly on the boundary $\partial G$, then it also converges uniformly in $G$ to a harmonic function. This theorem can be generalized to solutions of an elliptic equation,
$$\sum_{i,j=1}^na_{ij}(x)\frac{\partial^2u}{\partial x_i\partial y_i}+\sum_{i=1}^na_i(x)\frac{\partial u}{\partial x_i}+a(x)u=0,\label{*}\tag{*}$$
which has a unique solution of the Dirichlet problem for any continuous boundary function . If the sequence of solutions of equation \eqref{*} converges uniformly on $\partial G$, then it also converges uniformly in $G$ to a solution of equation \eqref{*}.
Harnack's second theorem, the Harnack principle: If a monotone sequence of harmonic functions in a bounded domain $G$ converges at some point in $G$, then it converges at all points of $G$ to a harmonic function, and this convergence is uniform on any closed subdomain of $G$. Harnack's second theorem can be generalized to monotone sequences of solutions of the elliptic equation .