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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,

$$\sum_{i,j=1}^na_{ij}(x)\frac{\partial^2u}{\partial x_i\partial y_i}+\sum_{i=1}^na_i(x)\frac{\partial u}{\partial x_i}+a(x)u=0,\tag{*}$$

which has a unique solution of the [[Dirichlet problem|Dirichlet problem]] for any continuous boundary function . If the sequence of solutions of equation \ref{*} converges uniformly on $\partial G$, then it also converges uniformly in $G$ to a solution of equation \ref{*}.

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 .

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.G. [I.G. Petrovskii] Petrowski,   "Vorlesungen über partielle Differentialgleichungen" , Teubner  (1965)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Friedman,   "Partial differential equations of parabolic type" , Prentice-Hall  (1964)</TD></TR></table>



====Comments====
In the axiomatic theory of harmonic spaces (cf. [[Harmonic space|Harmonic space]]) the first Harnack theorem is known as the Bauer convergence property and the second Harnack theorem as the Brélot convergence property, see [[#References|[a3]]] and [[#References|[a1]]]. The following properties are equivalent to the Brélot convergence property (see [[#References|[a4]]]): 1) each positive harmonic function $u$ on a domain $U$ is either strictly positive or $u=0$. Moreover, the set of positive harmonic functions on $U$, equal to 1 in a given point $u\in U$, is equicontinuous (cf. [[Equicontinuity|Equicontinuity]]); and 2) for any domain $U$ and any compact subset $K$ of $U$ there exists a constant $c>0$ such that $u(x)\leq cu(y)$ for any $x,y\in K$ and any positive harmonic function $u$ on $U$ (the [[Harnack inequality|Harnack inequality]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.-M. Bony,   "Opérateurs elliptiques dégénérés associés aux axiomatiques de la théorie du potentiel"  M. Brelot (ed.)  H. Bauer (ed.)  J.-M. Bony (ed.)  J. Deny (ed.)  G. Mokobodzki (ed.) , ''Potential theory (CIME, Stresa, 1969)'' , Cremonese  (1970)  pp. 69–119</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Brélot,   "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris  (1959)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  C. Constantinescu,   A. Cornea,   "Potential theory on harmonic spaces" , Springer  (1972)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P. Loeb,   B. Walsh,   "The equivalence of Harnack's principle and Harnack's inequality in the axiomatic system of Brélot"  ''Ann. Inst. Fourier'' , '''15''' :  2  (1965)  pp. 597–600</TD></TR></table>