Harnack theorem

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 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

[1]	I.G. [I.G. Petrovskii] Petrowski, "Vorlesungen über partielle Differentialgleichungen" , Teubner (1965) (Translated from Russian)
[2]	A. Friedman, "Partial differential equations of parabolic type" , Prentice-Hall (1964)

Comments

In the axiomatic theory of harmonic spaces (cf. Harmonic space) the first Harnack theorem is known as the Bauer convergence property and the second Harnack theorem as the Brélot convergence property, see [a3] and [a1]. The following properties are equivalent to the Brélot convergence property (see [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); 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).

References

[a1]	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
[a2]	M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1959)
[a3]	C. Constantinescu, A. Cornea, "Potential theory on harmonic spaces" , Springer (1972)
[a4]	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