In the mathematical field ofpartial differential equations,Harnack's principle orHarnack's theorem is a corollary ofHarnack's inequality which deals with the convergence of sequences ofharmonic functions.

Given a sequence ofharmonic functionsu₁,u₂, ... on anopenconnectedsubsetG of theEuclidean spaceRⁿ, which are pointwise monotonically nondecreasing in the sense that

${\displaystyle u_1(x)\le u_2(x)\le \dots }$

for every pointx ofG, then thelimit

${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{u}_n(x)}$

automatically exists in theextended real number line for everyx. Harnack's theorem says that the limit either is infinite at every point ofG or it is finite at every point ofG. In the latter case, the convergence isuniform on compact sets and the limit is a harmonic function onG.^[1]

The theorem is a corollary of Harnack's inequality. Ifu_n(y) is aCauchy sequence for any particular value ofy, then the Harnack inequality applied to the harmonic functionu_m −u_n implies, for an arbitrary compact setD containingy, thatsup_D |u_m −u_n| is arbitrarily small for sufficiently largem andn. This is exactly the definition of uniform convergence on compact sets. In words, the Harnack inequality is a tool which directly propagates the Cauchy property of a sequence of harmonic functions at a single point to the Cauchy property at all points.

Having established uniform convergence on compact sets, the harmonicity of the limit is an immediate corollary of the fact that themean value property (automatically preserved by uniform convergence) fully characterizes harmonic functions among continuous functions.^[2]

The proof of uniform convergence on compact sets holds equally well for any linear second-orderelliptic partial differential equation, provided that it is linear so thatu_m −u_n solves the same equation. The only difference is that the more generalHarnack inequality holding for solutions of second-order elliptic PDE must be used, rather than that only for harmonic functions. Having established uniform convergence on compact sets, the mean value property is not available in this more general setting, and so the proof of convergence to a new solution must instead make use of other tools, such as theSchauder estimates.

Sources

• Courant, R.;Hilbert, D. (1962).Methods of mathematical physics. Volume II: Partial differential equations. New York–London:Interscience Publishers.doi:10.1002/9783527617234.ISBN 9780471504399.MR 0140802.Zbl 0099.29504.{{cite book}}:ISBN / Date incompatibility (help)
• Gilbarg, David;Trudinger, Neil S. (2001).Elliptic partial differential equations of second order. Classics in Mathematics (Reprint of the 1998 ed.). Berlin:Springer-Verlag.doi:10.1007/978-3-642-61798-0.ISBN 3-540-41160-7.MR 1814364.Zbl 1042.35002.
• Protter, Murray H.;Weinberger, Hans F. (1984).Maximum principles in differential equations (Corrected reprint of the 1967 original ed.). New York:Springer-Verlag.doi:10.1007/978-1-4612-5282-5.ISBN 0-387-96068-6.MR 0762825.Zbl 0549.35002.

• Kamynin, L.I. (2001) [1994],"Harnack theorem",Encyclopedia of Mathematics,EMS Press

• Harmonic functions
• Theorems in complex analysis
• Mathematical principles

• Articles with short description
• Short description is different from Wikidata
• CS1 errors: ISBN date