Harnack's principle
Template:Short description In the mathematical field of partial differential equations, Harnack's principle or Harnack's theorem is a corollary of Harnack's inequality which deals with the convergence of sequences of harmonic functions.
Given a sequence of harmonic functions Template:Math on an open connected subset Template:Mvar of the Euclidean space Template:Math, which are pointwise monotonically nondecreasing in the sense that
- <math>u_1(x) \le u_2(x) \le \dots</math>
for every point Template:Mvar of Template:Mvar, then the limit
- <math> \lim_{n\to\infty}u_n(x)</math>
automatically exists in the extended real number line for every Template:Mvar. Harnack's theorem says that the limit either is infinite at every point of Template:Mvar or it is finite at every point of Template:Mvar. In the latter case, the convergence is uniform on compact sets and the limit is a harmonic function on Template:Mvar.Template:Sfnm
The theorem is a corollary of Harnack's inequality. If Template:Math is a Cauchy sequence for any particular value of Template:Mvar, then the Harnack inequality applied to the harmonic function Template:Math implies, for an arbitrary compact set Template:Mvar containing Template:Mvar, that Template:Math is arbitrarily small for sufficiently large Template:Mvar and Template:Mvar. 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 the mean value property (automatically preserved by uniform convergence) fully characterizes harmonic functions among continuous functions.Template:Sfnm
The proof of uniform convergence on compact sets holds equally well for any linear second-order elliptic partial differential equation, provided that it is linear so that Template:Math solves the same equation. The only difference is that the more general Harnack 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 the Schauder estimates.
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