In the mathematical field of

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...

s, Harnack's principle or Harnack's theorem is a corollary of

Harnack's inequality In mathematics, Harnack's inequality is an inequality relating the values of a positive harmonic function at two points, introduced by . Harnack's inequality is used to prove Harnack's theorem about the convergence of sequences of harmonic functions ...

which deals with the convergence of sequences of

harmonic function In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f\colon U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that i ...

s. Given a sequence of

s on an

open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gerd Dudek, Buschi Niebergall, and Edward Vesala album), 1979 * ''Open'' (Go ...

connected

subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...

of the

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

, which are pointwise monotonically nondecreasing in the sense that :

u_1(x) \le u_2(x) \le \dots

for every point of , then the limit :

\lim_u_n(x)

automatically exists in the

extended real number line In mathematics, the extended real number system is obtained from the real number system \R by adding two elements denoted +\infty and -\infty that are respectively greater and lower than every real number. This allows for treating the potential ...

for every . Harnack's theorem says that the limit either is infinite at every point of or it is finite at every point of . In the latter case, the convergence is uniform on compact sets and the limit is a harmonic function on . The theorem is a corollary of Harnack's inequality. If is a

Cauchy sequence In mathematics, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all excluding a finite number of elements of the sequence are le ...

for any particular value of , then the Harnack inequality applied to the harmonic function implies, for an arbitrary compact set containing , that is arbitrarily small for sufficiently large and . 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. The proof of uniform convergence on compact sets holds equally well for any linear second-order

elliptic partial differential equation In mathematics, an elliptic partial differential equation is a type of partial differential equation (PDE). In mathematical modeling, elliptic PDEs are frequently used to model steady states, unlike parabolic PDE and hyperbolic PDE which gene ...

, provided that it is linear so that 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.

References

Sources * * *

External links

*{{springer, id=h/h046620, title=Harnack theorem, first=L.I., last= Kamynin Harmonic functions Theorems in complex analysis Mathematical principles