In the mathematical field of

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to h ...

s, Harnack's principle or Harnack's theorem is a corollary of Harnack's inequality 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: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is, : \f ...

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), ''Open'' (Blues Image album), 1969 * Open (Gotthard album), ''Open'' (Gotthard album), 1999 * Open (C ...

connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...

subset In mathematics, set ''A'' is a subset of a set ''B'' if all 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 are unequal, then ''A'' is a proper subset of ...

of the

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

, which are pointwise monotonically nondecreasing in the sense that :

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

for every point of , then the

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\lim_u_n(x)

automatically exists in the

extended real number line In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra on ...

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 (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...

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 Second-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic. Any second-order linear PDE in two variables can be written in the form :Au_ + 2Bu_ + Cu_ + Du_x + Eu_y + Fu +G= 0,\, whe ...

, 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 In mathematics, the Schauder estimates are a collection of results due to concerning the regularity of solutions to linear, uniformly elliptic partial differential equations. The estimates say that when the equation has appropriately smooth terms ...

References

Sources * * *

External Links

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