Harnack's inequality
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In mathematics, Harnack's inequality is an inequality relating the values of a positive harmonic function at two points, introduced by A. Harnack (1887). J. Serrin (1955) and J. Moser (1961, 1964) generalized Harnack's inequality to solutions of elliptic or parabolic partial differential equations. Perelman's solution of the Poincare conjecture uses a version of the Harnack inequality, found by R. Hamilton (1993), for the Ricci flow. Harnack's inequality is used to prove Harnack's theorem about the convergence of sequences of harmonic functions.
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[edit] Harmonic functions in the plane
Let D = D(z0,R) be an open disk and let f be a harmonic function on D such that f(z) is non-negative for all 
. Then the following inequality holds for all :
For general domains Ω in 
the inequality can be stated as follows: If ω is a bounded domain with 
, then there is a constant C such that
for every twice differentiable, harmonic and nonnegative function u(x). (The constant C is independent of u.)
[edit] Harmonic functions in Euclidean space
[edit] Elliptic partial differential equations
For elliptic partial differential equations, Harnack's inequality states that the value of a positive solution in some connected open region at some point is bounded by some constant times the value at a different point. The constant depends on the equation, the points, and the connected open region.
[edit] Parabolic partial differential equations
For parabolic partial differential equations, such as the heat equation, Harnack's inequality states that for positive solutions y in some connected open set, there is an inequality
for some constant c depending on the equation, the two times, the two points, and the open region, provided the time t2 is later than time t1. It is not possible to bound y in terms of the value at some earlier time without extra information. Informally, this means that there is a limit to how fast things can cool down, but there is no limit to how fast they can heat up.
[edit] References
- Hamilton, Richard S. (1993), “The Harnack estimate for the Ricci flow”, Journal of Differential Geometry 37 (1): 225–243, MR1198607, ISSN 0022-040X
- Harnack, A. (1887), Die Grundlagen der Theorie des logarithmischen Potentiales und der eindeutigen Potentialfunktion in der Ebene, Leipzig: V. G. Teubner, <http://www.archive.org/details/vorlesunganwend00weierich>
- Kamynin, L.I. (2001), “Harnack theorem”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
- Kamynin, L.I. & Kuptsov, L.P. (2001), “Harnack's inequality”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
- Moser, Jürgen (1961), “On Harnack's theorem for elliptic differential equations”, Communications on Pure and Applied Mathematics 14: 577–591, MR0159138, ISSN 0010-3640
- Moser, Jürgen (1964), “A Harnack inequality for parabolic differential equations”, Communications on Pure and Applied Mathematics 17: 101–134, MR0159139, ISSN 0010-3640
- Serrin, James (1955), “On the Harnack inequality for linear elliptic equations”, Journal d'Analyse Mathématique 4: 292–308, MR0081415, ISSN 0021-7670
