By Poisson's formula
where ωn − 1 is the area of the unit sphere in Rn and r = |x − x0|.
Since
the kernel in the integrand satisfies
Harnack's inequality follows by substituting this inequality in the above integral and using the fact that the average of a harmonic function over a sphere equals its value at the center of the sphere:
For elliptic partial differential equations, Harnack's inequality states that the supremum of a positive solution in some connected open region is bounded by some constant times the infimum, possibly with an added term containing a functional norm of the data:
The constant depends on the ellipticity of the equation and the connected open region.
There is a version of Harnack's inequality for linear parabolic PDEs such as heat equation.
Let be a smooth (bounded) domain in and consider the linear elliptic operator
with smooth and bounded coefficients and a positive definite matrix . Suppose that is a solution of
- in
such that
Let be compactly contained in and choose . Then there exists a constant C > 0 (depending only on K, , , and the coefficients of ) such that, for each ,
- Caffarelli, Luis A.; Cabré, Xavier (1995), Fully Nonlinear Elliptic Equations, Providence, Rhode Island: American Mathematical Society, pp. 31–41, ISBN 0-8218-0437-5
- Folland, Gerald B. (1995), Introduction to partial differential equations (2nd ed.), Princeton University Press, ISBN 0-691-04361-2
- Gilbarg, David; Trudinger, Neil S. (1988), Elliptic Partial Differential Equations of Second Order, Springer, ISBN 3-540-41160-7
- Hamilton, Richard S. (1993), "The Harnack estimate for the Ricci flow", Journal of Differential Geometry, 37 (1): 225–243, doi:10.4310/jdg/1214453430, ISSN 0022-040X, MR 1198607
- Harnack, A. (1887), Die Grundlagen der Theorie des logarithmischen Potentiales und der eindeutigen Potentialfunktion in der Ebene, Leipzig: V. G. Teubner
- John, Fritz (1982), Partial differential equations, Applied Mathematical Sciences, vol. 1 (4th ed.), Springer-Verlag, ISBN 0-387-90609-6
- Kamynin, L.I. (2001) [1994], "Harnack theorem", Encyclopedia of Mathematics, EMS Press
- Kassmann, Moritz (2007), "Harnack Inequalities: An Introduction" Boundary Value Problems 2007:081415, doi: 10.1155/2007/81415, MR 2291922
- Moser, Jürgen (1961), "On Harnack's theorem for elliptic differential equations", Communications on Pure and Applied Mathematics, 14 (3): 577–591, doi:10.1002/cpa.3160140329, MR 0159138
- Moser, Jürgen (1964), "A Harnack inequality for parabolic differential equations", Communications on Pure and Applied Mathematics, 17 (1): 101–134, doi:10.1002/cpa.3160170106, MR 0159139
- Serrin, James (1955), "On the Harnack inequality for linear elliptic equations", Journal d'Analyse Mathématique, 4 (1): 292–308, doi:10.1007/BF02787725, MR 0081415
- L. C. Evans (1998), Partial differential equations. American Mathematical Society, USA. For elliptic PDEs see Theorem 5, p. 334 and for parabolic PDEs see Theorem 10, p. 370.