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 Poincaré 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. Harnack's inequality can also be used to show the interior regularity of weak solutions of partial differential equations.
Harnack's inequality applies to a non-negative function f defined on a closed ball in Rn with radius R and centre x0. It states that, if f is continuous on the closed ball and harmonic on its interior, then for every point x with |x − x0| = r < R,
In the plane R2 (n = 2) the inequality can be written:
For general domains in the inequality can be stated as follows: If is a bounded domain with , then there is a constant such that
for every twice differentiable, harmonic and nonnegative function . The constant is independent of ; it depends only on the domains and .
Proof of Harnack's inequality in a ball
where ωn − 1 is the area of the unit sphere in Rn and r = |x − x0|.
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:
Elliptic partial differential equations
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.
Parabolic partial differential equations
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 parabolic operator
with smooth and bounded coefficients and a positive definite matrix . Suppose that is a solution of
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 ,
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- 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.