Harnack's inequality
In mathematics, Harnack's inequality is an inequality relating the values of a positive harmonic function at two points, introduced by . , and 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 , 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. The statement. Harnack's inequality  applies to a non-negative function "f" defined on a closed ball in R with radius "R" and centre "x". It states that, if "f" is continuous on the closed ball and harmonic on its interior, then for every point "x" with |"x" − "x"| = "r" < "R", In the plane R ("n" = 2) the inequality can be written: For general domains formula_3 in formula_4 the inequality can be stated as follows: If formula_5 is a bounded domain with formula_6, then there is a constant formula_7 such that for every twice differentiable, harmonic and nonnegative function formula_9. The  constant formula_7 is independent of formula_11; it depends only on the domains formula_3 and formula_5. Proof of Harnack's inequality in a ball. By Poisson's formula where "ω" is the area of the unit sphere in R and "r" = |"x" − "x"|. 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: 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 formula_19 be a smooth (bounded) domain in formula_20 and consider the linear elliptic operator with smooth and bounded coefficients and a positive definite matrix formula_22. Suppose that formula_23 is a solution of such that  Let formula_27 be compactly contained in formula_19 and choose formula_29. Then there exists a constant "C" > 0 (depending only on "K", formula_30, formula_31, and the coefficients of formula_32) such that, for each formula_33,