Harnack's inequality

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 ${\displaystyle \Omega }$ in ${\displaystyle \mathbf {R} ^{n}}$ the inequality can be stated as follows: If ${\displaystyle \omega }$ is a bounded domain with ${\displaystyle {\bar {\omega }}\subset \Omega }$, then there is a constant ${\displaystyle C}$ such that

...
Wikipedia