Harnack's inequality
（重定向自Harnack inequality）
![A harmonic function (green) over a disk (blue) is bounded from above by a function (red) that coincides with the harmonic function at the disk center and approaches infinity towards the disk boundary.](/Images/godic/202501/17/Harnack3828.png")
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.