Harnack's inequality

Yau developed the method of gradient estimates for Harnack's inequalities.

This is a consequence of Harnack's inequality.

Harnack's inequality also implies the regularity of the function in the interior of its domain.

Harnack's inequality applied to harmonic functions.

In addition to these basic inequalities, one has Harnack's inequality, which states that positive harmonic functions on bounded domains are roughly constant.

In mathematics, Harnack's inequality is an inequality relating the values of a positive harmonic function at two points, introduced by .

(An article on Harnack's inequality that contains a biography of Axel Harnack in the introduction).

There is a version of Harnack's inequality for linear parabolic PDEs such as heat equation.

Harnack's inequality is used to prove Harnack's theorem about the convergence of sequences of harmonic functions.

(with M. Aizenman) Brownian motion and Harnack's inequality for Schrödinger operators, Commun.

In complex analysis, Harnack's principle or Harnack's theorem is one of several closely related theorems about the convergence of sequences of harmonic functions, that follow from Harnack's inequality.

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 it 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: