Harnack's inequality is an inequality relating the values of a positive harmonic function at two points, introduced by A. Harnack (1887). Harnack's inequality...

of partial differential equations, Harnack's principle or Harnack's theorem is a corollary of Harnack's inequality which deals with the convergence of...

theory. Harnack's inequality applied to harmonic functions. He also worked on the real algebraic geometry of plane curves, proving Harnack's curve theorem...

inequalities Hardy's inequality Hardy–Littlewood inequality Hardy–Littlewood–Sobolev inequality Harnack's inequality Hausdorff–Young inequality Hermite–Hadamard...

bounded domain Ω. Then for every non-negative harmonic function u, Harnack's inequality sup V u ≤ C inf V u {\displaystyle \sup _{V}u\leq C\inf _{V}u} holds...

solution of the heat equation. These inequalities, known as differential Harnack inequalities or Li–Yau inequalities, are useful since they can be integrated...

differential Harnack inequality for the curvature operator of certain Ricci flows, and Grigori Perelman proved a differential Harnack inequality for the solutions...

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

f(x) is zero for some unit vector x and f(y) < 0 if |y| < 1. By Harnack's inequality applied to −f − f ( r x ) ≥ − 1 − r ( 1 + r ) n − 1 f ( 0 ) , {\displaystyle...

De Giorgi and Nash's results, he was able to use it to prove a new Harnack inequality. In his original work, a key role was played by an extension of the...