Harnack's inequality

In mathematics give Harnack inequalities estimates for the upper bounds of solutions of various differential equations, especially the heat conduction equation. They are named after the mathematician Axel Harnack.

Classical Harnack inequality

Statement

It is a non-negative solution of the heat equation

Where the Laplace operator called on the compact Riemannian manifold.

Then there is a only abhängende constant, so that

Applies to all.

Determining the optimal constant as a function of the geometry of a difficult problem.

Harmonic Functions

In particular, for any non-negative harmonic functions.

Example

Be the ball with radius and center point in Euclidean space. Then for any non-negative harmonic function ( with continuous boundary values ​​)

The inequality

With for all.

Hence the Harnack 's inequality yields for with.

Differential Harnack inequality

Be an n-dimensional Riemannian manifold with nonnegative Ricci curvature and convex boundary, then for any positive solution of the heat equation, the inequality

From this inequality one can often derive optimal constants for the classical Harnack inequality.