Harnack inequality

In mathematics , Harnack inequalities give estimates for the upper bounds of solutions to various differential equations . In the classical case of the heat conduction equation, they limit the diffusion of heat. They are named after the mathematician Axel Harnack .

Classic Harnack inequality

statement

Let it be a nonnegative solution of the heat conduction equation ${\displaystyle u\colon M\times \left[0,T\right]\rightarrow [0,\infty )}$

${\displaystyle {\frac {\partial u}{\partial t}}=\Delta u}$,

where denotes the Laplace operator on the compact Riemann manifold . ${\displaystyle \Delta }$ ${\displaystyle M}$

Then there is a constant that only depends on , so that ${\displaystyle M}$ ${\displaystyle C}$

${\displaystyle \sup _{x\in M}u(x,t)\leq C\inf _{x\in M}u(x,t)}$

applies to all . ${\displaystyle 0\leq t\leq T}$

Determining the optimal constant as a function of the geometry of is a difficult problem. ${\displaystyle C}$ ${\displaystyle M}$

Harmonic functions

In particular, applies to all non-negative harmonic functions . ${\displaystyle \sup _{x\in M}u(x)\leq C\inf _{x\in M}u(x)}$ ${\displaystyle u\colon M\rightarrow [0,\infty )}$

example

Let be the ball with radius and center in Euclidean space. Then for every nonnegative harmonic function (with continuous boundary values) ${\displaystyle M=B(x_{0},R)\subset \mathbb {R} ^{n}}$ ${\displaystyle R}$ ${\displaystyle x_{0}}$

${\displaystyle u\colon B(x_{0},R)\rightarrow [0,\infty )}$

the inequality

${\displaystyle \displaystyle {{1-(r/R) \over [1+(r/R)]^{n-1}}u(x_{0})\leq u(x)\leq {1+(r/R) \over [1-(r/R)]^{n-1}}u(x_{0})}}$

with for everyone . ${\displaystyle r=\parallel x-x_{0}\parallel }$ ${\displaystyle x\in B(x_{0},R)}$

This results in the Harnack inequality for with . ${\displaystyle M=B(x_{0},R)}$ ${\displaystyle C={\frac {1+(r/R)}{[1-(r/R)]^{n-1}}}{\frac {[1+(r/R)]^{n-1}}{1-(r/R)}}}$

Differential Harnack Inequality

Let be an n-dimensional Riemann manifold with non-negative Ricci curvature and convex edge , then the inequality holds for every positive solution of the heat conduction equation ${\displaystyle M}$

${\displaystyle {\frac {\partial _{t}u}{u}}-{\frac {\mid \nabla u\mid ^{2}}{u^{2}}}+{\frac {n}{2t}}\geq 0.}$

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

literature

• Axel Harnack : The basics of the theory of the logarithmic potential and the unique potential function in the plane. VG Teubner, Leipzig 1887.
• Peter Li; Shing-Tung Yau : On the parabolic kernel of the Schrödinger operator. Acta Math. 156 (1986), No. 3-4, pp. 153-201.
• Reto Müller: Differential Harnack inequalities and the Ricci flow (= EMS Series of Lectures in Mathematics. ) 1st edition. European Mathematical Society (EMS), Zurich 2006, ISBN 3-03719-030-2 .