Harnack's inequality

The statement

Harnack's inequality applies to a non-negative function f defined on a closed ball in Rⁿ with radius R and centre x₀. It states that, if f is continuous on the closed ball and harmonic on its interior, then for every point x with |x − x₀| = r < R,

In the plane R² (n = 2) the inequality can be written:

For general domains Ω in R n the inequality can be stated as follows: If ω is a bounded domain with ω ¯ ⊂ Ω , then there is a constant C such that

for every twice differentiable, harmonic and nonnegative function u ( x ) . The constant C is independent of u ; it depends only on the domains Ω and ω .

Proof of Harnack's inequality in a ball

By Poisson's formula

where ω_{n − 1} is the area of the unit sphere in Rⁿ and r = |x − x₀|.

Since

the kernel in the integrand satisfies

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 its value at the center of the sphere:

Elliptic partial differential equations

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:

The constant depends on the ellipticity of the equation and the connected open region.

Parabolic partial differential equations

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

Let M be a smooth (bounded) domain in R n and consider the linear parabolic operator

with smooth and bounded coefficients and a positive definite matrix ( a i j ) . Suppose that u ( t , x ) ∈ C 2 ( ( 0 , T ) × M ) is a solution of

such that

Let K be compactly contained in M and choose τ ∈ ( 0 , T ) . Then there exists a constant C > 0 (depending only on K, τ and the coefficients of L ) such that, for each t ∈ ( τ , T ) ,