In mathematics, Harnack's inequality is an inequality  relating the values of a positive harmonic function at two points, introduced by . and generalized Harnack's inequality to solutions of elliptic or parabolic partial differential equation

Partial differential equation

In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

s. Perelman

Grigori Perelman

Grigori Yakovlevich Perelman is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology.In 1992, Perelman proved the soul conjecture. In 2002, he proved Thurston's geometrization conjecture...

's solution of the Poincaré conjecture

Poincaré conjecture

In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space...

 uses a version of the Harnack inequality, found by , for the Ricci flow

Ricci flow

In differential geometry, the Ricci flow is an intrinsic geometric flow. It is a process that deforms the metric of a Riemannian manifold in a way formally analogous to the diffusion of heat, smoothing out irregularities in the metric....

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

Hölder condition

In mathematics, a real or complex-valued function ƒ on d-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants C, \alpha , such that...

 of the function in the interior of its domain.

The statement

Let be an open

Open set

The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

 disk

Ball (mathematics)

In mathematics, a ball is the space inside a sphere. It may be a closed ball or an open ball ....

 in the plane and let f be a harmonic function

Harmonic function

In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R which satisfies Laplace's equation, i.e....

 on D such that f(z) is non-negative for all . Then the following inequality holds for all :


For general domains in the inequality can be stated as follows: If is a bounded domain with , then there is a constant such that


for every twice differentiable, harmonic and nonnegative function . The constant is independent of ; it depends only on the domain.

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

Norm (mathematics)

In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...

 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

Heat equation

The heat equation is an important partial differential equation which describes the distribution of heat in a given region over time...

.

Let be a smooth domain in and consider the linear parabolic operator

with smooth and bounded coefficients and a nondegenerate matrix . Suppose that is a solution of

in

such that

in

Let be a compact subset of and choose . Then for each there exists a constant (depending only on , and the coefficients of ) such that