In mathematics, Harnack's inequality is an inequality relating the values of a positive
harmonic function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is,
: \f ...
at two points, introduced by . Harnack's inequality is used to prove
Harnack's theorem
In the mathematical field of partial differential equations, Harnack's principle or Harnack's theorem is a corollary of Harnack's inequality which deals with the convergence of sequences of harmonic functions.
Given a sequence of harmonic function ...
about the convergence of sequences of harmonic functions. , and generalized Harnack's inequality to solutions of elliptic or parabolic
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
s. Such results can be used to show the interior
regularity of
weak solution
In mathematics, a weak solution (also called a generalized solution) to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless deemed to satisfy the equation in some precis ...
s.
Perelman
Perelman ( he, פרלמן) is an Ashkenazi Jewish surname. Notable people with the surname include:
* Bob Perelman (b. 1947), American poet
* Chaim Perelman (1912-1984), Polish-born Belgian philosopher of law
* Eliezer Ben-Yehuda () (1858-1922), ...
's solution of the Poincaré conjecture uses a version of the Harnack inequality, found by , for the
Ricci flow
In the mathematical fields of differential geometry and geometric analysis, the Ricci flow ( , ), sometimes also referred to as Hamilton's Ricci flow, is a certain partial differential equation for a Riemannian metric. It is often said to be ana ...
.
The statement
Harnack's inequality applies to a non-negative function ''f'' defined on a closed ball in R
''n'' with radius ''R'' and centre ''x''
0. It states that, if ''f'' is continuous on the closed ball and
harmonic
A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', the ...
on its interior, then for every point ''x'' with , ''x'' − ''x''
0, = ''r'' < ''R'',
:
In the plane R
2 (''n'' = 2) the inequality can be written:
:
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 domains
and
.
Proof of Harnack's inequality in a ball
By
Poisson's formula
:
where ''ω''
''n'' − 1 is the area of the unit sphere in R
''n'' and ''r'' = , ''x'' − ''x''
0, .
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
Second-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic. Any second-order linear PDE in two variables can be written in the form
:Au_ + 2Bu_ + Cu_ + Du_x + Eu_y + Fu +G= 0,\,
whe ...
, 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
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
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
In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for t ...
.
Let
be a smooth (bounded) domain in
and consider the linear elliptic operator
:
with smooth and bounded coefficients and a
positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular:
* Positive-definite bilinear form
* Positive-definite f ...
matrix
. Suppose that
is a solution of
:
in
such that
:
Let
be compactly contained in
and choose
. Then there exists a constant ''C'' > 0 (depending only on ''K'',
,
, and the coefficients of
) such that, for each
,
:
See also
*
Harnack's theorem
In the mathematical field of partial differential equations, Harnack's principle or Harnack's theorem is a corollary of Harnack's inequality which deals with the convergence of sequences of harmonic functions.
Given a sequence of 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 \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is,
: \f ...
References
*
*
*
*
*
*
*
*Kassmann, Moritz (2007), "Harnack Inequalities: An Introduction" Boundary Value Problems 2007:081415,
doi:
10.1155/2007/81415,
MRbr>
2291922*
*
*
*L. C. Evans (1998), ''Partial differential equations''. American Mathematical Society, USA. For elliptic PDEs see Theorem 5, p. 334 and for parabolic PDEs see Theorem 10, p. 370.
{{Authority control
Harmonic functions
Inequalities