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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, ...
at two points, introduced by . Harnack's inequality is used to prove Harnack's theorem 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 function. The function is often thought of as an "unknown" to be solved for, similarly to ...
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 precise ...
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'', ...
on its interior, then for every point ''x'' with , ''x'' − ''x''0,  = ''r'' < ''R'', : \frac f(x_0)\le f(x) \le f(x_0). In the plane R2 (''n'' = 2) the inequality can be written: : f(x_0)\le f(x)\le f(x_0). For general domains \Omega in \mathbf^n the inequality can be stated as follows: If \omega is a bounded domain with \bar \subset \Omega, then there is a constant C such that : \sup_ u(x) \le C \inf_ u(x) for every twice differentiable, harmonic and nonnegative function u(x). The constant C is independent of u; it depends only on the domains \Omega and \omega.


Proof of Harnack's inequality in a ball

By Poisson's formula : f(x) = \frac 1 \int_ \frac \cdot f(y) \, dy, where ''ω''''n'' − 1 is the area of the unit sphere in R''n'' and ''r'' = , ''x'' − ''x''0, . Since : R-r \le , x-y, \le R+r, the kernel in the integrand satisfies : \frac \le \frac\le \frac. 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: : f(x_0)= \frac 1 \int_ f(y)\, dy.


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: : \sup u \le C ( \inf u + \, f\, ) 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 \mathcal be a smooth (bounded) domain in \mathbb^n and consider the linear elliptic operator : \mathcalu=\sum_^n a_(t,x)\frac + \sum_^n b_i(t,x)\frac + c(t,x)u with smooth and bounded coefficients and a positive definite matrix (a_). Suppose that u(t,x)\in C^2((0,T)\times\mathcal) is a solution of : \frac-\mathcalu=0 in (0,T)\times\mathcal such that : \quad u(t,x)\ge0 \text (0,T)\times\mathcal. Let K be compactly contained in \mathcal and choose \tau\in(0,T). Then there exists a constant ''C'' > 0 (depending only on ''K'', \tau, t-\tau, and the coefficients of \mathcal) such that, for each t\in(\tau,T), : \sup_K u(t-\tau,\cdot)\le C \inf_K u(t,\cdot).


See also

* Harnack's theorem *
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, ...


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