# Harnack inequality

(dual Harnack inequality)

An inequality that gives an estimate from above and an estimate from below for the ratio $u( x)/u( y)$ of two values of a positive harmonic function; obtained by A. Harnack . Let $u \geq 0$ be a harmonic function in a domain $G$ of an $n$- dimensional Euclidean space; let $E _ {r} ( y)$ be the ball $\{ {x } : {| x - y | < r } \}$ with radius $r$ and centre at the point $y$. If the closure $\overline{ {E _ {r} ( y) }}\; \subset G$, then the Harnack inequality

$$\tag{1 } \left ( { \frac{r}{r + \rho } } \right ) ^ {n - 2 } \frac{r - \rho }{r + \rho } u ( y) \leq u ( x) \leq \ \left ( { \frac{r}{r - \rho } } \right ) ^ {n - 2 } \frac{r + \rho }{r - \rho } u ( y) ,$$

or

$$\max _ {x \in E _ \rho ( y) } \ u ( x) \leq \left ( \frac{r + \rho }{r - \rho } \right ) ^ {n} \ \min _ {x \in E _ \rho ( y) } u ( x),$$

is valid for all $x \in E _ \rho ( y)$, $0 \leq \rho < r$. If $g$ is a compactum, $\overline{g}\; \subset G$, then there exists a number $M = M( G, g)$ such that

$$\tag{2 } M ^ {-} 1 u ( y) \leq u ( x) \leq Mu ( y)$$

for any $x, y \in \overline{g}\;$. In particular,

$$\max _ {x \in g } u ( x) \leq \ M \min _ {x \in g } u ( x).$$

Harnack's inequality has the following corollaries: the strong maximum principle, the Harnack theorem on sequences of harmonic functions, compactness theorems for families of harmonic functions, the Liouville theorem (cf. Liouville theorems), and other facts. Harnack's inequality can be generalized ,  to non-negative solutions of a wide class of linear elliptic equations of the form

$$Lu \equiv \ \sum _ {i, j = 1 } ^ { n } { \frac \partial {\partial x ^ {i} } } \left ( a ^ {ij} ( x) \frac{\partial u }{\partial x ^ {j} } \right ) + \sum _ {i = 1 } ^ { n } b ^ {i} ( x) \frac{\partial u }{\partial x ^ {i} } + c ( x) u = 0$$

with a uniformly positive-definite matrix $\| a ^ {ij} \|$:

$$\lambda \sum _ {i = 1 } ^ { n } \xi _ {i} ^ {2} \leq \ \sum _ {i, j = 1 } ^ { n } a ^ {ij} ( x) \xi _ {i} \xi _ {j} \leq \ \Lambda \sum _ {i = 1 } ^ { n } \xi _ {i} ^ {2} ,$$

where $\Lambda \geq \lambda > 0$ are numbers, $\xi = ( \xi _ {1} \dots \xi _ {n} )$ is any $n$- dimensional vector and $x \in G$. The constant $M$ in inequality (2) depends only on $\lambda$, $\Lambda$, certain norms of the lower coefficients of the operator $L$, and the distance between the boundaries of $G$ and of $g$. Figure: h046600a

The analogue of Harnack's inequality is also applicable  to non-negative solutions $u( x, t)$ of uniformly-parabolic equations of the form $u _ {t} + Lu= 0$( the coefficients of the operator $L$ may also depend on $t$). In such a case only a one-sided inequality

$$u ( x, t) \leq Mu ( y, \tau )$$

is possible for points $( x, t)$ lying inside the paraboloid

$$\{ {( x, t) } : {| x - y | ^ {2} \leq \mu ^ {2} ( \tau - t),\ \tau - v ^ {2} \leq t \leq \tau } \} ,$$

which is concave downwards with apex at $( y, \tau )$( Fig., left part). Here $M$ depends on $y$, $\tau$, $\lambda$, $\Lambda$, $\mu$, $\nu$, on certain norms of the lower coefficients of the operator $L$, and on the distance between the boundary of the paraboloid and the boundary of the domain on which $u \geq 0$. If, for instance, $u \geq 0$ in the cylinder

$$Q = G \times ( a, b],\ \ \overline{g}\; \subset G,$$

if the distance between $\partial G$ and $\partial g$ is at least $d > 0$ and if $d$ is sufficiently small, then the inequality 

$$\mathop{\rm ln} \frac{u ( x, t) }{u ( y, \tau ) } \leq M \left ( \frac{| x - y | ^ {2} }{\tau - t } + \frac{\tau - t }{d ^ {2} } + 1 \right )$$

is valid in $g \times ( a - d ^ {2} , b ]$. In particular, if $u \geq 0$ in $Q$( Fig., right part), if the compacta $Q _ {1}$ and $Q _ {2}$ are situated in $Q$ and if

$$\delta = \ \min _ {\begin{array}{c} ( x, t) \in Q _ {1} , \\ ( y, \tau ) \in Q _ {2} \end{array} } \ ( t - \tau ) > 0,$$

then

$$\max _ {( x, t) \in Q _ {2} } u ( x, t) \leq M \ \min _ {( x, t) \in Q _ {1} } u ( x, t),$$

where

$$M = M ( \delta , Q, Q _ {1} , Q _ {2} , L).$$

The example of the function

$$u ( x, t) = \mathop{\rm exp} \left ( \sum _ {i = 1 } ^ { n } k _ {i} x ^ {i} + t \sum _ {i = 1 } ^ { n } k _ {i} ^ {2} \right ) ,$$

which is a solution of the heat equation $u _ {t} - \Delta u = 0$ for any $k _ {1} \dots k _ {n}$, shows that in the parabolic case it is impossible to have two-sided estimates.

How to Cite This Entry:
Harnack inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Harnack_inequality&oldid=47187
This article was adapted from an original article by L.I. KamyninL.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article