# Harnack inequality

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(dual Harnack inequality)

An inequality that gives an estimate from above and an estimate from below for the ratio of two values of a positive harmonic function; obtained by A. Harnack . Let be a harmonic function in a domain of an -dimensional Euclidean space; let be the ball with radius and centre at the point . If the closure , then the Harnack inequality (1)

or is valid for all , . If is a compactum, , then there exists a number such that (2)

for any . In particular, 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 with a uniformly positive-definite matrix : where are numbers, is any -dimensional vector and . The constant in inequality (2) depends only on , , certain norms of the lower coefficients of the operator , and the distance between the boundaries of and of . Figure: h046600a

The analogue of Harnack's inequality is also applicable  to non-negative solutions of uniformly-parabolic equations of the form (the coefficients of the operator may also depend on ). In such a case only a one-sided inequality is possible for points lying inside the paraboloid which is concave downwards with apex at (Fig., left part). Here depends on , , , , , , on certain norms of the lower coefficients of the operator , and on the distance between the boundary of the paraboloid and the boundary of the domain on which . If, for instance, in the cylinder if the distance between and is at least and if is sufficiently small, then the inequality is valid in . In particular, if in (Fig., right part), if the compacta and are situated in and if then where The example of the function which is a solution of the heat equation for any , shows that in the parabolic case it is impossible to have two-sided estimates.