Harnack's inequality

In mathematics, Harnack's inequality is an inequality relating the values of a positive harmonic function at two points, introduced by A. Harnack (1887). J. Serrin (1955), and J. Moser (1961, 1964) generalized Harnack's inequality to solutions of elliptic or parabolic partial differential equations. Perelman's solution of the Poincaré conjecture uses a version of the Harnack inequality, found by R. Hamilton (1993), for the Ricci flow. Harnack's inequality is used to prove Harnack's theorem about the convergence of sequences of harmonic functions. Harnack's inequality can also be used to show the interior regularity of weak solutions of partial differential equations. In mathematics, Harnack's inequality is an inequality relating the values of a positive harmonic function at two points, introduced by A. Harnack (1887). J. Serrin (1955), and J. Moser (1961, 1964) generalized Harnack's inequality to solutions of elliptic or parabolic partial differential equations. Perelman's solution of the Poincaré conjecture uses a version of the Harnack inequality, found by R. Hamilton (1993), for the Ricci flow. Harnack's inequality is used to prove Harnack's theorem about the convergence of sequences of harmonic functions. Harnack's inequality can also be used to show the interior regularity of weak solutions of partial differential equations. Harnack's inequality applies to a non-negative function f defined on a closed ball in Rn with radius R and centre x0. It states that, if f is continuous on the closed ball and harmonic on its interior, then for every point x with |x − x0| = r < R, In the plane R2 (n = 2) the inequality can be written: For general domains Ω {displaystyle Omega } in R n {displaystyle mathbf {R} ^{n}} the inequality can be stated as follows: If ω {displaystyle omega } is a bounded domain with ω ¯ ⊂ Ω {displaystyle {ar {omega }}subset Omega } , then there is a constant C {displaystyle C} such that for every twice differentiable, harmonic and nonnegative function u ( x ) {displaystyle u(x)} . The constant C {displaystyle C} is independent of u {displaystyle u} ; it depends only on the domains Ω {displaystyle Omega } and ω {displaystyle omega } . By Poisson's formula where ωn − 1 is the area of the unit sphere in Rn and r = |x − x0|.