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 also implies the regularity of the function in the interior of its domain.

The statement

Let $D=D(z_0,R)$ be an open disk in the plane and let f be a harmonic function on D such that f(z) is non-negative for all $z \in D$. Then the following inequality holds for all $z \in D$:

$0\le f(z)\le \left( \frac{R}{R-\left|z-z_0\right|}\right)^2f(z_0).$

For general domains $\Omega$ in $\mathbf{R}^n$ the inequality can be stated as follows: If $\omega$ is a bounded domain with $\bar{\omega} \subset \Omega$, then there is a constant $C$ such that

$\sup_{x \in \omega} u(x) \le C \inf_{x \in \omega} u(x)$

for every twice differentiable, harmonic and nonnegative function $u(x)$. The constant $C$ is independent of $u$; it depends only on the domain.

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 $\mathcal{M}$ be a smooth domain in $\mathbb{R}^n$ and consider the linear parabolic operator

$\mathcal{L}u=\sum_{i,j=1}^n a_{ij}(t,\xi)\frac{\partial^2 u}{\partial x_i\,\partial x_j}+\sum_{i=1}^n b_i(t,\xi)\frac{\partial u}{\partial x_i} + c(t,\xi)u$

with smooth and bounded coefficients and a nondegenerate matrix $(a_{ij})$. Suppose that $u(t,x)\in C^2((0,T)\times\mathcal{M})$ is a solution of

$\frac{\partial u}{\partial t}-\mathcal{L}u\ge0$ in $(0,T)\times\mathcal{M}$

such that

$\quad u(t,x)\ge0$ in $\quad(0,T)\times\mathcal{M}.$

Let $K$ be a compact subset of $\mathcal{M}$ and choose $\tau\in(0,T)$. Then for each $\quad t\in(\tau,T)$ there exists a constant $\quad C>0$ (depending only on $K$, $\tau$ and the coefficients of $\mathcal{L}$) such that

$\sup_K u(t-\tau,\cdot)\le C\inf_K u(t,\cdot).\,$

See also

• Harnack's theorem
• Harmonic function

References

• Caffarelli, Luis A.; Xavier Cabre (1995), Fully Nonlinear Elliptic Equations, Providence, Rhode Island: American Mathematical Society, pp. 31–41, ISBN 0-8218-0437-5 
• Gilbarg, David; Neil S. Trudinger (1988), Elliptic Partial Differential Equations of Second Order, Springer, ISBN 3-540-41160-7 
• Hamilton, Richard S. (1993), "The Harnack estimate for the Ricci flow", Journal of Differential Geometry 37 (1): 225–243, ISSN 0022-040X, MR1198607 
• Harnack, A. (1887), Die Grundlagen der Theorie des logarithmischen Potentiales und der eindeutigen Potentialfunktion in der Ebene, Leipzig: V. G. Teubner, http://www.archive.org/details/vorlesunganwend00weierich 
• Kamynin, L.I. (2001), "Harnack theorem", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=h/h046620 
• Kamynin, L.I.; Kuptsov, L.P. (2001), "Harnack's inequality", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=H/h046600 
• Moser, Jürgen (1961), "On Harnack's theorem for elliptic differential equations", Communications on Pure and Applied Mathematics 14 (3): 577–591, doi:10.1002/cpa.3160140329, MR0159138 
• Moser, Jürgen (1964), "A Harnack inequality for parabolic differential equations", Communications on Pure and Applied Mathematics 17 (1): 101–134, doi:10.1002/cpa.3160170106, MR0159139 
• Serrin, James (1955), "On the Harnack inequality for linear elliptic equations", Journal d'Analyse Mathématique 4 (1): 292–308, doi:10.1007/BF02787725, MR0081415 
• 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.