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| In mathematics, '''Harnack's inequality''' is an [[inequality (mathematics)|inequality]] relating the values of a positive [[harmonic function]] at two points, introduced by {{harvs|txt|authorlink=Carl Gustav Axel Harnack|first=A.|last=Harnack|year=1887}}. {{harvs|txt|first=J. |last=Serrin|authorlink=James Serrin|year=1955}} and {{harvs|txt|last=Moser|first=J.|authorlink=Jürgen Moser |year1=1961|year4=1964}} generalized Harnack's inequality to solutions of elliptic or parabolic [[partial differential equation]]s. [[Grigori Perelman|Perelman]]'s solution of the [[Poincaré conjecture]] uses a version of the Harnack inequality, found by {{harvs|txt|first=R.|last=Hamilton|authorlink=Richard Hamilton (mathematician)|year=1993|txt}}, 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 [[Hölder condition|regularity]] of the function in the interior of its domain.
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| ==The statement==
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| [[Image:Harnack.png|thumb|200px|A harmonic function (green) over a disk (blue) is bounded from above by a function (red) that coincides with the harmonic function at the disk center and approaches infinity towards the disk boundary.]]
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| '''Harnack's inequality''' applies to a non-negative function ''f'' defined on a closed ball in '''R'''<sup>''n''</sup> with radius ''R'' and centre ''x''<sub>0</sub>. It states that, if ''f'' is continuous on the closed ball and [[harmonic function|harmonic]] on its interior, then for any point ''x'' with |''x'' - ''x''<sub>0</sub>| = ''r'' < ''R''
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| :<math>\displaystyle{{1-(r/R)\over [1+(r/R)]^{n-1}}f(x_0)\le f(x) \le {1+(r/R)\over [1-(r/R)]^{n-1}} f(x_0).}</math>
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| In the plane ''R''<sup>2</sup> (''n'' = 2) the inequality can be written:
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| :<math>{R-r\over R+r} f(x_0)\le f(x)\le {R+r\over R-r}f(x_0).</math>
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| For general domains <math>\Omega</math> in <math>\mathbf{R}^n</math> the inequality can be stated as follows: If <math>\omega</math> is a bounded domain with <math>\bar{\omega} \subset \Omega</math>, then there is a constant <math>C</math> such that
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| :<math> \sup_{x \in \omega} u(x) \le C \inf_{x \in \omega} u(x)</math>
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| for every twice differentiable, harmonic and nonnegative function <math>u(x)</math>. The constant <math>C</math> is independent of <math>u</math>; it depends only on the domain.
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| ==Proof of Harnack's inequality in a ball==
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| By [[Poisson kernel|Poisson's formula]]
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| :<math>\displaystyle{f(x) = {1\over \omega_{n-1}} \int_{|y-x_0|=R} {R^2 -r^2\over R|x-y|^n}\cdot f(y)\, dy,}</math>
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| where ω<sub>''n'' − 1</sub> is the area of the unit sphere in '''R'''<sup>''n''</sup> and ''r'' = |''x'' - ''x''<sub>0</sub>|.
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| Since
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| :<math>\displaystyle{R-r \le |x-y| \le R+r,}</math>
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| the kernel in the integrand satisfies | |
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| :<math>\displaystyle{{R -r\over R (R+r)^{n-1}} \le {R^2 -r^2\over R|x-y|^n}\le {R+r\over R(R-r)^{n-1}}.}</math>
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| 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 it value at the center of the sphere:
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| :<math>\displaystyle{f(x_0)={1\over R^{n-1}\omega_{n-1}} \int_{|y-x_0|=R} f(y)\, dy.}</math>
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| ==Elliptic partial differential equations==
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| 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 (mathematics)|norm]] of the data: | |
| :<math>\sup u \le C ( \inf u + ||f||)</math>
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| The constant depends on the ellipticity of the equation and the connected open region.
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| ==Parabolic partial differential equations==
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| There is a version of Harnack's inequality for linear parabolic PDEs such as [[heat equation]].
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| Let <math>\mathcal{M}</math> be a smooth domain in <math>\mathbb{R}^n</math> and consider the linear parabolic operator
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| : <math>\mathcal{L}u=\sum_{i,j=1}^n a_{ij}(t,x)\frac{\partial^2 u}{\partial x_i\,\partial x_j}+\sum_{i=1}^n b_i(t,x)\frac{\partial u}{\partial x_i} + c(t,x)u</math>
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| with smooth and bounded coefficients and a nondegenerate matrix <math>(a_{ij})</math>. Suppose that <math>u(t,x)\in C^2((0,T)\times\mathcal{M})</math> is a solution of
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| : <math>\frac{\partial u}{\partial t}-\mathcal{L}u\ge0</math> in <math>(0,T)\times\mathcal{M}</math>
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| such that
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| : <math>\quad u(t,x)\ge0</math> in <math>\quad(0,T)\times\mathcal{M}.</math> | |
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| Let <math>K</math> be a compact subset of <math>\mathcal{M}</math> and choose <math>\tau\in(0,T)</math>. Then there exists a constant <math>\quad C>0</math> (depending only on <math>K</math>, <math>\tau</math> and the coefficients of <math>\mathcal{L}</math>) such that, for each <math>\quad t\in(\tau,T)</math>,
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| : <math>\sup_K u(t-\tau,\cdot)\le C\inf_K u(t,\cdot).\,</math>
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| ==See also==
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| *[[Harnack's theorem]]
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| *[[Harmonic function]]
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| ==References==
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| *{{Citation |title=Fully Nonlinear Elliptic Equations |last=Caffarelli |first=Luis A. |coauthors=Xavier Cabre |year=1995 |publisher=American Mathematical Society |location=Providence, Rhode Island |pages=31–41 |isbn=0-8218-0437-5}}
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| *{{citation|last=Folland|first= Gerald B.|title= Introduction to partial differential equations|edition=2nd|publisher=Princeton University Press|year=1995|id= ISBN 0-691-04361-2}}
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| *{{Citation |title= Elliptic Partial Differential Equations of Second Order |last=Gilbarg |first=David |coauthors=Neil S. Trudinger | year=1988| publisher=Springer |isbn=3-540-41160-7}}
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| *{{Citation | last1=Hamilton | first1=Richard S. | title=The Harnack estimate for the Ricci flow | id={{MathSciNet | id = 1198607}} | year=1993 | journal=Journal of Differential Geometry | issn=0022-040X | volume=37 | issue=1 | pages=225–243}}
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| *{{citation|first=A. |last=Harnack|title=Die Grundlagen der Theorie des logarithmischen Potentiales und der eindeutigen Potentialfunktion in der Ebene|publisher=V. G. Teubner|place= Leipzig |year=1887|url=http://www.archive.org/details/vorlesunganwend00weierich}}
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| *{{citation|last=John|first= Fritz|title=Partial differential equations|edition=4th|series= Applied Mathematical Sciences|volume= 1|publisher= Springer-Verlag|year= 1982|id= ISBN 0-387-90609-6}}
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| *{{springer|id=h/h046620|title=Harnack theorem|first=L.I.|last= Kamynin}}
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| *{{springer|id=H/h046600|first1=L.I.|last1= Kamynin|first2=L.P.|last2= Kuptsov}}
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| *{{Citation | last1=Moser | first1=Jürgen | title=On Harnack's theorem for elliptic differential equations | id={{MathSciNet | id = 0159138}} | year=1961 | journal=[[Communications on Pure and Applied Mathematics]] | volume=14 | issue=3 | pages=577–591 | doi=10.1002/cpa.3160140329}}
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| *{{Citation | last1=Moser | first1=Jürgen | title=A Harnack inequality for parabolic differential equations | id={{MathSciNet | id = 0159139}} | year=1964 | journal=[[Communications on Pure and Applied Mathematics]] | volume=17 | issue=1 | pages=101–134 | doi=10.1002/cpa.3160170106}}
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| *{{Citation | last1=Serrin | first1=James | title=On the Harnack inequality for linear elliptic equations | id={{MathSciNet | id = 0081415}} | year=1955 | journal=Journal d'Analyse Mathématique | volume=4 | issue=1 | pages=292–308 | doi=10.1007/BF02787725}}
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| *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.
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| [[Category:Harmonic functions]]
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| [[Category:Inequalities]]
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