Harnack's inequality and Echinochloa caudata: Difference between pages
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In mathematics, '''Harnack's inequality''' is an [[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|year=1955}} and {{harvs|txt|last=Moser|first=J.|authorlink=Jurgen 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 [[Poincare conjecture]] uses a version of the Harnack inequality, found by {{harvs|txt|first=R.|last=Hamilton|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 [[Holder condition|regularity]] of the function in the interior of its domain. |
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| name = ''Echinochloa caudata'' |
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| regnum = [[Plant]]ae |
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| divisio = [[Magnoliophyta]] |
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| classis = [[Liliopsida]] |
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| ordo = [[Poales]] |
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| familia = [[Poaceae]] |
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| subfamilia = [[Panicoideae]] |
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| genus = ''[[Echinochloa]]'' |
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| species = '''''E. caudata''''' |
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| binomial = '''''Echinochloa caudata''''' |
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}} |
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'''''Echinochloa caudata''''', is a species of ''[[Echinochloa]]''. |
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==Harmonic functions== |
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Let <math>D=D(z_0,R)</math> be an [[open_set|open]] [[ball (mathematics)|disk]] in the plane and let ''f'' be a [[harmonic function]] on ''D'' such that ''f(z)'' is non-negative for all <math>z \in D</math>. Then the following inequality holds for all <math>z \in D</math>: |
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[[Category:Underutilized crops]] |
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:<math>0\le f(z)\le \left( \frac{R}{R-\left|z-z_0\right|}\right)^2f(z_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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{{Poales-stub}} |
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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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==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: |
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:<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,\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} u + c(t,\xi)u</math> |
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with smooth and bounded coefficients. 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=0\quad</math> in <math>\quad(0,T)\times\mathcal{M}</math> such that <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 for each <math>\quad t\in(\tau,T)</math> 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 |
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<math>\sup_K u(t-\tau,\cdot)\le C\inf_K u(t,\cdot)\,.</math> |
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==References== |
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*{{cite book |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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*{{cite book |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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*{{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]] | issn=0010-3640 | volume=14 | pages=577–591}} |
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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]] | issn=0010-3640 | volume=17 | pages=101–134}} |
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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 | issn=0021-7670 | volume=4 | pages=292–308}} |
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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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[[vi:Bất đẳng thức Harnack]] |
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Revision as of 02:40, 14 October 2008
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| Echinochloa caudata | |
Echinochloa caudata, is a species of Echinochloa.
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