Verma module - Revision history

en>R'n'B: Disambiguated: Harish-Chandra theorem → Harish-Chandra isomorphism

2014-08-13T00:03:04Z

Disambiguated: Harish-Chandra theorem → Harish-Chandra isomorphism

← Older revision		Revision as of 02:03, 13 August 2014
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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.		Hello, my title is Andrew and my spouse doesn't like it at all. Credit authorising is how he makes money. She is really fond of caving but she doesn't have the time recently. For many years he's been residing in Mississippi and he doesn't strategy on altering it.<br><br>Feel free to visit my webpage; accurate psychic predictions ([https://www-Ocl.gist.ac.kr/work/xe/?document_srl=605236 click])

	~~==The statement==~~

	~~[[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.]]~~
	'''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''

	~~:<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>~~

	~~In the plane ''R''<sup>2</sup> (''n'' = 2) the inequality can be written:~~

	~~:<math>{R-r\over R+r} f(x_0)\le f(x)\le {R+r\over R-r}f(x_0)~~.~~</math>~~

	~~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~~

	~~:<math> \sup_{x \in \omega} u(x) \le C \inf_{x \in \omega} u(x)</math>~~

	~~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~~.

	~~==Proof of Harnack's inequality in a ball==~~
	~~By [[Poisson kernel\|Poisson's formula]]~~

	~~:<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>~~

	~~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>\|.~~

	~~Since~~

	~~:<math>\displaystyle{R-r \le \|x-y\| \le R+r,}</math>~~

	the ~~kernel in the integrand satisfies~~

	~~:<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>~~

	~~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:~~

	~~:<math>\displaystyle{f(x_0)={1\over R^{n-1}\omega_{n-1}} \int_{\|y-x_0\|=R} f(y)\, dy.}</math>~~

	~~==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 (mathematics)\|norm]] of the data:~~
	~~:<math>\sup u \le C ( \inf u + \|\|f\|\|)</math>~~
	~~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 <math>\mathcal{M}</math> be a smooth domain in <math>\mathbb{R}^n</math> and consider the linear parabolic operator~~

	~~: <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>~~

	~~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~~

	~~: <math>\frac{\partial u}{\partial t}-\mathcal{L}u\ge0</math> in <math~~>(~~0,T)\times\mathcal{M}</math>~~

	~~such that~~

	: ~~<math>\quad u(t,x)\ge0<~~/~~math> in <math>\quad(0,T)\times\mathcal{M}.<~~/~~math>~~

	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>,

	~~: <math>\sup_K u(t~~-~~\tau,\cdot)\le C\inf_K u(t,\cdot)~~.~~\,</math>~~

	~~==See also==~~

	*[[Harnack's theorem]]
	*[[Harmonic function]]

	~~==References==~~

	*{{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}}~~
	*{{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}}~~
	*{{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}}
	*{{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}}
	*{{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}}~~
	*{{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}}~~
	*{{springer\|id=h/h046620\|title=Harnack theorem\|first=L.I.\|last= Kamynin}}
	*{{springer\|id=H/h046600\|first1=L.I.\|last1= Kamynin\|first2=L.P.\|last2= Kuptsov}}
	*{{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}}
	*{{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}}~~
	*{{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}}
	*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.~~

	~~[[Category:Harmonic functions]]~~
	~~[[Category:Inequalities]]~~

71.95.34.230: /* Definition of Verma modules */

2013-11-21T17:37:25Z

Definition of Verma modules

← Older revision		Revision as of 19:37, 21 November 2013
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	The ~~author~~ is ~~recognized~~ by the ~~title of Numbers Wunder~~. ~~To collect coins~~ is 1 of the ~~things I love most~~. ~~For years he~~'s ~~been operating as~~ a ~~meter reader~~ and it's ~~something he truly appreciate~~. ~~Puerto Rico~~ is ~~where he~~'s ~~usually been living but she requirements to transfer simply because~~ of ~~her family~~.<br><br>~~My blog~~ [http://~~test~~.~~gydch~~.~~com~~/~~praise~~/~~919726 at home std testing~~]		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.

			==The statement==

			[[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.]]
			'''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''

			:<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>

			In the plane ''R''<sup>2</sup> (''n'' = 2) the inequality can be written:

			:<math>{R-r\over R+r} f(x_0)\le f(x)\le {R+r\over R-r}f(x_0).</math>

			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

			:<math> \sup_{x \in \omega} u(x) \le C \inf_{x \in \omega} u(x)</math>

			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.

			==Proof of Harnack's inequality in a ball==
			By [[Poisson kernel\|Poisson's formula]]

			:<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>

			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>\|.

			Since

			:<math>\displaystyle{R-r \le \|x-y\| \le R+r,}</math>

			the kernel in the integrand satisfies

			:<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>

			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:

			:<math>\displaystyle{f(x_0)={1\over R^{n-1}\omega_{n-1}} \int_{\|y-x_0\|=R} f(y)\, dy.}</math>

			==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 (mathematics)\|norm]] of the data:
			:<math>\sup u \le C ( \inf u + \|\|f\|\|)</math>
			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 <math>\mathcal{M}</math> be a smooth domain in <math>\mathbb{R}^n</math> and consider the linear parabolic operator

			: <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>

			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

			: <math>\frac{\partial u}{\partial t}-\mathcal{L}u\ge0</math> in <math>(0,T)\times\mathcal{M}</math>

			such that

			: <math>\quad u(t,x)\ge0</math> in <math>\quad(0,T)\times\mathcal{M}.</math>

			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>,

			: <math>\sup_K u(t-\tau,\cdot)\le C\inf_K u(t,\cdot).\,</math>

			==See also==

			*[[Harnack's theorem]]
			*[[Harmonic function]]

			==References==

			*{{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}}
			*{{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}}
			*{{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}}
			*{{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}}
			*{{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}}
			*{{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}}
			*{{springer\|id=h/h046620\|title=Harnack theorem\|first=L.I.\|last= Kamynin}}
			*{{springer\|id=H/h046600\|first1=L.I.\|last1= Kamynin\|first2=L.P.\|last2= Kuptsov}}
			*{{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}}
			*{{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}}
			*{{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}}
			*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.

			[[Category:Harmonic functions]]
			[[Category:Inequalities]]

en>FrescoBot: Bot: fixing section wikilinks

2012-07-19T16:04:17Z

Bot: fixing section wikilinks

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