Scalar weak maximum principle - Revision history

Vipul: 6 revisions

2008-05-18T20:08:22Z

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Vipul: Started the page

2007-04-08T05:02:33Z

Started the page

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	==Definition==		==Definition==

			===Basic definition===

	Let <math>M</math> be a [[differential manifold]] and <math>F</math> be a differential operator that acts on functions <math>M \to \R</math>. Consider the [[flow equation]] associated with <math>f</math>, namely the equation for <math>u:\R \times M \to \R</math> given as:		Let <math>M</math> be a [[differential manifold]] and <math>F</math> be a differential operator that acts on functions <math>M \to \R</math>. Consider the [[flow equation]] associated with <math>f</math>, namely the equation for <math>u:\R \times M \to \R</math> given as:
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	<math>\frac{\partial u}{\partial t} = F(u)</math>		<math>\frac{\partial u}{\partial t} = F(u)</math>

	An ''initial value problem'' corresponding to this differential equation is a specification of <math>~~u(x,0)</math> for each <math>x \in M</math>.~~		An ''initial value problem'' corresponding to this differential equation is a specification of <math>u(0,x)</math> for each <math>x \in M</math>.

	Such a differential equation is said to satisfy the '''scalar weak maximum principle''' if whenever <math>u</math> is a solution for which there are constants <math>C_1</math> and <math>C_2</math> such that <math>C_1 \le u(0,x) ~~\le C_2~~</math> for ~~all~~ <math>x \in M~~</math>, then <math>C_1 \le u(t,x) \le C_2</math> for all <math>x \in M, t \in \R^+~~</math>.

	~~In other words, any bounded set in~~ which ~~the range of~~ <math>u(x,0)</math> ~~lies also contains the image of~~ <math>u(t,x)</math> for all <math>t</math>.		The differential operator <math>F</math> is said to satisfy the '''scalar weak maximum principle''' if whenever <math>u</math> is a solution for which there are constants <math>C_1</math> and <math>C_2</math> such that <math>C_1 \le u(0,x) \le C_2</math> for all <math>x \in M</math>, then <math>C_1 \le u(t,x) \le C_2</math> for all <math>x \in M, t \in \R^+</math>.

	~~==Significance==~~		In other words, any bounded set in which the range of <math>u(0,x)</math> lies also contains the image of <math>u(t,x)</math> for all <math>t</math>.

	~~The maximum principle makes sense for diffusion processes, where we are~~ in ~~general trying to equalize the value~~ of <math>u</math> across the manifold. Thus, there is no reason for the value at a point to go up unless the value in its neighbourhood is higher than it. In particular, the value at no point can exceed the maximum.		===Definition in terms of trajectory properties===

	~~In fact, this can also be used~~ to ~~rigourously establish that diffusion equations (such as the heat equation)~~ satisfy the maximum principle.		A differential operator <math>F</math> is said to satisfy the scalar weak maximum principle if the trajectories of the corresponding [[flow equation]] are all [[bound-narrowing trajectory\|bound-narrowing]].

	==Relation with other properties==		==Relation with other properties===

	===One-sided maximum principles===		===One-sided maximum principles===

	* The flow equation is said to satisfy a		* The flow equation is said to satisfy a one-sided scalar weak maximum principle if the trajectories of the flow equation are only min-increasing

Vipul at 02:50, 8 April 2007

2007-04-08T02:50:29Z

← Older revision		Revision as of 02:50, 8 April 2007
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	{{~~diffeq~~ property}}		{{flow equation property}}

	==Definition==		==Definition==

	~~Consider a differential equation involving one dependent variable~~ <math>u</math> ~~that is~~ a ~~function of~~ <math>~~\R \times M~~</math> ~~where~~ <math>M</math> ~~denotes a manifold (whose generic point will be denoted as~~ <math>x</math>~~) and~~ <math>\R</math> ~~corresponds to the time axis (whose generic point will be denoted~~ as <math>t</math>).		Let <math>M</math> be a [[differential manifold]] and <math>F</math> be a differential operator that acts on functions <math>M \to \R</math>. Consider the [[flow equation]] associated with <math>f</math>, namely the equation for <math>u:\R \times M \to \R</math> given as:

			<math>\frac{\partial u}{\partial t} = F(u)</math>

	An ''initial value problem'' corresponding to this differential equation is a specification of <math>u(x,0)</math> for each <math>x \in M</math>.		An ''initial value problem'' corresponding to this differential equation is a specification of <math>u(x,0)</math> for each <math>x \in M</math>.

	Such a differential equation is said to satisfy the '''maximum principle''' if whenever <math>u</math> is a solution for which there are constants <math>C_1</math> and <math>C_2</math> such that <math>C_1 \le u(0,x) \le C_2</math> for all <math>x \in M</math>, then <math>C_1 \le u(t,x) \le C_2</math> for all <math>x \in M, t \in \R^+</math>.		Such a differential equation is said to satisfy the '''scalar weak maximum principle''' if whenever <math>u</math> is a solution for which there are constants <math>C_1</math> and <math>C_2</math> such that <math>C_1 \le u(0,x) \le C_2</math> for all <math>x \in M</math>, then <math>C_1 \le u(t,x) \le C_2</math> for all <math>x \in M, t \in \R^+</math>.

	In other words, any bounded set in which the range of <math>u(x,0)</math> lies also contains the image of <math>u(t,x)</math> for all <math>t</math>.		In other words, any bounded set in which the range of <math>u(x,0)</math> lies also contains the image of <math>u(t,x)</math> for all <math>t</math>.
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	In fact, this can also be used to rigourously establish that diffusion equations (such as the heat equation) satisfy the maximum principle.		In fact, this can also be used to rigourously establish that diffusion equations (such as the heat equation) satisfy the maximum principle.

			==Relation with other properties==

			===One-sided maximum principles===

			* The flow equation is said to satisfy a

Vipul: Maximum principle moved to Scalar weak maximum principle: The term "maximum principle" is too general and applies to many results

2007-04-08T02:41:40Z

Maximum principle moved to Scalar weak maximum principle: The term "maximum principle" is too general and applies to many results

← Older revision	Revision as of 02:41, 8 April 2007
(No difference)

Vipul at 02:34, 8 April 2007

2007-04-08T02:34:46Z

← Older revision		Revision as of 02:34, 8 April 2007
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	==Definition==		==Definition==

	Consider a differential equation involving one dependent variable <math>u</math> that is a function of <math>M \times \R</math> where <math>M</math> denotes a manifold (whose generic point will be denoted as <math>x</math>) and <math>\R</math> corresponds to the ~~tiem~~ axis (whose generic point will be denoted as <math>t</math>).		Consider a differential equation involving one dependent variable <math>u</math> that is a function of <math>\R \times M</math> where <math>M</math> denotes a manifold (whose generic point will be denoted as <math>x</math>) and <math>\R</math> corresponds to the time axis (whose generic point will be denoted as <math>t</math>).

	An ''initial value problem'' corresponding to this differential equation is a specification of <math>u(x,0)</math> for each <math>x \in M</math>.		An ''initial value problem'' corresponding to this differential equation is a specification of <math>u(x,0)</math> for each <math>x \in M</math>.

	Such a differential equation is said to satisfy the '''maximum principle''' if whenever <math>u</math> is a solution for which there are constants <math>C_1</math> and <math>C_2</math> such that <math>C_1 \le u(x,0) \le C_2</math> for all <math>x \in M</math>, then <math>C_1 \le u(x,t) \le C_2</math> for all <math>x \in M, t \in \R^+</math>.		Such a differential equation is said to satisfy the '''maximum principle''' if whenever <math>u</math> is a solution for which there are constants <math>C_1</math> and <math>C_2</math> such that <math>C_1 \le u(0,x) \le C_2</math> for all <math>x \in M</math>, then <math>C_1 \le u(t,x) \le C_2</math> for all <math>x \in M, t \in \R^+</math>.

	In other words, any bounded set in which the range of <math>u(x,0)</math> lies also contains the image of <math>u(x,t)</math> for all <math>t</math>.		In other words, any bounded set in which the range of <math>u(x,0)</math> lies also contains the image of <math>u(t,x)</math> for all <math>t</math>.

	==Significance==		==Significance==

Vipul: /* Definition */

2007-04-06T00:35:03Z

Definition

← Older revision		Revision as of 00:35, 6 April 2007
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	An ''initial value problem'' corresponding to this differential equation is a specification of <math>u(x,0)</math> for each <math>x \in M</math>.		An ''initial value problem'' corresponding to this differential equation is a specification of <math>u(x,0)</math> for each <math>x \in M</math>.

	Such a differential equation is said to satisfy the '''maximum principle''' if ~~for any solution~~ <math>u</math>~~, if~~ there are constants <math>C_1</math> and <math>C_2</math> such that <math>C_1 \le u(x,0) \le C_2</math> for all <math>x \in M</math>, then <math>C_1 \le u(x,t) \le C_2</math> for all <math>x \in M, t \in \R</math>.		Such a differential equation is said to satisfy the '''maximum principle''' if whenever <math>u</math> is a solution for which there are constants <math>C_1</math> and <math>C_2</math> such that <math>C_1 \le u(x,0) \le C_2</math> for all <math>x \in M</math>, then <math>C_1 \le u(x,t) \le C_2</math> for all <math>x \in M, t \in \R^+</math>.

			In other words, any bounded set in which the range of <math>u(x,0)</math> lies also contains the image of <math>u(x,t)</math> for all <math>t</math>.

			==Significance==

			The maximum principle makes sense for diffusion processes, where we are in general trying to equalize the value of <math>u</math> across the manifold. Thus, there is no reason for the value at a point to go up unless the value in its neighbourhood is higher than it. In particular, the value at no point can exceed the maximum.

			In fact, this can also be used to rigourously establish that diffusion equations (such as the heat equation) satisfy the maximum principle.

Vipul at 00:28, 6 April 2007

2007-04-06T00:28:44Z

New page

{{diffeq property}}

==Definition==

Consider a differential equation involving one dependent variable <math>u</math> that is a function of <math>M \times \R</math> where <math>M</math> denotes a manifold (whose generic point will be denoted as <math>x</math>) and <math>\R</math> corresponds to the tiem axis (whose generic point will be denoted as <math>t</math>).

An ''initial value problem'' corresponding to this differential equation is a specification of <math>u(x,0)</math> for each <math>x \in M</math>.

Such a differential equation is said to satisfy the '''maximum principle''' if for any solution <math>u</math>, if there are constants <math>C_1</math> and <math>C_2</math> such that <math>C_1 \le u(x,0) \le C_2</math> for all <math>x \in M</math>, then <math>C_1 \le u(x,t) \le C_2</math> for all <math>x \in M, t \in \R</math>.