Scalar weak maximum principle: Difference between revisions

Latest revision as of 20:08, 18 May 2008

Definition

Basic definition

Let $M$ be a differential manifold and $F$ be a differential operator that acts on functions $M\to \mathbb {R}$ . Consider the flow equation associated with $f$ , namely the equation for $u:\mathbb {R} \times M\to \mathbb {R}$ given as:

${\frac {\partial u}{\partial t}}=F(u)$

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

The differential operator $F$ is said to satisfy the scalar weak maximum principle if whenever $u$ is a solution for which there are constants $C_{1}$ and $C_{2}$ such that $C_{1}\leq u(0,x)\leq C_{2}$ for all $x\in M$ , then $C_{1}\leq u(t,x)\leq C_{2}$ for all $x\in M,t\in \mathbb {R} ^{+}$ .

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

Definition in terms of trajectory properties

A differential operator $F$ is said to satisfy the scalar weak maximum principle if the trajectories of the corresponding flow equation are all bound-narrowing.

Relation with other properties=

One-sided maximum principles

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

@@ Line 1: / Line 1: @@
-{{diffeq property}}
+{{flow equation property}}
 ==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>).
+===Basic definition===
-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>.
+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:
-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>.
+<math>\frac{\partial u}{\partial t} = F(u)</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>.
+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>.
-==Significance==
+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>.
-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 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>.
-In fact, this can also be used to rigourously establish that diffusion equations (such as the heat equation) satisfy the maximum principle.
+===Definition in terms of trajectory properties===
+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===
+===One-sided maximum principles===
+* 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