Scalar weak maximum principle

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 $C_1\le u(0,x)\le C_2$ for all $x\in M$, then $C_1\le u(t,x)\le C_2$ 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