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 $C_{1}\leq u(0,x)\leq C_{2}$ 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