# Scalar weak maximum principle

From Diffgeom

(Redirected from Maximum principle)

Template:Flow equation property

## Contents

## Definition

### Basic definition

Let be a differential manifold and be a differential operator that acts on functions . Consider the flow equation associated with , namely the equation for given as:

An *initial value problem* corresponding to this differential equation is a specification of for each .

The differential operator is said to satisfy the **scalar weak maximum principle** if whenever is a solution for which there are constants and such that for all , then for all .

In other words, any bounded set in which the range of lies also contains the image of for all .

### Definition in terms of trajectory properties

A differential operator 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