Scalar weak maximum principle

Definition

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

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

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

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

Significance

The maximum principle makes sense for diffusion processes, where we are in general trying to equalize the value of $u$ 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.