Contents
Maximum Principle in Analysis
Author: Alex Beers
Table of Contents
- Introduction and Overview
- What is a partial differential equations
- What are parabolic and elliptic PDEs
- Intuition for the maximum principle
- Minimums
- Applications
Introduction and Overview
The Maximum Principle in analysis describes where maximum values can be found for functions described by parabolic and elliptic partial differential equations. These partial differential equations (PDEs) will be explained later on, but for now they can be thought of as equations that describe how a function’s values change, as opposed to the actual values at any one point. The Maximum Principle states that the maximum value for a function described by a PDE occurs at either the edge of the domain, or at any point inside the domain at the beginning of when the function is described. These facts are only true if the value of the function is not constant, in which the maximum value would be everywhere.
This principle can also be applied to finding minimums, which makes it a practical tool in mathematics for finding the max or min of a function described by a PDE, as only the boundary of the domain has to be checked. It also has many applications in physics and engineering, one of the most common of which has to do with quantities that tend to disperse or approach some equilibrium over time, such as temperature, concentration, or pressure, among many others. This also has nice implications for learners, in that the principle has many intuitive properties.
What is a partial differential equations
Partial differential equations describe the rates of change of functions with multiple independent variables. They are used because it is often easier to describe how a value is changing at some point, rather than what the value actually is. For the best possible introduction to PDEs, I would check out the 17 minute video by 3 Blue 1 Brown on the subject. It is the first link in the references below and provides wonderful graphical intuition via animations, which simply cannot be described here. An explanation is provided here as well, of course.
To first understand Partial Differential Equations, we have to understand Partial Derivatives. Partial derivatives are basically derivatives for one independent variable (normal derivatives) but for multiple independent variables. The trick is that we consider all of the variables as constant except for one. An example is shown below for a function with independent variables x and y.
The partial derivative with respect to x (called partial x) is the rate of change of the function as we walk along a line parallel to the x axis. If y = 0, the function is always zero, so it makes sense that partial x is always 0. If y is positive, the function positively increases as we walk parallel to the y axis, as shown by the partial equation. Higher order partials can be taken in the same manner.
Partials for each independent variable are added together to create the PDE of a function. For example, the PDE for the equations above would be NEED A LITTLE FINAL EXPLANATION FOR PDEs This PDE, coupled with values for initial velocity (dx/dt) and position (x) would
What are parabolic and elliptic PDEs
Parabolic PDEs, for our purposes, will be considered to deal with time and and tend towards some equilibrium. This is a big generalization, but will work for explaining the basics of the Maximum Principle. More information can be found in the refrences below about parabolic PDEs. These equations describe a range of time dependent processes such as heat dispersion, diffusion of particles, and even economic models. All of these processes can be described by what is called “the heat equation” which was created to describes how heat flows throughout an object over time. We will focus on the heat equation here, as it leads to an incredibly intuitive explanation of the Maximum Principle.
Suppose a rod is heated up and has a temperature distribution as shown below in Fig. 1, where the graph represents the temperature at any point along the rod. Intuitively, which parts along the rod would cool down the fastest? It would seem that hot portions surrounded by cold neighbors would cool down the fastest, and cold portions with hot neighbors would warm up the fastest. In addition, a portion whose neighbor is say, 10 degrees colder on one side, and 10 degrees warmer on the other side would stay the same temperature — the average of the two neighbors. Looking back at Fig. 1, we may begin to recognize a pattern — portions with high concavity experience the highest temperature change over time. Since concavity can be described by the second derivative of position here, we can say
where ALPHA is a proportionality constant, and can be expanded to
to describe objects in more than one dimension. This is the heat equation, and an example of a parabolic PDE.
