Maximum principle (mathematics)

Illustration of the maximum principle: Both the maxima and the minima of this function (red) lie on the edge of the definition range (blue).

In mathematics, the maximum principle is a property that is fulfilled by solutions of certain partial differential equations. If the maximum principle applies to a function, extensive statements can be made about its behavior even if this function is not known. Roughly speaking, a function satisfies the maximum principle if and only if it assumes its (global) maximum on the edge of its domain of definition.

The strong maximum principle states that a function that assumes its maximum in the interior of its domain must be constant. The weak maximum principle states that the maximum is assumed on the edge, but further maximum points can exist inside the domain of definition. In addition, there are other, even weaker maximum principles. As a rule, statements analogous to the maximum principle also apply to the minimum of a function; these are then referred to as the minimum principle.

The maximum principle can be defined not only for real-valued functions , but also for complex-valued or vector-valued functions . In these cases the maximum for the amount or the norm of the function values ​​is considered. The best known example of this is the class of holomorphic functions .

history

The first maximum principle was established by Bernhard Riemann in his dissertation for the class of harmonic functions . Eberhard Hopf then extended this to the solutions of elliptic differential equations of the second order. For harmonic functions, the maximum principle can be derived quickly from the mean value property of these functions. Heinz Bauer expanded this idea into a general maximum principle for convex cones with functions that are semi-continuous upwards in compact spaces. From this abstract maximum principle it follows, among other things, that upward semi-continuous, convex functions on compact, convex sets assume their maximum on the extreme points of the convex set.

Physical motivation

Solution of a two-dimensional heat equation

Let be a function that gives the temperature of a solid as a function of location and time, so . is time-dependent because the thermal energy spreads over the material over time. The physical self-evident fact that heat does not arise out of nowhere is mathematically reflected in the maximum principle: The maximum value over time and space of the temperature is either at the beginning of the considered time interval ( see also: initial value problem ) or at the edge of the considered spatial area ( see also: Boundary value problem ). ${\displaystyle u}$ ${\displaystyle u=u(x_{1},x_{2},x_{3},t)}$ ${\displaystyle u}$

Applications

In the case of partial differential equations, the maximum principle is of particular interest with regard to Dirichlet boundary conditions . In particular, the uniqueness and stability with respect to small disturbances of the solutions to this problem follow from this.

In addition, the maximum principle applies to:

• Solutions of parabolic differential equations of the 2nd order (e.g. the heat conduction equation ),
• non-linear elliptical systems with quadratic growth in the gradient under the assumption of a smallness condition.

Function theory

The mathematical formulation of the maximum principle is:

Let it be holomorphic in the area . Let there be a point such that in has a local maximum, i.e. i.e., there is a neighborhood of with . Then constant is in . ${\displaystyle f}$ ${\displaystyle G}$ ${\displaystyle c\in G}$ ${\displaystyle |f|}$ ${\displaystyle c}$ ${\displaystyle U\subset G}$ ${\displaystyle c}$ ${\displaystyle |f(c)|=\sup _{z\in U}|f(z)|}$ ${\displaystyle f}$ ${\displaystyle G}$

The other variant of the sentence is:

Let it be a bounded area and let it be a continuous and holomorphic function. Then the function assumes its maximum on the edge of : for all . ${\displaystyle G}$ ${\displaystyle f}$ ${\displaystyle {\bar {G}}=G\cup \partial G}$ ${\displaystyle G}$ ${\displaystyle |f|}$ ${\displaystyle G}$ ${\displaystyle |f(z)|\leq \sup _{\zeta \in \partial G}|f(\zeta )|}$ ${\displaystyle z\in {\bar {G}}}$

The application of the theorem leads directly to the minimum principle: ${\displaystyle 1/f}$

Let it be holomorphic in . Let there be a point such that in has a local minimum, i.e. i.e., there is a neighborhood of with . Then either holds or is constant in . ${\displaystyle f}$ ${\displaystyle G}$ ${\displaystyle c\in G}$ ${\displaystyle f}$ ${\displaystyle c}$ ${\displaystyle U\subset G}$ ${\displaystyle c}$ ${\displaystyle |f(c)|=\inf _{z\in U}|f(z)|}$ ${\displaystyle f(c)=0}$ ${\displaystyle f}$ ${\displaystyle G}$

literature

• Lawrence C. Evans : Partial Differential Equations. Reprinted with corrections. American Mathematical Society, Providence RI 2008, ISBN 978-0-8218-0772-9 ( Graduate studies in mathematics 19).
• David Gilbarg , Neil S. Trudinger: Elliptic Partial Differential Equations of Second Order. 2nd edition, revised 3rd printing. Springer, Berlin et al. 1998 ISBN 3-540-13025-X ( Basic Teachings of Mathematical Sciences 224).
• Erhard Heinz , Günter Hellwig: Partial differential equations. 25.2. until 3/3/1973. Lecture at the Georg-August-Universität Göttingen. Mathematical Research Institute, Oberwolfach 1973 ( Mathematical Research Institute Oberwolfach. Conference Report 1973, 7, ZDB -ID 529790-4 ).
• Murray H. Protter , Hans F. Weinberger: Maximum principles in differential equations. Prentice-Hall, Englewood Cliffs NJ 1967 ( Prentice-Hall partial differential equations series ).
• Friedrich Sauvigny : Partial differential equations of geometry and physics. 2 volumes. Springer, Berlin et al. 2004–2005, ISBN 3-540-20453-9 .
• IN Vekua : Generalized Analytical Functions. Akademie Verlag, Berlin 1963.
• Reinhold Remmert , Georg Schumacher: Theory of functions 1st 5th edition, Springer Verlag, 2001.