Editing Maximum principle (section)

{{Short description|Theorem in complex analysis}}
{{hatnote|This article describes the maximum principle in the theory of partial differential equations. For the maximum principle in optimal control theory, see [[Pontryagin's maximum principle]]. For the theorem in complex analysis, see [[Maximum modulus principle]].}}

In the mathematical fields of [[differential equations]] and [[geometric analysis]], the '''maximum principle''' is one of the most useful and best known tools of study. Solutions of a differential inequality in a domain ''D'' satisfy the '''maximum principle''' if they achieve their maxima at the boundary of ''D''.

The maximum principle enables one to obtain information about solutions of differential equations without any explicit knowledge of the solutions themselves. In particular, the maximum principle is a useful tool in the numerical approximation of solutions of ordinary and partial differential equations and in the determination of bounds for the errors in such approximations.<ref>{{Cite book |last=Protter |first=Murray H. |title=Maximum principles in differential equations |last2=Weinberger |first2=Hans Felix |date=1984 |publisher=Springer |isbn=978-3-540-96068-3 |location=New York Berlin Heidelberg [etc.]}}</ref> 

In a simple two-dimensional case, consider a function of two variables {{math|''u''(''x'',''y'')}} such that
:<math>\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=0.</math>
The '''weak maximum principle''', in this setting, says that for any open precompact subset {{mvar|M}} of the domain of {{mvar|u}}, the maximum of {{mvar|u}} on the closure of {{mvar|M}} is achieved on the boundary of {{mvar|M}}. The '''strong maximum principle''' says that, unless {{mvar|u}} is a constant function, the maximum cannot also be achieved anywhere on {{mvar|M}} itself.

Such statements give a striking qualitative picture of solutions of the given differential equation. Such a qualitative picture can be extended to many kinds of differential equations. In many situations, one can also use such maximum principles to draw precise quantitative conclusions about solutions of differential equations, such as control over the size of their [[gradient]]. There is no single or most general maximum principle which applies to all situations at once.

In the field of [[convex optimization]], there is an analogous statement which asserts that the maximum of a [[convex function]] on a [[compact set|compact]] [[convex set]] is attained on the [[boundary (topology)|boundary]].<ref>Chapter 32 of [[R. Tyrrell Rockafellar|Rockafellar]] (1970).</ref>