- This article describes the maximum principle in the theory of partial differential equations. For the maximum principle in optimal control theory, see Pontryagin's minimum principle.
In
mathematics, the
maximum principle is a property of solutions to certain
partial differential equations, of the
elliptic and
parabolic types. Roughly speaking, it says that the
maximum of a function in a
domain is to be found on the boundary of that domain. Specifically, the
strong maximum principle says that if a function achieves its maximum in the interior of the domain, the function is uniformly a constant. The
weak maximum principle says that the maximum of the function is to be found on the boundary, but may re-occur in the interior as well. Other, even weaker maximum principles exist which merely bound a function in terms of its maximum on the boundary.