Maximum principle

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.

In convex optimization, the maximum principle states that the maximum of a convex function on a compact convex set is attained on the boundary.[1]

The classical example

Harmonic functions are the classical example to which the strong maximum principle applies. Formally, if f is a harmonic function, then f cannot exhibit a true local maximum within the domain of definition of f. In other words, either f is a constant function, or, for any point $x_0\,$ inside the domain of f, there exist other points arbitrarily close to $x_0\,$ at which f takes larger values.[2]

Let f be an harmonic function defined on some connected open subset D of the Euclidean space Rn. If $x_0\,$ is a point in D such that

$f(x_0)\ge f(x) \,$

for all x in a neighborhood of $x_0\,$, then the function f is constant on D.

By replacing "maximum" with "minimum" and "larger" with "smaller", one obtains the minimum principle for harmonic functions.

The maximum principle also holds for the more general subharmonic functions, while superharmonic functions satisfy the minimum principle.[3]

Heuristics for the proof

The weak maximum principle for harmonic functions is a simple consequence of facts from calculus. The key ingredient for the proof is the fact that, by the definition of a harmonic function, the Laplacian of f is zero. Then, if $x_0\,$ is a non-degenerate critical point of f(x), we must be seeing a saddle point, since otherwise there is no chance that the sum of the second derivatives of f is zero. This of course is not a complete proof, and we left out the case of $x_0\,$ being a degenerate point, but this is the essential idea.

The strong maximum principle relies on the Hopf lemma, and this is more complicated.