# 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 maximum 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 inside the domain of *f*, there exist other points arbitrarily close to at which *f* takes larger values.^{[2]}

Let *f* be a harmonic function defined on some connected open subset *D* of the Euclidean space **R**^{n}. If is a point in *D* such that

for all *x* in a neighborhood of , 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 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 being a degenerate point, but this is the essential idea.

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

## See also

## References

- ↑ Chapter 32 of Rockafellar (1970).
- ↑ Berenstein and Gay.
- ↑ Evans.

- Berenstein, Carlos A.; Roger Gay (1997).
*Complex Variables: An Introduction*. Springer (Graduate Texts in Mathematics). ISBN 0-387-97349-4. - Caffarelli, Luis A.; Xavier Cabre (1995).
*Fully Nonlinear Elliptic Equations*. Providence, Rhode Island: American Mathematical Society. pp. 31–41. ISBN 0-8218-0437-5. - Evans, Lawrence C. (1998).
*Partial Differential Equations*. Providence, Rhode Island: American Mathematical Society. ISBN 0-8218-0772-2. - Rockafellar, R. T. (1970).
*Convex analysis*. Princeton: Princeton University Press. - Gilbarg, D.; Trudinger, Neil (1983).
*Elliptic Partial Differential Equations of Second Order*. New York: Springer. ISBN 3-540-41160-7.