Let us start our notes with a very fundamental maximum principle for any strongly second order elliptic operator. We have

Theorem(Maximum principle). Let satisfy the differential inequalityin a domain where is uniformly elliptic. Suppose the coefficients and are uniformly bounded. If attains a maximum at a point of , then in .

In order to memorize the above result, let us think about the parabola with and . In this one-dimentional case, which confirms that only achieves its maximum at .

As you may know the operator is only assumed to be strongly elliptic which only effects the coefficients . Regarding to the coefficients , we only assume these are uniformly bounded. However, the uniform ellipticity of the operator L and the boundedness of the coefficients are not really essential as you can check in the proof. Besides, the domain need not be bounded in this version.

Now for operators of the form , we still have a result analogous to the above.

Theorem(Maximum principle). Let satisfy the differential inequalitywith , with uniformly elliptic in , and with the coefficients of and bounded. If attains a non-negative maximum at an interior point of , then .

Clearly, the assumption and are crucial as we may face some difficulty as raised in this note. Counterexamples are easily obtained if . For example, the function has an absolute maximum at .

Now we turn to the so-called strong maximum principle, or the Hopf maximum principle. We know from the maximum principle that the maximum point only occurs on the boundary. The following tells us further the behavior of the function at that maximum point.

Theorem(Hopf maximum principle). Let satisfy the differential inequalityin a domain in which is uniformly elliptic. Suppose that in and that at a boundary point . Assume that lies on the boundary of a ball in . If is continuous in and an outward directional derivative exists at , then

at unless .

The above result also works for the operator with . We have

Theorem(Hopf maximum principle). Let satisfy the differential inequalityin a domain in which is uniformly elliptic and . Suppose that in and that at a boundary point , and that . Assume that lies on the boundary of a ball in . If is continuous in and an outward directional derivative exists at , then

at unless .

We are now in a position to talk about the generalized maximum principle which is the main point of this notes.

Keep in mind that from now on we do not assume that is nonpositive. Let be a given positive function on and define

A computation shows that

in . Then, if satisfies the inequality

in , we may apply the maximum principle above to the function to obtain the following result

Theorem(Generalized maximum principle). Let satisfy the differential inequalityin a domain in which is uniformly elliptic. If there exists a function such that

then cannot attain a non-negative maximum in unless it is a constant. If attains its non-negative maximum at a point on which lies on the boundary of a ball in and if is not constant, then

at where is any outward directional derivative.

As an application of the generalized maximum principle, we can prove the following non-existence for some Yamabe-type equations.

Theorem. Suppose that is a complete, non-compact Riemannian manifold of dimension . Let satisfyand

Then the differential equation

has no nontrivial non-negative ground states, i.e., is entire and

See also:

- A note on the maximum principle
- The Payne’s Maximum Principles
- An upper bound for solutions via the maximum principle
- The maximum principles

**Source**: Murray H. Protter, Hans F. Weinberger, *Maximum Principles in Differential Equations*, Springer, 1984

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