Today, let us discuss a very interesting stuff. Let say is a compact manifold without boundary of dimention .

On , we consider the following simple PDE

where and are smooth functions with and . Since , it is well-known that the operator is coercive, see here. A standard variation method tells us that there exists a weak solution to the above PDE. By regularity theorem, is at least a function, thus, a strong solution (in the classical sense).

Next, we claim that . To this purpose, assume that the solution achieves its minimum at some point . In particular, there holds

.

This, together with the fact that and , implies that . Thus, we have shown that in .

Once we have the non-negativity of , in view of the strong maximum principle, either in or . In other words, the solution cannot achieves its minimum inside the manifold. Since the manifold has no boundary, it is natural to think that the solution cannot achieve its minimum although . This is clear a contradiction to the fact that the manifold is compact and is of class .

So something went wrong but what and why?

In fact, we have made a small mistake. In view of the strong maximum principle, we can only claim that the solution can only achieve its non-positive minimum value on the boundary. Therefore, there are cases so that may achieve its positive minimum inside . Thus, there is no contradiction here.

In order to see this, let us go back to a proof of the strong maximum principle. Roughly speaking, it starts with the following simple one.

**Lemma 1**. If at any point in , then cannot have non-positive minimum value in .

*Proof*. The proof is standard. Assume that at , the function realizes its minimum, besides, . In particular, and . These force at . A contradiction.

Form the proof above, if , we cannot get any contradiction. This is why we can claim either or since can achieve its positive minimum in .

Notice that, if we don’t have any (in the operator), the non-positivity can be dropped.

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I have a question.

If is not smooth but only bounded, say . Then does the result still hold?

Comment by baotangquoc — September 1, 2016 @ 16:27