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.