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.