In this small note, we aim to derive some uniqueness property of solutions of the following PDE
on a compact manifold where , , and .
We assume that and are solutions of the above PDE. Setting . We wish to prove that .
Let us consider the following trick basically due to David Maxwell, see this paper. Let . Then the well-known formula for the Laplace-Beltrami operator , which is
helps us to write
We now calculate as follows.
where we have used the fact that solves the PDE. Making use of the formula for gives
Since solves the PDE, we further have
If we denote , we then have
Therefore, we can rewrite the resulting identity as follows
Dividing both sides by gives
We now use the test functions . Indeed, we first have
Clearly, if then the right hand side of the preceding equality is non-positive. Therefore, we must have
A similar argument using as a test function shows
Hence, in particular,
which says that is constant. If then it follows from
that and .
Assume that we have some boundary condition, say
where is the outward normal vector field on . Then we have
Using the equation for , we conclude that