3 Heat Kernel on an Unbounded Domain
This section is completely unrelated to the remainder of the article. Here we are going to further specialize to the case of the Laplacian acting on functions. Our goal is to extend our results to all open manifolds.
The strategy we shall adopt is to consider an exhaustion by relatively compact open submanifolds 1 . We take the heat kernels on and consider the limit as . We will show that this gives us a fundamental solution to the heat equation. Note that everything we have done in the previous sections apply to manifolds with boundary as well, as long as we require the initial conditions and solutions to vanish at the boundary.
We begin with a note on terminology. On an open manifold, the heat kernel is not necessarily unique. To avoid confusion, we will say a function is a fundamental solution if for any continuous bounded , the function
is a solution to the heat equation, and as for all . The label “heat kernel” will be reserved for the one we explicitly construct, which we will show to be the smallest positive heat kernel.
The main property of the Laplacian that we will use is the maximum principle:
Let be a Riemannian manifold and a precompact open subset. Let be a continuous solution to the heat equation on . Let be the subset of the boundary consisting of . Then
It suffices to show the statement for the supremum, as the infimum case follows by considering .
The idea is that at a maximum, the first derivatives all vanish, and the operator picks out information about the second derivative. The second derivative test then prevents the existence of maxima or minima.
After picking local coordinates, ellipticity means we can write
where is a negative definite matrix for all (one convinces oneself that there is no constant term since kills all constant functions).
The second derivative test is not very useful if the second derivatives vanish. Thus, we perform a small perturbation. For , we set
Then we instead have
It suffices to show that
The result then follows from taking the limit .
First suppose the supremum is achieved at some interior point . Then we know
But we know that is negative definite, so by elementary linear algebra, there must be some such that (e.g. apply Sylvester's law of inertia to ). So moving in the direction increases , hence . This contradicts the fact that is a maximum.
It remains to exclude the possibility that the supremum is attained when . But if it is attained at , then , or else going back slightly in time will increase and hence . So the same argument as above applies.□
Note that here we only get a weak form of the maximum principle. We do not preclude the possibility that the supremum is attained at both the boundary and the interior. This is the price we have to pay for cheating by perturbing a bit to apply the second derivative test.
From this, we deduce the following bounds on the heat kernel:
The heat kernel for a compact Riemannian manifold satisfies
For every fixed and , we have
Moreover, the integral as .
If is open, and has heat kernel , then
for all and . In particular, taking , the heat kernel is unique.
By the maximum principle, convolving any non-negative function with gives a non-negative function. So must itself be non-negative. (We would like to apply the maximum principle directly to but unfortunately is not continuous at )
is the solution to the heat equation with initial condition everywhere.
Extend by zero outside of . For any non-negative function , the convolution
is a solution to the heat equation on with initial conditions . Moreover, when is on , the function vanishes, so . So by the maximum principle, everywhere. It follows that everywhere.
These bounds allow us to carry out our initial strategy. Pick a sequence of exhausting relatively compact open submanifolds . Let be the heat kernel of , extended to all of by zero. We then seek to define a fundamental solution
By (iii) above, we see that the limit exists pointwise, since it is an increasing sequence. To say anything more substantial than that, we need a result that controls the limit of solutions to the heat equation:
Let be any Riemannian manifold and . Suppose is a non-decreasing non-negative sequence of solutions to the heat equation on such that
for some constant independent of and . Then
is a smooth solution to the heat equation and uniformly on compact subsets together with all derivatives of all orders.
We only prove the case where is compact. Let be the heat kernel. Fix . Then for any and , we can write
By monotone convergence, we also have
So is a solution to the heat equation and is smooth. To show the convergence is uniform, simply observe that
for some constant , and the right-hand side by dominated convergence.
In the non-compact case, we multiply by a compactly supported bump function to reduce to the compact (with boundary) case, and replace by Duhamel's principle.□
If we define
then is a smooth fundamental solution to the heat equation. Moreover, is independent of the choices of . In fact,
Moreover, is the smallest positive heat kernel, i.e. for any other positive heat kernel .
Since are increasing, we know the pointwise limit exists, but can possibly be infinite.
To see that is in fact smooth, we apply Lemma 3.3. To do this, we observe that on any open subset of , the function is a solution to
which, after rescaling , is the heat equation.
So Lemma 3.3 tells us is smooth on . Since was arbitrary, is a smooth function.
To see this is a fundamental solution, we again apply Lemma 3.3. By adding a constant, we may assume is positive. So by monotone convergence,
Moreover, we see that
So is bounded in the supremum, hence locally bounded in . So is a solution to the heat equation.
The it remains to show that is in fact continuous as . This will follow if we can show that for any and any (relatively compact) open subset containing , we have
We know this limit is bounded above by by monotone convergence, and that it is equal to if we replace by by Theorem 3.2(ii). But this replacement only makes the integral smaller by the maximum principle applied to the difference. So we are done.
The rest is clear from the maximum principle.□