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 Δ acting on functions. Our goal is to extend our results to all open manifolds.
The strategy we shall adopt is to consider an exhaustion Ω1⊆Ω2⊆⋯⊆M by relatively compact open submanifolds. We take the heat kernels on Ωˉi and consider the limit as i→∞. 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 Kt(x,y) is a fundamental solution if for any continuous bounded f0(x), the function
f(x,t)=∫MKt(x,y)f0(y)dy
is a solution to the heat equation, and f(x,t)→f0(x) as t→0 for all x∈M. 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:
Theorem
3.1
(Maximum principle)
Let M be a Riemannian manifold and U⊆M a precompact open subset. Let f be a continuous solution to the heat equation on UT=Uˉ×[0,T]. Let ∂∗UT be the subset of the boundary consisting of Uˉ×{0}∪(∂U)×[0,1]. Then
UTsupf=∂∗UTsupf,UTinff=∂∗UTinff.
It suffices to show the statement for the supremum, as the infimum case follows by considering
−f.
The idea is that at a maximum, the first derivatives all vanish, and the operator Δ+∂t∂ 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
Δ+∂t∂=∑aij(x)∂xi∂xj∂2+bi(x)∂xi∂+∂t∂,
where aij(x) is a negative definite matrix for all x (one convinces oneself that there is no constant term since Δ+∂t∂ kills all constant functions).
The second derivative test is not very useful if the second derivatives vanish. Thus, we perform a small perturbation. For δ>0, we set
gδ=f−tδ.
Then we instead have
(Δ+∂t∂)gδ=−δ.
It suffices to show that
UTsupgδ=∂∗UTsupgδ.
The result then follows from taking the limit δ→0.
First suppose the supremum is achieved at some interior point (x,t)∈UT∖∂UT. Then we know
∑aij(x)∂xi∂xj∂2f=−δ.
But we know that aij(x) is negative definite, so by elementary linear algebra, there must be some v such that vivj∂i∂jf>0 (e.g. apply Sylvester's law of inertia to aij). So moving x in the direction v increases f, hence gδ. This contradicts the fact that (x,t) is a maximum.
It remains to exclude the possibility that the supremum is attained when t=T. But if it is attained at (x,T), then ∂t∂f(x,T)≥0, or else going back slightly in time will increase f and hence gδ. 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
u a bit to apply the second derivative test.
From this, we deduce the following bounds on the heat kernel:
Theorem
3.2
The heat kernel Ht for a compact Riemannian manifold M satisfies
Ht(x,y)≥0.
For every fixed x∈M and t>0, we have
∫MHt(x,y)dy≤1.
Moreover, the integral →1 as t→0.
If U⊆M is open, and Uˉ has heat kernel Kt(x,y), then
Kt(x,y)≤Ht(x,y)
for all x,y∈U and t>0. In particular, taking Uˉ=M, the heat kernel is unique.
By the maximum principle, convolving any non-negative function with Ht gives a non-negative function. So Ht must itself be non-negative. (We would like to apply the maximum principle directly to Ht but unfortunately Ht is not continuous at t=0)
f(x,t)=∫MHt(x,y)dy is the solution to the heat equation with initial condition 1 everywhere.
Extend Kt(x,y) by zero outside of U. For any non-negative function f0, the convolution
f(x,t)=∫M(Ht(x,y)−Kt(x,y))f0(y)dy
is a solution to the heat equation on U with initial conditions 0. Moreover, when x is on ∂U, the function Kt(x,y) vanishes, so f(x,t)≥0. So by the maximum principle, f≥0 everywhere. It follows that Ht(x,y)≥Kt(x,y) everywhere.
These bounds allow us to carry out our initial strategy. Pick a sequence of exhausting relatively compact open submanifolds Ω1⊆Ω2⊆⋯⊆M. Let HtΩi(x,y) be the heat kernel of Ωˉi, extended to all of M×M by zero. We then seek to define a fundamental solution
Ht(x,y)=i→∞limHtΩi(x,y).
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:
Lemma
3.3
Let M be any Riemannian manifold and a,b∈R. Suppose {fi} is a non-decreasing non-negative sequence of solutions to the heat equation on M×(a,b) such that
∫Mfi(x,t)dx≤C
for some constant C independent of i and t. Then
f=i→∞limfi
is a smooth solution to the heat equation and fi→f uniformly on compact subsets together with all derivatives of all orders.
[Proof sketch] We only prove the case where
M is compact. Let
Ht be the heat kernel. Fix
[t1,t2]⊆(a,b). Then for any
x∈M and
t∈(t1,t2), we can write
fi(x,t)=∫MHt−t1(x,y)fi(y,t1)dy.(∗)
By monotone convergence, we also have
f(x,t)=∫MHt−t1(x,y)f(y,t1)dy.
So f is a solution to the heat equation and is smooth. To show the convergence is uniform, simply observe that
0≤(f−fi)(x,t)≤D∫M[f(y,t1)−fi(y,t1)]dy
for some constant D, and the right-hand side →0 by dominated convergence.
In the non-compact case, we multiply fi by a compactly supported bump function to reduce to the compact (with boundary) case, and replace (∗) by Duhamel's principle.
Theorem
3.4
If we define
Ht(x,y)=i→∞limHtΩi(x,y),
then Ht is a smooth fundamental solution to the heat equation. Moreover, Ht(x,y) is independent of the choices of Ωi. In fact,
Ht(x,y)=Ω⊆MsupHtΩ(x,y).
Moreover, Ht(x,y) is the smallest positive heat kernel, i.e. Ht(x,y)≤Ht′(x,y) for any other positive heat kernel Ht′.
Since
HtΩi are increasing, we know the pointwise limit
Ht(x,y) exists, but can possibly be infinite.
To see that Ht(x,y) is in fact smooth, we apply Lemma 3.3. To do this, we observe that on any open subset of Ωi, the function HtΩi(x,y) is a solution to
(Δx+Δy+2∂t∂)HtΩ(x,y)=0,
which, after rescaling t, is the heat equation.
If we fix any relatively compact open U⊆M, then Theorem 3.2(ii) gives us a uniform bound
∫U×UHtΩi(x,y)dxdy≤volU.
So Lemma 3.3 tells us Ht(x,y) is smooth on U. Since U was arbitrary, Ht(x,y) is a smooth function.
To see this is a fundamental solution, we again apply Lemma 3.3. By adding a constant, we may assume f0 is positive. So by monotone convergence,
f(x,t)=i→∞lim∫HtΩi(x,y)f0(y)dy,t>0.
Moreover, we see that
∫HtΩi(x,y)f0(y)dy≤(supf0)∫HtΩi(x,y)dy≤supf0.
So f(x,t) is bounded in the supremum, hence locally bounded in L1. So f is a solution to the heat equation.
The it remains to show that f is in fact continuous as t→0. This will follow if we can show that for any x∈M and any (relatively compact) open subset U containing x, we have
t→0lim∫UHt(x,y)dy=1.
We know this limit is bounded above by 1 by monotone convergence, and that it is equal to 1 if we replace Ht(x,y) by HtU(x,y) 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.