Heat Content in Non-compact Riemannian Manifolds

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Introduction
Let (M, g) be a complete, smooth, non-compact, m-dimensional Riemannian manifold without boundary, and let Δ be the Laplace-Beltrami operator acting in L 2 (M ). It is well known (see [1][2][3][4]) that the heat equation Δu(x; t) = ∂u(x; t) ∂t , x ∈ M, t > 0, (1.1) M is compact and without boundary, (ii) (M, g) = (R m , g e ) where g e is the usual Euclidean metric on R m , (iii) M is a compact submanifold of R m with smooth boundary and g = g e | M , then there exists a complete asymptotic series such that where J ∈ N is arbitrary, and where the β j : j = 0, 1, 2, . . . are locally computable geometric invariants. In particular, we have that where |Ω| is the measure of Ω, and Per(Ω) is the perimeter of Ω. For earlier results in the Euclidean setting we refer to [6][7][8], and subsequently to [9,10], and [11].
Define u Ω : Ω × (0, ∞) → R by It can be shown that if |Ω| < ∞, then the convergence is also in L 1 (M ). If Ω has infinite measure and |∂Ω| = 0, then the convergence is also in L 1 loc (M ) (Section 7.4 in [12]). In this paper we obtain bounds for the heat content in the case where Ω has possibly infinite measure or infinite perimeter, and where M satisfies the following condition.
There exists C ∈ [2, ∞) such that for all x ∈ M, y ∈ M, t > 0, R > 0, e −Cd(x,y) 2 /t C |B(x; t 1/2 )||B(y; and where B(x; R) = {y ∈ M : d(x, y) < R}, and d(x, y) denotes the geodesic distance between x and y. It was shown independently in [13] and [14] that M satisfying a volume doubling property and a Poincaré inequality is equivalent to M satisfying a parabolic Harnack principle, and is also equivalent to the Li-Yau bound (1.6) above. See for example Theorem 5.4.12 in [3]. We included (1.7) in the definition of the constant C, even though the volume doubling property is implied by (1.6).
We recall a few basic facts (i) Volume doubling implies that for x ∈ M, r 0 > 0,  Hence u Ω , defined by (1.4), is the unique, bounded solution of (1.1) with initial condition (1.5) in the sense of L 1 loc (M ). Moreover stochastic completeness holds. That is for all x ∈ M, t > 0, M dy p M (x, y; t) = 1. (1.8) We refer to Chapter 9 in [2].
Our main result is the following.
for all t > 0. (ii) If (1.11) holds for some t = T > 0, then for all t > 0, where If Ω has finite Lebesgue measure, then we define the heat loss of Ω in M at t by F Ω (t) = |Ω| − H Ω (t). (1.14) We have that the heat loss t → F Ω (t) of Ω in M is increasing, concave, subadditive, and continuous. If Ω is bounded and ∂Ω is smooth, then, by (1.3), there exists an asymptotic series of which the first few coefficients are known explicitly. Theorem 1.3 below concerns the general situation |Ω| < ∞, and gives bounds in non-classical geometries where e.g. either Ω has infinite perimeter, and/or ∂Ω is not smooth.
This paper is organised as follows. In Sect. 2 we give the proofs of Theorems 1.2 and 1.3. In Sect. 3 we analyse an example of Ω in R m where precise analysis of H Ω (t) is possible.

Proofs
The main idea in the proof of Theorem 1.2 is to use the Li-Yau bound (1.6), and (1. . This is possible for c sufficiently large (in terms of C). A similar bootstrap argument features in the proof of Theorem 1.3. There, the stochastic completeness of M , (1.8), is also exploited.
The choice R = t 1/2 implies, by (2.1) and (2.2), that with K 1 given in (1.13). Next suppose that 0 < t ≤ T . By (1.9), and (2.3) for t = T , we have that This completes the proof of the assertion in part (i).
(ii) Let n > 0, p ∈ Ω, R > 0, and Ω n = Ω ∩ B(p; n), and suppose that (1.11) holds for some t = T > 0. Then |Ω n | ≤ |B(p; n)| < ∞. Reversing the roles of x and y in (2.2) we have that for d(x, y) < R, We have that (2.5) To bound the second term in the right-hand side of (2.5), we note that Hence, where we have used (2.7), (1.9), the lower bound in (1.6), and (1.10). We now choose R 2 such that the coefficient of H Ωn (t) in the right-hand side of (2.8) is equal to 1 2 . That is Rearranging and bootstrapping gives, by (2.5)-(2.9), and the fact that We choose t = t T such that R * = T , and take the limit n → ∞ in the righthand side of (2.10). This limit is finite by the hypothesis at the beginning of the proof. We conclude that By monotone convergence, (2.12) for all t > 0 with R * given by (2.9). Since H Ω (t) is decreasing in t, and since R * ≥ t 1/2 we conclude from (2.10) that (2.14) Rescaling t gives the upper bound in (1.12) with K 2 given in (1.13). This completes the proof of the assertion in part (ii).
Proof of Theorem 1.3. To prove the lower bound in (1.15), we have by definition of F Ω (t) in (1.14), and by (1.8) that Hence by (1.6) we have for R > 0 that Since B(y; t 1/2 ) ⊂ B(x; R + t 1/2 ), for y ∈ B(x; R), we have by (2.2) that The choice R = t 1/2 gives the lower bound in (1.15), with L 1 given in (1.16).

Analysis of an Example
In this section we present the asymptotic analysis of H Ω (t) as t ↓ 0, of an open set Ω in M = R m consisting of disjoint balls with centres in Z m , and decreasing radii.
Below we consider four main regimes: The latter regime is absent for m = 2. In the first regime Ω has infinite measure, and Theorem 1.2 (iii) gives the order of magnitude as t ↓ 0. This has been refined in (3.5)-(3.6) below. In the second regime Ω has infinite perimeter, and Theorem 1.3 gives the order of magnitude as t ↓ 0. This has been refined in (3.12)-(3.13) below. In the third and fourth regimes Ω has finite perimeter. Theorem 1.3 gives two-sided bounds of order t 1/2 . In (3.9) and (3.11) below we show that the perimeter term appears with the usual numerical constant. The remainder estimates depend on whether i∈N r m−2 i is infinite or finite. Furthermore there are several They all involve logarithmic corrections in the heat content. We only analyse, as an example, the case α = 1 m−2 . The latter case is again absent for m = 2.
Theorem 3.2. Let 0 < a ≤ 1 4 , m ≥ 2, and let r i = ai −α , i ∈ N. where where If m > 2 and 1 m−2 < α then Proof of Theorem 3.1. To prove part (i) we first suppose that H Ω (t) < ∞ for some t > 0. Then which implies the reverse implication by Theorem 1.2 (ii). This proves the assertion under (i).
To prove part (ii) we note that the lower bound in (3.4) follows from the first inequality in (3.12). To prove the upper bound we observe that if which gives the bound in (3.4).

Proof of Theorem 3.2.
We first consider the case 1 2m < α < 1 m . By (3.4), it suffices to consider the sum in the left-hand side of (3.8).
(3.14) A straightforward application of Tonelli's theorem gives the formulae under (3.5) and (3.6). To obtain a lower bound for the left-hand side of (3.14), we use the monotonicity of i → H B(0;ai −α ) (t) once more, and obtain that The last term in the right-hand side of (3.15) is bounded in absolute value by This completes the proof of the assertion under (3.5) and (3.6).
Consider the case 1 m < α < 1 m−1 . By (3.4), and scaling we have that In a similar way to the proof of (3.5),(3.6), we approximate the sum with respect to i by an integral. However, i → F B(0;1) (a −2 i 2α t) is increasing, whereas i → (ai −α ) m is decreasing.
To obtain an upper bound, we let J ∈ N, and note that by (3.4),