Short-Time Heat Content Asymptotics via the Wave and Eikonal Equations

In this short paper, we derive an alternative proof for some known (van den Berg & Gilkey 2015) short-time asymptotics of the heat content in a compact full-dimensional submanifolds S with smooth boundary. This includes formulae like ∫Sexp(tΔ)(f1S)dV=∫SfdV-tπ∫∂SfdA+o(t),t→0+,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{S} \exp (t\Delta ) (f \mathbb {1}_{S}) \,\mathrm {d}V= \int _S f \,\mathrm {d}V- \sqrt{\frac{t}{\pi }} \int _{\partial S} f \,\mathrm {d}A+ o(\sqrt{t}),\quad t \rightarrow 0^+, \end{aligned}$$\end{document}and explicit expressions for similar expansions involving other powers of t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{t}$$\end{document}. By the same method, we also obtain short-time asymptotics of ∫Sexp(tmΔm)(f1S)dV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _S \exp (t^m\Delta ^m)(f \mathbb {1}_S)\,\mathrm {d}V$$\end{document}, m∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m \in \mathbb N$$\end{document}, and more generally for one-parameter families of operators t↦k(-tΔ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \mapsto k(\sqrt{-t\Delta })$$\end{document} defined by an even Schwartz function k.


Introduction
Let (M, g) be a complete, boundaryless, 1 oriented Riemannian manifold with Laplace-Beltrami operator , and volume dV . On a codimension-1 submanifold of M, we write d A for the induced surface (hyper)-area form. The heat semi-group T t := exp(t ) acting on L 2 (M, dV ) is well defined ( is essentially self-adjoint on C ∞ c (M) [2]) and its behaviour as t → 0 + has been extensively investigated in the literature. Specifically, for a set S ⊂ M, the heat content of the form 1 We assume that M has no boundary for the sake of simplicity, and the method presented here can be adapted to more general manifolds with boundary provided that S is compactly contained in the interior of M. If this is not the case, such as in the classical heat content setting as in [13], it should be possible to obtain similar results by modifying the geometrical optics construction used.
Extensions of this idea to abstract metric spaces are given in [6]. In the setting of compact manifolds M (or M = R n ) and S a full-dimensional submanifold with smooth boundary ∂ S, the authors of [12] show that where the coefficients β j depend on S, f and the geometry of M. The setting of [12] is more general, amongst other things it includes f which have singularities. Some of the coefficients obtained in [12, corollary 1.7] are Extensions to some non-compact manifolds M and certain non-compact S are in [11]. Both Eqs. (1) and (2) are proven with significant technical effort, yielding strong results. For example, in [7], explicit knowledge of the fundamental solution of the heat equation is used to obtain Eq. (1) for C 1,1 -smooth ∂ S, after which geometric measure theory is used. Similarly, [12] requires pseudo-differential calculus and invariance theory.
Our aim is to show that slightly weaker results can be obtained by considerably lower technical effort. In contrast to [7], we treat only compact S with smooth boundary, and do not allow f to have singularities like [12] does. On the other hand, we put no further restrictions than completeness on M. The proof presented here is simple, comparatively short, and provides an alternative differential geometric/functional analytic point of view to questions regarding heat content. Moreover, this approach is readily extended to some other PDEs including the semi-group generated by m .

Theorem 2
The coefficients of Theorem 1 satisfy β 0 = r 0 S f dV and Moreover, given the Lie-derivative L ν with respect to ν, similar expression can be found also for larger odd values of j (see Sect. 3).
The properties of the signed distance function ϕ may be used to express terms appearing in Theorem 2 using other quantities. For example, its Hessian ∇ 2 ϕ is the second fundamental form on the tangent space of ∂ S [3, Chap. 3], and thus 1 2 ϕ is the mean curvature.
Our approach to prove Theorems 1 and 2 is to combine 3 well-known facts: (A) The short-time behaviour of the heat flow is related to the short-time behaviour of the wave equation (cf. [1]). (B) The short-time behaviour of the wave equation with discontinuous initial data is related to the short-time behaviour of the eikonal equation (cf. 'geometrical optics' and the progressing wave expansion [10]). (C) The short-time behaviour of the wave and eikonal equations with initial data f 1 S is directly related to the geometry of M near ∂ S. Though points (A)-(C) are well known in the literature, they have (to the best of our knowledge) not been applied to the study of heat content so far.
A significant portion of (C) will rest on an application of the Reynolds transport theorem. Here, denote by s the time-s flow of the vector field ν = −∇ϕ. For small s, the (half) tubular neighbourhood The last equation is a consequence of Cartan's magic formula and Stokes' theorem, where we use that dV (ν, ·) = d A(·) on ∂ S.

Proof forˇ0,ˇ1
By Fourier theory (for non-Gaussian k, the formulae must be adapted), On the operator level, this yields the well-known formula [10, Sect. 6.2] The operator W s := cos(s √ − ) is the time-s solution operator for the wave equation with zero initial velocity, in particular u(s, (5), Similar reasoning has been used to great effect in [1] to derive heat-kernel bounds by making use of the finite propagation speed of the wave equation. As in [1], finite propagation speed yields for we have just seen that the inner product W s f 1 S , 1 M\S is nevertheless well defined. In [1], it is further observed that W s ≤ 1. Using the Cauchy-Schwarz inequality and assuming We conclude with some calculations (cf. Lemma 3), that This is weaker than the desired estimate, and restricts to f = 1 M . The problem is that the estimates in Eq. (6) are too crude. To improve them, we instead approximate the solution u to the wave equation with geometrical optics, using the "progressing wave" construction described in [10, Sect. 6.6], some details of which we recall here. The basic idea is that u is in general discontinuous, with an outward-and an inwardmoving discontinuity given by the zero level-set of functions ϕ + and ϕ − , respectively. The functions ϕ ± satisfy the eikonal equation ∂ t ϕ = ±|∇ϕ ± | with initial value ϕ ± (0, ·) = ϕ(·). Equivalently, using the (nonlinear) operator Ew := (∂ t w) 2 − |∇w| 2 , the functions ϕ ± satisfy E(ϕ ± ) = 0. Our analysis is greatly simplified by choosing the initial ϕ to (locally) be the signed distance function to ∂ S. The eikonal equation is The progressing wave construction further makes use of two (locally existing and smooth) solutions a ± 0 to the first-order transport equations ±∂ t a ± 0 (t, ·) + ν · ∇a ± 0 (t, x) = 1 2 a ± 0 ϕ ± . Observe that with the Heaviside function θ : R → R, and := ∂ 2 t − , the expression (a ± 0 θ(ϕ ± )) is given by The functions ϕ ± and a ± 0 have been chosen so the above simplifies to Thus (a ± 0 θ(ϕ ± )) is as smooth as θ is. We usẽ as an approximation to the discontinuity of the solution u to the wave equation. To maintain consistency with the initial values of u, the initial values of the approximatioñ u are chosen to coincide with those of u at t = 0, this is achieved by setting a ± 0 (0, ·) = As ∇ϕ = −ν, for sufficiently small t the sets {x ∈ M : ϕ + (t, x) = 0} (resp. {x : ϕ − (t, x) = 0}) are level sets of ϕ on the outside (resp. inside) of S (see also [10, Sect. 6.6]). By construction, θ(ϕ − ) vanishes outside of S for t > 0. Consequently, using Eq. (4), we see that as s → 0 + , Combining Eqs. (9) and (10), Calculations along the lines of Lemma 3 and Eq. (7) yield

With k(s) = exp(−s 2 ) and h(s)
Proof For even j, we obtain Eq. (11) by the Fourier-transform formula for jth derivatives. If j is odd, we also need to multiply by the sign function in frequency space, and then use that the inverse Fourier-transform (unnormalized) of the sign function is given by the principal value p.v. 2i x [10,Sect. 4], see also [9,Chap. 7]. Equation 11 holds more generally, e.g. if k is an even Schwarz function. Equation 12 may also be verified directly without Eq. (11).

Proof forˇ2,ˇ3, . . .
We now turn to calculating β j for j ≥ 2. We use the N th order progressing wave construction with sufficiently large N j. For the sake of simplicity, we write O(t ∞ ) for quantities that can be made O(t k ) for any k ∈ N by choosing sufficiently large N . As in the previous section, the construction is from [10, Sect. 6.6]. With t, x)).
Here the functions a ± 0 are defined as before, and for i ≥ 1 the ith order transport −1 (0, ·)). As in Eq. (8), one may verify that ũ ± = a i θ N (ϕ ± ). Writingũ =ũ + +ũ − and and on M \ S, provided that this expression is interpreted in a sufficiently weak sense. Formally, therefore where the last step is the divergence theorem. One may verify Eq. (14) rigorously by either doing the above steps in the sense of distributions, or by a (somewhat tedious) manual computation. Combining this with Eq. (13), The quantity h ( j) (0) may thus be seen to dependũ + (0, ·) at ∂ S, which in turn depends on a ± i at t = 0. Defining S i := a + i + a − i and D i := a + i − a − i for i = 0, 1, . . . , let L be the (spatial) differential operator defined for w ∈ C ∞ (M) by Lw := 1 2 ϕw − ν ·∇w. For i ∈ N 0 , the transport equations imply with initial values satisfying Lemma 4 For i , n ∈ N 0 it holds that ∂ 2n t D i (0, ·) = 0 (note that as a consequence, also a i+1 (0, ·), LD i (0, ·), and n D i (0, ·) are zero).
Proof We will proceed by induction over i and use the identities Eqs. (16)-(19). For i = 0, D 0 (0, ·) = 0 is trivially satisfied. Moreover, ∂ 2n t D 0 = R n D 0 , which is zero at t = 0. For i = 1, observe that a + 1 (0, ·) = − 1 2 ∂ t S 0 (0, ·) = − 1 2 LD 0 (0, ·) = 0, and thus As the operator L commutes with ∂ 2 t , this expression vanishes at t = 0. Induction over n proves the remainder of the statement for i = 1. For the general case, we assume the induction hypothesis for i and i + 1 and start by noting that D i+2 (0, which again vanishes at t = 0; the case n > 1 may again be proven by induction over n.
The odd coefficients are trickier, we only compute the case j = 3. We start with the observation that for x ∈ ∂ S, ϕ + (t, x) = t and thereforẽ Recall that the Lie-derivative acts on functions w ∈ C ∞ (M) by L ν w = ∇w · ν. Thus L ν θ i+1 (ϕ + (t, x)) = −θ i (ϕ + (t, x)), so for x ∈ ∂ S, . (0, x)), but the second term is zero as a + 1 and a + 2 vanish at t = 0 by Lemma 4. Substituting the transport equations and removing further zero terms leaves ∂ t L νũ . Thus (recall that L = −L ν + 1 2 ϕ) directly from Eq. (15), The formula established in the previous section, together with Lemma 3, yields the asymptotic behaviour of S, f (t) by taking the Taylor expansion of h using Corollary 5. This gives the remainder of the claims of theorem 2.

Discussion
The above-said is not specific to the heat equation. Taking k(x) = exp(−x 2m ), m ∈ N, we may, for example, study the one-parameter operator family exp(−t m m ). The wave equation estimates needed are the same. For m ≥ 2, a brief calculation yields the explicit t → 0 + asymptotics We conclude with the observation that the generalization of this paper to weighted Riemannian manifolds (cf. [4]) is straightforward.
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