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

Abstract

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

$$\begin{aligned} \int _{S} \exp (t\Delta ) (f \mathbb {1}_{S}) \,\mathrm {d}V= \int _S f \,\mathrm {d}V- \sqrt{\frac{t}{\pi }} \int _{\partial S} f \,\mathrm {d}A+ o(\sqrt{t}),\quad t \rightarrow 0^+, \end{aligned}$$

and explicit expressions for similar expansions involving other powers of $\sqrt{t}$. By the same method, we also obtain short-time asymptotics of $\int _S \exp (t^m\Delta ^m)(f \mathbb {1}_S)\,\mathrm {d}V$, $m \in \mathbb N$, and more generally for one-parameter families of operators $t \mapsto k(\sqrt{-t\Delta })$ defined by an even Schwartz function k.

1 Introduction

Let (M, g) be a complete, boundaryless,¹ oriented Riemannian manifold with Laplace–Beltrami operator $\Delta$, and volume $\,\mathrm {d}V$. On a codimension-1 submanifold of M, we write $\,\mathrm {d}A$ for the induced surface (hyper)-area form. The heat semi-group $T_t :=\exp (t\Delta )$ acting on $L^2(M,\,\mathrm {d}V)$ is well defined ($\Delta$ is essentially self-adjoint on $C^\infty _c(M)$ [2]) and its behaviour as $t \rightarrow 0^+$ has been extensively investigated in the literature. Specifically, for a set $S \subset M$, the heat content of the form $\Omega _{S,f}(t) :=\int _S T_t(f \mathbb {1}_S)\,\mathrm {d}V$, $f \in C^\infty (M)$, has recently received much attention; see, for instance, [7, 11, 12] and the references therein.

Let us briefly recall some known results. On ${\mathbb {R}}^n$, sets S of finite perimeter P(S) are characterized by [7, Thm. 3.3 ]

$$\begin{aligned} \lim \limits _{t \rightarrow 0^+ }\sqrt{\frac{\pi }{t}}\Big (\Omega _{S,\mathbb {1}_M}(0) - \Omega _{S,\mathbb {1}_M}(t)\Big ) = P(S). \end{aligned}$$

(1)

Extensions of this idea to abstract metric spaces are given in [6]. In the setting of compact manifolds M (or $M = {\mathbb {R}}^n$) and S a full-dimensional submanifold with smooth boundary $\partial S$, the authors of [12] show that

$$\begin{aligned} \Omega _{S,f}(t) = \sum _{j=0}^\infty \beta _j t^\frac{j}{2},\quad t\rightarrow 0^+, \end{aligned}$$

(2)

where the coefficients $\beta _j$ 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

$$\begin{aligned} \beta _0 = \int _S f \,\mathrm {d}V,\quad \beta _1 = -\frac{1}{\sqrt{\pi }}\int _{\partial S} f \,\mathrm {d}A,\quad \beta _2 = \frac{1}{2}\int _{S} \Delta f \,\mathrm {d}V. \end{aligned}$$

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 $\partial S$, 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 $\Delta ^m$. Observe that $T(t) = k(\sqrt{-t\Delta })$ with $k(x) = \exp (-x^2)$. We allow k to be an arbitrary even Schwarz function, with $\Omega _{S,f}(t) = \int _{S} k(\sqrt{-t\Delta })(f \mathbb {1}_S)\,\mathrm {d}V$ and will prove:

Theorem 1

Let M be a complete Riemannian manifold with Laplace–Beltrami operator $\Delta$, Riemannian volume $\,\mathrm {d}V$ and induced (hyper) area form $\,\mathrm {d}A$. Let $S \subset M$ be a compact full-dimensional submanifold with smooth boundary. For $f \in C^\infty (M)$ and $N \in {\mathbb {N}}$,

$$\begin{aligned} \Omega _{S,f}(t) = \sum _{j=0}^N \beta _j t^{\frac{j}{2}} + o(t^\frac{N}{2}),\quad t \rightarrow 0^+, \end{aligned}$$

for constants $(\beta _j)_{j=0}^N$ described further in the next theorem.

With the jth derivative $k^{(j)}$ (for $j \in {\mathbb {N}}_0$), let $r_{j} :=(-1)^{j/2} k^{(j)}(0)$ for j even and $r_{j} :=(-1)^{(j-1)/2} \int _0^\infty \frac{2k^{j}(s)}{-\pi s}\, \mathrm d s$ for j odd. Let $\varphi$ locally be the signed distance function (see also [8, Sect. 3.2.2]) to $\partial S$ with $S = \varphi ^{-1}([0,\infty ))$, and denote by $\nabla$ and $\cdot$ the gradient and (metric) inner product, respectively. The vector field $\nu :=-\nabla \varphi$ is outer unit normal at $\partial S$.

Theorem 2

The coefficients of Theorem 1 satisfy $\beta _0 = r_0 \int _S f \,\mathrm {d}V$ and $\beta _1 = -\frac{1}{2} r_1 \int _{\partial S} f \,\mathrm {d}A$. For even $j \in {\mathbb {N}}_{\ge 2}$,

$$\begin{aligned} \beta _{j} = \frac{r_j}{j!} \int _S\frac{1}{2} {\Delta ^{j/2} f}\,\mathrm {d}V. \end{aligned}$$

Moreover, given the Lie-derivative ${\mathcal {L}}_\nu$ with respect to $\nu$,

$$\begin{aligned} \beta _3 = \frac{r_3}{2\cdot 3!} \int _{\partial S}{\mathcal {L}}_\nu (-{\mathcal {L}}_\nu + \frac{1}{2}\Delta \varphi ) f - \frac{1}{2} \Delta f + \frac{1}{2} (-{\mathcal {L}}_\nu + \frac{1}{2}\Delta \varphi )^2 f\,\mathrm {d}A, \end{aligned}$$

similar expression can be found also for larger odd values of j (see Sect. 3).

The properties of the signed distance function $\varphi$ may be used to express terms appearing in Theorem 2 using other quantities. For example, its Hessian $\nabla ^2 \varphi$ is the second fundamental form on the tangent space of $\partial S$ [3, Chap. 3], and thus $\frac{1}{2}\Delta \varphi$ 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 $f \mathbb {1}_S$ is directly related to the geometry of M near $\partial S$.

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 $\Phi ^s$ the time-s flow of the vector field $\nu = - \nabla \varphi$. For small s, the (half) tubular neighbourhood

$$\begin{aligned} S^{-s} :=\{x \in M \setminus S : \mathrm {dist}(x,\partial S) \le s\} \end{aligned}$$

(3)

satisfies $S \cup S^{-s} = \Phi ^s(S)$. For $a \in C^\infty ((-\varepsilon , \varepsilon ) \times M)$, by [5, Chap. V, Prop. 5.2],

$$\begin{aligned} \nonumber \frac{\mathrm d}{\mathrm ds}\left. \int _{S^{-s}} a(s,\cdot )\,\mathrm {d}V\right| _{s=0}&=\frac{\mathrm d}{\mathrm ds}\left( \left. \int _{S^{-s} \cup S} a(s,\cdot )\,\mathrm {d}V- \int _{S} a(s,\cdot )\,\mathrm {d}V\right) \right| _{s=0} \\&=\int _{S} {\mathcal {L}}_{{\tilde{\nu }}} [ a(0,\cdot ) \,\mathrm {d}V]= \int _{\partial S} a(0,\cdot )\,\mathrm {d}A. \end{aligned}$$

(4)

The last equation is a consequence of Cartan’s magic formula and Stokes’ theorem, where we use that $\,\mathrm {d}V(\nu , \cdot ) = \,\mathrm {d}A(\cdot )$ on $\partial S$.

2 Proof for $\beta _0,\beta _1$

By Fourier theory (for non-Gaussian k, the formulae must be adapted),

$$\begin{aligned} k(t) = \exp (- t^2) = \int _0^\infty {\hat{k}}(s) \cos (ts) \,\mathrm {d}s \quad \mathrm {with}\quad {\hat{k}}(s) :=\frac{1}{\sqrt{\pi }}\exp \left( \frac{-s^2}{4 }\right) . \end{aligned}$$

On the operator level, this yields the well-known formula [10, Sect. 6.2]

$$\begin{aligned} T_t = \exp (t\Delta ) = \int _0^\infty {\hat{k}}(s) \cos ( s \sqrt{-t\Delta }) \,\mathrm ds. \end{aligned}$$

(5)

The operator $W^s :=\cos (s\sqrt{-\Delta })$ is the time-s solution operator for the wave equation with zero initial velocity, in particular $u(s,x) :=(W^s f \mathbb {1}_S)(x)$ (weakly) satisfies $(\partial _t^2 - \Delta )u = 0$. Let $\langle \cdot ,\cdot \rangle$ denote the $L^2(M,\,\mathrm {d}V)$ inner product. Using Eq. (5),

$$\begin{aligned} \langle T_t f \mathbb {1}_S, \mathbb {1}_S \rangle = \int _0^\infty {\hat{k}}(s)\langle W_{s\sqrt{t}}f\mathbb {1}_S, \mathbb {1}_{S} \rangle \, \mathrm {d} s. \end{aligned}$$

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 $s \ge 0$ that $\langle W_{s} f \mathbb {1}_S, \mathbb {1}_{M \setminus S} \rangle = \langle W_{s} f \mathbb {1}_{S^{s}}, \mathbb {1}_{S^{-s}} \rangle$, where $S^{s} :=(M \setminus S)^{-s}$ is defined like Eq. (3). Even if $\mathbb {1}_{M \setminus S} \notin L^2(M,\,\mathrm {d}V)$, we have just seen that the inner product $\langle W_s f \mathbb {1}_S, \mathbb {1}_{M \setminus S} \rangle$ is nevertheless well defined. In [1], it is further observed that ${\Vert W_s \Vert } \le 1$. Using the Cauchy–Schwarz inequality and assuming $f = \mathbb {1}_M$, Eq. (4) yields

$$\begin{aligned} h(s) :=\langle W_{s} f \mathbb {1}_{S^{s}}, \mathbb {1}_{S^{-s}} \rangle \le {\Vert \mathbb {1}_{S^s} \Vert }_2 {\Vert \mathbb {1}_{S^{-s}} \Vert }_2 \le s \int _{\partial S} \,\mathrm {d}A+ o(s), \quad s \rightarrow 0^+. \end{aligned}$$

(6)

In addition, $|\langle W_s f \mathbb {1}_S, \mathbb {1}_{S} \rangle | \le {\Vert f\mathbb {1}_S \Vert }_2{\Vert \mathbb {1}_S \Vert }_2$ for all $s\ge 0$, in particular as $s \rightarrow \infty$. We conclude with some calculations (cf. Lemma 3), that

$$\begin{aligned} \nonumber \langle T_t \mathbb {1}_S, \mathbb {1}_{S} \rangle&= \int _0^\infty {\hat{k}}(s) \left( \langle W_{s\sqrt{t} } \mathbb {1}_S, \mathbb {1}_M \rangle - \langle W_{s \sqrt{t}} \mathbb {1}_S, \mathbb {1}_{M\setminus S} \rangle \right) \,\mathrm {d} s\\&= \langle \mathbb {1}_S, \mathbb {1}_M \rangle - \int _0^\infty {\hat{k}}(s) h(s\sqrt{t})\, \,\mathrm d s\\&\ge \int _S \,\mathrm {d}V- 2\sqrt{\frac{t}{\pi }} \int _{\partial S} \,\mathrm {d}A+ o(\sqrt{t}), \qquad t \rightarrow 0^+.\nonumber \end{aligned}$$

(7)

This is weaker than the desired estimate, and restricts to $f = \mathbb {1}_M$. 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 inward—moving discontinuity given by the zero level-set of functions $\varphi ^+$ and $\varphi ^-$, respectively. The functions $\varphi ^\pm$ satisfy the eikonal equation $\partial _t \varphi = \pm |\nabla \varphi ^\pm |$ with initial value $\varphi ^\pm (0,\cdot ) = \varphi (\cdot )$. Equivalently, using the (nonlinear) operator $Ew :=(\partial _t w )^2 - |\nabla w|^2$, the functions $\varphi ^\pm$ satisfy $E(\varphi ^\pm )=0$. Our analysis is greatly simplified by choosing the initial $\varphi$ to (locally) be the signed distance function to $\partial S$. The eikonal equation is then $\partial _t \varphi ^\pm = \pm |\nabla \varphi | = \pm |-\nu |= \pm 1$, i.e.  $\varphi ^\pm (x,t) = \varphi (x) \pm t$.

The progressing wave construction further makes use of two (locally existing and smooth) solutions $a^\pm _0$ to the first-order transport equations $\pm \partial _t a^\pm _0(t,\cdot ) + \nu \cdot \nabla a^\pm _0(t,x) = \frac{1}{2} a^\pm _0\Delta \varphi ^\pm$. Observe that with the Heaviside function $\theta :{\mathbb {R}} \rightarrow {\mathbb {R}}$, and $\Box :=\partial _t^2 - \Delta$, the expression $\Box (a_0^\pm \theta (\varphi ^\pm ))$ is given by

$$\begin{aligned} (\theta ''(\varphi ^\pm )E\varphi ^\pm + \Box \varphi ^\pm \theta '(\varphi ^\pm ))a_0^\pm + 2\left( \partial _t a_0^\pm \partial _t \varphi ^\pm - \nabla a_0^\pm \cdot \nabla \varphi ^\pm \right) \theta '(\varphi ^\pm ) + \Box a_0^\pm \theta (\varphi ^\pm ). \end{aligned}$$

The functions $\varphi ^\pm$ and $a_0^\pm$ have been chosen so the above simplifies to

$$\begin{aligned} \Box (a_0^\pm \theta (\varphi ^\pm ))&= 2\left( \pm \partial _t a_0^\pm + \nabla a_0^\pm \cdot \nu -\frac{1}{2}\Delta \varphi a^\pm _0 \right) \theta '(\varphi ^\pm ) + \Box a_0^\pm \theta (\varphi ^\pm )\nonumber \\&= \Box a_0^\pm \theta (\varphi ^\pm ) . \end{aligned}$$

(8)

Thus $\Box (a_0^\pm \theta (\varphi ^\pm ))$ is as smooth as $\theta$ is. We use

$$\begin{aligned} {\tilde{u}}(t,x) :=a^+_0(t,x)\theta (\varphi ^+(t,x)) + a^-_0(t,x)\theta (\varphi ^-(t,x)) \end{aligned}$$

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 approximation ${\tilde{u}}$ are chosen to coincide with those of u at $t=0$, this is achieved by setting $a_0^\pm (0,\cdot ) = \frac{1}{2} f$ so that (at least formally) $\partial _t {\tilde{u}}(0,\cdot ) = 0$ and also ${\tilde{u}}(0,\cdot ) = \mathbb {1}_S f$.

The function ${\tilde{u}}$ approximates the discontinuous solution u of the wave equation well enough that the function $(s,x) \mapsto u(s,x) - {\tilde{u}}(s,x)$ is continuous on $[-T,T] \times M$, see [10, Sect. 6.6, eq. 6.35]. By construction, ${\tilde{u}}(0,\cdot ) = u(0,\cdot )$. Hence $|(u(s,x) - {\tilde{u}} (s,x)| = o(1)$ as $s \rightarrow 0^+$, which implies

$$\begin{aligned} |\langle u(s,\cdot ), \mathbb {1}_{S^{-s}} \rangle - \langle {\tilde{u}}(s,\cdot ),\mathbb {1}_{S^{-s}} \rangle | = o(s)\,\quad s \rightarrow 0^+. \end{aligned}$$

(9)

As $\nabla \varphi = -\nu$, for sufficiently small t the sets $\{x \in M : \varphi ^+(t,x) = 0\}$ (resp. $\{x : \varphi ^-(t,x) = 0\}$) are level sets of $\varphi$ on the outside (resp. inside) of S (see also [10, Sect. 6.6]). By construction, $\theta (\varphi ^-)$ vanishes outside of S for $t > 0$. Consequently, using Eq. (4), we see that as $s \rightarrow 0^+$,

$$\begin{aligned} \langle {\tilde{u}} (s,\cdot ),\mathbb {1}_{S^{-s}} \rangle&= \int _{S^{-s}} a^+_0(s,x) \mathbb {1}_{ \{\varphi ^+(s,\cdot ) \ge 0 \}} + a^-_0(s,x) \mathbb {1}_{\{\varphi ^-(s,x) \ge 0 \}} \,\mathrm {d}V(x)\nonumber \\&= s\int _{\partial S} a_0^+(0,x) \,\mathrm {d}A(x) + o(s) = \frac{s}{2}\,\int _{\partial S} f \,\mathrm {d}A+ o(s). \end{aligned}$$

(10)

Combining Eqs. (9) and (10),

$$\begin{aligned} h(s) = \langle W_s f \mathbb {1}_S, \mathbb {1}_{S^{-s}} \rangle = \langle u(s,\cdot ),\mathbb {1}_{S^{-s}} \rangle = \frac{s}{2} \int _{\partial S} f \,\mathrm {d}A+ o(s),\quad s \rightarrow 0^+. \end{aligned}$$

Calculations along the lines of Lemma 3 and Eq. (7) yield

$$\begin{aligned}\langle T_t f \mathbb {1}_S, \mathbb {1}_S \rangle =\int _S f \,\mathrm {d}V- \sqrt{\frac{t}{\pi }} \int _{\partial S}f \,\mathrm {d}A+ o(\sqrt{t}),\qquad t \rightarrow 0^+, \end{aligned}$$

as claimed.

Lemma 3

Let $j \in {\mathbb {N}}$ and $\gamma : {\mathbb {R}}_{\ge 0} \rightarrow {\mathbb {R}}$. Let $\gamma (s) = s^j + o(s^j)$ for $s \rightarrow 0$ and $\gamma (s) = O(1)$ for $s \rightarrow \infty$. Then for $t \rightarrow 0^+$,

$$\begin{aligned} \int _0^\infty \gamma (s \sqrt{t}){\hat{k}}(s) \,\mathrm ds = t^\frac{j}{2}{\left\{ \begin{array}{ll} (-1)^{\frac{j}{2}}\,k^{(j)}(0) &{} \text {{ j} even } \\ (-1)^{\frac{j-1}{2}}\int _0^\infty \frac{2\, k^{(j)}(s)}{-\pi s}\, \mathrm d s &{} \text {{ j} odd} \end{array}\right. } \quad + o\left( t^\frac{j}{2}\right) . \end{aligned}$$

(11)

With $k(s) = \exp (-s^2)$ and $h(s) = c_0 + c_1s + c_2 s^2 + o(s^2)$, this implies

$$\begin{aligned} \int _0^\infty h(s\sqrt{t}) {\hat{k}}(s) \,\mathrm d s = c_0 + \frac{2c_1}{\sqrt{\pi }} \sqrt{t} + 2 c_2 t + o(t). \end{aligned}$$

(12)

Proof

For even j, we obtain Eq. (11) by the Fourier-transform formula forjth 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 $\mathrm {p.v.}\left( \frac{2i}{x}\right)$ [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). $\square$

3 Proof for $\beta _2,\beta _3,\ldots$

We now turn to calculating $\beta _j$ for $j \ge 2$. We use the Nth order progressing wave construction with sufficiently large $N \gg j$. For the sake of simplicity, we write $O(t^\infty )$ for quantities that can be made $O(t^k)$ for any $k\in {\mathbb {N}}$ by choosing sufficiently large N. As in the previous section, the construction is from [10, Sect. 6.6]. With $\theta _0 :=\theta$, and $\theta _i(t) :=\int _{-\infty }^t \theta _{i-1}(s)\mathrm ds$ we write

$$\begin{aligned} {\tilde{u}}^\pm (t,x) :=\sum _{i=0}^N a^\pm _i(t,x) \theta _i(\varphi ^\pm (t,x)). \end{aligned}$$

Here the functions $a_0^\pm$ are defined as before, and for $i \ge 1$ the ith order transport equations $\pm \partial _t a^\pm _i = -\nu \cdot \nabla a^\pm _i + \frac{1}{2} a_i^\pm \Delta \varphi ^\pm - \frac{1}{2}\Box a_{i-1}^\pm$ define $a^\pm _i$ together with initial data $a^\pm _i(0,\cdot ) = -\frac{1}{2}(\partial _t a^+_{i-1}(0,\cdot ) + \partial _t a^-_{i-1}(0,\cdot ))$. As in Eq. (8), one may verify that $\Box {\tilde{u}}^\pm = \Box a_{i} \theta _N(\varphi ^\pm )$. Writing ${\tilde{u}} = {\tilde{u}}^+ + {\tilde{u}}^-$ and

$$\begin{aligned}u(t,x) = {\tilde{u}}^+(t,x) + {\tilde{u}}^-(t,x) + R_N(t,x), \end{aligned}$$

the remainder satisfies $R_N \in C^{(N,1)}([-T,T] \times M)$ and $R_N(t,\cdot )$ vanishes at $t=0$, see [10, Sect. 6.6, eq. 6.35]. Moreover, $R_N$ is supported on $\{(x,t) : {{\,\mathrm{{\mathrm {dist}}}\,}}(x,S) \le |t|\}$, all of this implies that, as $t \rightarrow 0^+$,

$$\begin{aligned} h(t) = \int _{M \setminus S} u(t,x)\,\mathrm {d}V(x) = \int _{M \setminus S} {\tilde{u}}^+(t,x)\,\mathrm {d}V(x) + O(t^\infty )\, \end{aligned}$$

(13)

and moreover $h \in C^\infty ([0,T])$. The structure of $R_N$ implies that $\Box {\tilde{u}}^+(t,x) = O(t^\infty )$ on $M \setminus S$, provided that this expression is interpreted in a sufficiently weak sense. Formally, therefore

$$\begin{aligned} \partial _t^2\int _{M\setminus S} {\tilde{u}}^+(\cdot ,t)\,\mathrm {d}V&= \int _{M \setminus S} \Delta {\tilde{u}}^+(\cdot ,t) \,\mathrm {d}V+ O(t^\infty ) \nonumber \\&= - \int _{\partial S} \nabla {\tilde{u}}^+(\cdot ,t) \cdot \nu \,\mathrm {d}A+ O(t^\infty ), \end{aligned}$$

(14)

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),

$$\begin{aligned} h''(t) = -\int _{\partial S} \nabla {\tilde{u}}^+(\cdot ,t) \cdot \nu \,\mathrm {d}A+ O(t^\infty ). \end{aligned}$$

(15)

The quantity $h^{(j)}(0)$ may thus be seen to depend ${\tilde{u}}^+(0,\cdot )$ at $\partial S$, which in turn depends on $a_i^\pm$ at $t=0$. Defining ${\mathbf {S}}_i :=a_i^+ + a_i^-$ and ${\mathbf {D}}_i :=a_i^+ - a_i^-$ for $i=0,1,\dots$, let L be the (spatial) differential operator defined for $w \in C^\infty (M)$ by $Lw :=\frac{1}{2} \Delta \varphi w - \nu \cdot \nabla w$. For $i \in {\mathbb {N}}_0$, the transport equations imply

$$\begin{aligned} \partial _t {\mathbf {S}}_0= & {} L{\mathbf {D}}_0, \quad \quad \partial _t {\mathbf {D}}_0 = L {\mathbf {S}}_0, \quad \quad \end{aligned}$$

(16)

$$\begin{aligned} \partial _t {\mathbf {S}}_{i+1}= & {} L{\mathbf {D}}_{i+1} - \frac{1}{2}\Box {\mathbf {D}}_{i}, \quad \quad \partial _t {\mathbf {D}}_{i+1} = L{\mathbf {S}}_{i+1} - \frac{1}{2}\Box S_{i} \quad \mathrm {for} \quad i \ge 0, \end{aligned}$$

(17)

with initial values satisfying

$$\begin{aligned} a_{0}^+(0,\cdot )= & {} \ \frac{1}{2} {\mathbf {S}}_0(0,\cdot ) = \frac{1}{2} f(\cdot ),\quad \quad {\mathbf {D}}_0(0,\cdot ) = 0, \end{aligned}$$

(18)

$$\begin{aligned} a_{i+1}^+(0,\cdot )= & {} \frac{1}{2} {\mathbf {D}}_{i+1}(0,\cdot ) = -\frac{1}{2} \partial _t {\mathbf {S}}_{i}(0,\cdot ),\quad {\mathbf {S}}_{i+1}(0,\cdot ) = 0. \end{aligned}$$

(19)

Lemma 4

For $i,n \in {\mathbb {N}}_0$ it holds that $\partial _t^{2n}{\mathbf {D}}_i(0,\cdot ) = 0$ (note that as a consequence, also $a_{i+1}(0,\cdot )$, $L{\mathbf {D}}_i(0,\cdot )$, and $\Box ^n {\mathbf {D}}_i(0,\cdot )$ are zero).

Proof

We will proceed by induction over i and use the identities Eqs. (16)–(19). For $i=0$, ${\mathbf {D}}_0(0,\cdot ) = 0$ is trivially satisfied. Moreover, $\partial _t^{2n} {\mathbf {D}}_0 = R^n{\mathbf {D}}_0$, which is zero at $t=0$. For $i=1$, observe that $a_1^+(0,\cdot ) = -\frac{1}{2} \partial _t {\mathbf {S}}_0(0,\cdot ) = -\frac{1}{2} L{\mathbf {D}}_0(0,\cdot ) = 0$, and thus ${\mathbf {D}}_1(0,\cdot ) = 0$. Likewise, $\partial _t^2 {\mathbf {D}}_1 = \partial _t(L{\mathbf {S}}_1 - \frac{1}{2}\Box {\mathbf {S}}_0) = L(L{\mathbf {D}}_1 - \frac{1}{2}\Box {\mathbf {D}}_0) - \frac{1}{2}\Box L{\mathbf {D}}_0$. As the operator L commutes with $\partial _t^2$, 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 ${\mathbf {D}}_{i+2}(0,\cdot ) = 2a_{i+2}^+(0,\cdot ) = -\partial _t {\mathbf {S}}_{i+1}(0,\cdot ) = -\left( L {\mathbf {D}}_{i+1}(0,\cdot ) - \frac{1}{2}\Box {\mathbf {D}}_{i}(0,\cdot ) \right) = 0$. Moreover, $\partial _t^2 {\mathbf {D}}_{i+2} = \partial _t ( L {\mathbf {S}}_{i+2} - \frac{1}{2} \Box {\mathbf {S}}_{i+1}) = L(L{\mathbf {D}}_{i+2} - \frac{1}{2} \Box {\mathbf {D}}_{i+1}) - \frac{1}{2} \Box \left( L {\mathbf {D}}_{i+1} - \frac{1}{2} \Box {\mathbf {D}}_i\right)$, which again vanishes at $t=0$; the case $n > 1$ may again be proven by induction over n. $\square$

Corollary 5

For even $j \in {\mathbb {N}}_{\ge 2}$, the jth derivative of h satisfies

$$\begin{aligned}h^{(j)}(0) = -\frac{1}{2} \int _S \Delta ^{j/2} f\,\mathrm {d}V. \end{aligned}$$

Proof

Lemma 4 shows that for $i \ge 1$, $a^+_i(0,x) = 0$. Together with Eq. (15), thus $h''(0) = -\int _{\partial S} \nabla a^+_0(0,\cdot ) \cdot \nu \,\mathrm {d}A= -\frac{1}{2}\int _{\partial S} \nabla f \cdot \nu \,\mathrm {d}A$. This is the case $j=2$. More generally, for $j = 2k$ with $k \in {\mathbb {N}}_{\ge 2}$, we use that (for $x \in \partial S$), ${\tilde{u}}^+$ satisfies $\partial _t^2 {\tilde{u}}^+(t,x) = \Delta {\tilde{u}}^+(t,x) + O(t^\infty )$. Equation 15 ensures that as $t \rightarrow 0^+$,

$$\begin{aligned} h^{(2k)}(t) = \int _{\partial S} \nabla (\Delta ^{k-1} {\tilde{u}}^+(t,\cdot )) \cdot \nu \,\mathrm {d}A+ O(t^\infty ). \end{aligned}$$

As for the case $k=1$, it follows that $h^{(2k)}(0) = -\int _{\partial S} \nabla (\Delta ^{k-1} a^+_0) \cdot \nu \,\mathrm {d}A$, the divergence theorem yields the claim. $\square$

The odd coefficients are trickier, we only compute the case $j=3$. We start with the observation that for $x\in \partial S$, $\varphi ^+(t,x) = t$ and therefore

$$\begin{aligned} {\tilde{u}}^+(t,x)&= \sum _{i=0}^N \frac{1}{i!} t^i a^+_i(t,x)\quad \mathrm { for }\ \ t \ge 0,\ \ x \in \partial S. \end{aligned}$$

Recall that the Lie-derivative acts on functions $w \in C^\infty (M)$ by ${\mathcal {L}}_\nu w = \nabla w \cdot \nu$. Thus ${\mathcal {L}}_\nu \theta _{i+1}(\varphi ^+(t,x)) = -\theta _{i}(\varphi ^+(t,x))$, so for $x \in \partial S$,

$$\begin{aligned} {\mathcal {L}}_\nu {\tilde{u}}^+(t,x)&= \sum _{i=0}^{N-1} \frac{t^i}{i!}({\mathcal {L}}_\nu a_i^+(t,x) - a_{i+1}(t,x)) + O(t^\infty ). \end{aligned}$$

Therefore $\partial _t {\mathcal {L}}_\nu {\tilde{u}}^+(0,x) = \partial _t({\mathcal {L}}_\nu a_0^+(0,x) - a_1^+(t,x)) + ({\mathcal {L}}_\nu a_1^+(0,x) - a_2^+(0,x))$, 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 $\partial _t {\mathcal {L}}_\nu {\tilde{u}}^+(0,x) = {\mathcal {L}}_\nu La_0^+(0,x) + \frac{1}{2} \Box a_0(0,x) = \frac{1}{2}\left( {\mathcal {L}}_\nu L f(x) - \frac{1}{2} \Delta f(x) + \frac{1}{2} L^2 f(x) \right)$. Thus (recall that $L = - {\mathcal {L}}_\nu + \frac{1}{2}\Delta \varphi$) directly from Eq. (15),

$$\begin{aligned} h^{(3)}(0)&= -\frac{1}{2}\int _{\partial S}{\mathcal {L}}_\nu L f(x) - \frac{1}{2} \Delta f(x) + \frac{1}{2} L^2 f(x)\,\mathrm {d}A(x). \end{aligned}$$

The formula

$$\begin{aligned} \Omega _{S,f}(t) = \int _0^\infty {\hat{k}}(s)\left( \int _S f \,\mathrm {d}V- h(s \sqrt{t} ) \right) \,\mathrm {d} s \end{aligned}$$

(20)

established in the previous section, together with Lemma 3, yields the asymptotic behaviour of $\Omega _{S,f}(t)$ by taking the Taylor expansion of h using Corollary 5. This gives the remainder of the claims of theorem 2.

4 Discussion

The above-said is not specific to the heat equation. Taking $k(x) = \exp (-x^{2m})$, $m \in {\mathbb {N}}$, we may, for example, study the one-parameter operator family $\exp (-t^m \Delta ^m)$. The wave equation estimates needed are the same. For $m \ge 2$, a brief calculation yields the explicit $t \rightarrow 0^+$ asymptotics

$$\begin{aligned} \langle \exp (t^m\Delta ^m)f \mathbb {1}_S, \mathbb {1}_S \rangle = \int _S f\,\mathrm {d}V- \left( \pi ^{-1} \Gamma \left( \frac{2m-1}{2m}\right) \int _{\partial S} f\,\mathrm {d}A\right) \sqrt{t} + o(t). \end{aligned}$$

We conclude with the observation that the generalization of this paper to weighted Riemannian manifolds (cf. [4]) is straightforward.