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 compact full-dimensional submanifolds $S$ with smooth boundary. This includes formulae like \begin{equation*} \int_{S} \exp(t\Delta) f \mathbb 1_S\, \mathrm{d}x = \int_S f \,\mathrm{d}x - \sqrt{\frac{t}{\pi}} \int_{\partial S} f \,\mathrm{d}A + o(\sqrt t),\quad t \rightarrow 0\,. \end{equation*} and (partially new) explicit expressions for similar expansions involving arbitrary powers of $\sqrt t$. By the same method, we also obtain short-time asymptotics of $\int_S \exp(t^m\Delta^m)f \mathbb 1_S\, \mathrm{d}x$, $m \in \mathbb N$, and more generally for one-parameter families of operators $t \mapsto k(\sqrt{-t\Delta})$ defined by an even Schwartz function $k$.


Introduction
Let (M, g) be a complete, boundaryless 1 , oriented Riemannian manifold with Laplace-Beltrami operator ∆, and volume dx. On a codimension-1 submanifold of M, we write dA for the induced surface (hyper)-area form. The heat semigroup T t := exp(t∆) acting on L 2 (M, dx) 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 Ω S,f (t) := S T t f 1 S dx, f ∈ C ∞ (M), has recently received much attention; see, for instance, [8,13,12] and the references therein.
Let us briefly recall some known results. On R n , sets S of finite perimeter P (S) are characterized by [8,Thm. 3.3 ] lim t→0 π t Ω S,f (t) = P (S) . (1) Extensions of this idea to abstract metric spaces are given in [7]. In the setting of compact manifolds M (or M = R n ) and S a full-dimensional submanifold with smooth boundary ∂S, the authors of [13] show that where the coefficients β j depend on S, f and the geometry of M. The setting of [13] is more general, amongst other things it includes f which have singularities. Some of the cofficients obtained in [13, corollary 1.7] are Extensions to some non-compact manifolds M and certain non-compact S are in [12]. Both eqs. (1) and (2) are proven with significant technical effort, yielding strong results. For example, in [8], 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, [13] requires pseudodifferential calculus and invariance theory.
Our aim is to show that slightly weaker results can be obtained by considerably lower technical effort. In contrast to [8], we treat only compact S with smooth boundary, and do not allow f to have singularities like [13] 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 . We prove: Theorem 1. Let M be a complete Riemannian manifold with Laplace-Beltrami operator ∆, Riemannian volume dx and induced (hyper) area form dA. Let S ⊂ M be a compact full-dimensional submanifold with smooth boundary. For f ∈ C ∞ (M) and N ∈ N, for constants (β j ) N j=0 which we characterize in the next theorem. To connect the constants (β j ) ∞ j=0 to the heat equation, observe that . With the j-th derivative k (j) (for j ∈ N 0 ), we define r j := (−1) j/2 k (j) (0) for j even and r j := (−1) (j−1)/2 ∞ 0 2k j (s) −πs ds for j odd. Let ϕ locally be the signed distance function (see also [9, section 3.2.2]) to ∂S, and denote by ∇ and · the gradient and (metric) inner product respectively. The vector field ν := −∇ϕ is outer unit normal at ∂S. Theorem 2. The coefficients of theorem 1 satisfy β 0 = r 0 S f dx and In particular, if f = 1 M , as t → 0, 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 [4, ch. 3].
Our approach 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 discontinous initial data is related to the short-time behaviour of the eikonal equation (cf. 'geometrical optics' [11]).
(C) The short-time behaviour of the wave and eikonal equations with initial data f 1 S are 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.
The specifics of point (C) will rest on an application of the Reynolds transport theorem. Here, letν be a smooth vector field with time-s flow Φ s andν = ν near ∂S. For small s, the (half) tubular neighborhood The last equation is a consequence of Cartan's magic formula and Stokes' theorem, where we use that dx(ν, ·) = dA(·) on ∂S.
2 Proof for β 0 , β 1 By Fourier theory, (6) On the operator level, this yields the well-known formula [11, section 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, Let ·, · denote the L 2 (M) inner product. Using eq. (7), 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 f = 1 M , eq. (5) yields In addition, | W s f 1 S , 1 S | ≤ f 1 S 2 1 S 2 for all s ≥ 0, in particular as s → ∞. We conclude with some calculations (cf. lemma 3 below), that This is weaker than the desired estimate, and restricts to f = 1 M . The problem is that the estimates in eq. (9) are too crude. To improve them, we instead approximate the solution u to the wave equation with geometrical optics, using the construction from [11, section 6.6]. The basic idea here is that u is in general discontinuous, with an outward-and an inward-moving front given by the zero level-set of functions ϕ + and ϕ − respectively. The functions ϕ ± satisfy the eikonal equation ∂ t ϕ = ±|∇ϕ ± | with inital value ϕ ± (0, ·) = ϕ(·). Our analysis is greatly simplified by choosing the initial ϕ to (locally) be the signed distance function to ∂S. In particular, ν = −∇ϕ is outward-pointing unit-normal to the level-sets of the distance function. The eikonal equation then reads ∂ t ϕ ± = ±|∇ϕ| = ±| − ν| = ±1.
The geometrical optics approximation 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) = a ± 0 ∆ϕ ± . With the Heaviside function θ : R → R, we defineũ The functionũ will serve as an approximation to the discontinuous part of the solution u to the wave-equation. To maintain consistency with the initial values of u, the initial values of the approximationũ are chosen to coincide with those of u at t = 0. Formally differentiating eq. (11) at t = 0, and equating the coefficients of θ ′ (ϕ(x)), shows that a ± 0 (0, ·) = 1 2 f (x) on ∂S. By looking at the transport equations, we see that the choice of a ± (0, ·) = 1 2 f (·) also in the interior of S has as a result that ∂ tũ (t, ·)| t=0 = 0 on all of S.
3 Proof for β 2 , β 3 , · · · We now briefly describe how the higher order terms can be calculated. Instead of working with a sufficiently high-order geometrical optics construction and tracking the error at every step, we write the solution u of the wave equation directly as the sum of an inward-and and outward-moving front, i.e. u(t, x) = a + (t, x)θ(ϕ + (t, x)) + a − (t, x)θ(ϕ − (t, x)) .
Much of this section is adapted with minor changes from [5, ch. 12] to the more general case considered here, see also [3] and [15, section 7.7].
The remainder of theorems 1 and 2 follows by combining Taylor's theorem, eq. (19) and lemma 3.

Discussion
Little of the above-said is 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 Similarly to [1], other even functions k can also be considered -given some smoothness and decay-at-infinity-type requirements, lemma 3 applies.