Heat content asymptotics

Let be a compact Riemannian manifold with boundary . Assume given a decomposition of the boundary as the disjoint union of two closed sets and . Impose Neumann boundary conditions on and Dirichlet boundary conditions on . Let be the temperature distribution of the manifold corresponding to an initial temperature ; is the solution to the equations:

Here, denotes differentiation with respect to the inward unit normal. Let be a smooth function giving the specific heat. The total heat energy content of is given by

As , there is an asymptotic expansion

The coefficients are the heat content asymptotics and are locally computable.

These coefficients were first studied with empty and with . Planar regions with smooth boundaries were studied in [a5], [a6], the upper hemisphere was studied in [a4], [a3], and polygonal domains in the plane were studied in [a7]. See [a11], [a12] for recursive formulas on a general Riemannian manifold.

More generally, let be the second fundamental form and let be the Riemann curvature tensor. Let indices , , range from to and index an orthonormal frame for the tangent bundle of the boundary. Let ":" (respectively, ";" ) denote covariant differentiation with respect to the Levi-Civita connection of (respectively, of ) summed over repeated indices. The first few coefficients have the form:

;

;

The coefficient is known.

The coefficients and have been determined if is empty.

One can replace the Laplace operator by an arbitrary operator of Laplace type as the evolution operator [a1], [a2], [a10], [a9]. One can study non-minimal operators as the evolution operator, inhomogeneous boundary conditions, and time-dependent evolution operators of Laplace type. A survey of the field is given in [a8].

References

[a1]	M. van den Berg, S. Desjardins, P. Gilkey, "Functoriality and heat content asymptotics for operators of Laplace type" Topol. Methods Nonlinear Anal. , 2 (1993) pp. 147–162
[a2]	M. van den Berg, P. Gilkey, "Heat content asymptotics of a Riemannian manifold with boundary" J. Funct. Anal. , 120 (1994) pp. 48–71
[a3]	M. van den Berg, P. Gilkey, "Heat invariants for odd dimensional hemispheres" Proc. R. Soc. Edinburgh , 126A (1996) pp. 187–193
[a4]	M. van den Berg, "Heat equation on a hemisphere" Proc. R. Soc. Edinburgh , 118A (1991) pp. 5–12
[a5]	M. van den Berg, E.M. Davies, "Heat flow out of regions in " Math. Z. , 202 (1989) pp. 463–482
[a6]	M. van den Berg, J.-F. Le Gall, "Mean curvature and the heat equation" Math. Z. , 215 (1994) pp. 437–464
[a7]	M. van den Berg, S. Srisatkunarajah, "Heat flow and Brownian motion for a region in with a polygonal boundary" Probab. Th. Rel. Fields , 86 (1990) pp. 41–52
[a8]	P. Gilkey, "Heat content asymptotics" Booss (ed.) Wajciechowski (ed.) , Geometric Aspects of Partial Differential Equations , Contemp. Math. , 242 , Amer. Math. Soc. (1999) pp. 125–134
[a9]	D.M. McAvity, "Surface energy from heat content asymptotics" J. Phys. A: Math. Gen. , 26 (1993) pp. 823–830
[a10]	D.M. McAvity, "Heat kernel asymptotics for mixed boundary conditions" Class. Quant. Grav , 9 (1992) pp. 1983–1998
[a11]	A. Savo, "Uniform estimates and the whole asymptotic series of the heat content on manifolds" Geom. Dedicata , 73 (1998) pp. 181–214
[a12]	A. Savo, "Heat content and mean curvature" J. Rend. Mat. Appl. VII Ser. , 18 (1998) pp. 197–219