zbMATH — the first resource for mathematics

Heat content asymptotics of a Riemannian manifold with boundary. (English) Zbl 0809.53047
Let \(M\) be an open bounded subset of \(\mathbb{R}^ m\) with smooth boundary and \(\Delta\) the classical Laplacian. If \(f_ 1 \in C^ \infty (M,\mathbb{R})\) represents the initial temperature distribution of \(M\), \(h = \exp (-t \Delta)f_ 1\) and \(f_ 2 \in C^ \infty (M,\mathbb{R})\), we can define \[ \beta (f_ 1, f_ 2)(t) = \int_ M h(x,t) f_ 2(x) dx. \] There is an asymptotic series as \(t \to 0^ +\) of the form: \[ \beta(f_ 1, f_ 2) (t) \simeq \sum^ \infty_{n = 0} \beta_ n (f_ 1, f_ 2) t^{n/2}. \] If we set \(f_ 1 = f_ 2 = 1\) then \[ \beta(t)= \beta(1,1) (t) \simeq \sum^ \infty_{n = 0} \beta_ n (M)t^{n/2} \] gives the asymptotic expansion of the total heat content of \(M\) with an initial temperature one. The first two coefficients \(\beta_ 0(M)\), \(\beta_ 1(M)\) were computed by the first author and E. B. Davies [Math. Z. 202, 463-482 (1989; Zbl 0687.35045)] and the third by the first author and J. F. Le Gall [Math. Z. 215, No. 3, 437-464 (1994; Zbl 0791.58089)]. Similar results hold for polygonal domains in \(\mathbb{R}^ 2\), the first author and S. Srisatkunarajah [Probab. Theory Relat. Fields 86, No. 1, 41-52 (1990; Zbl 0705.60068)] and for the upper hemisphere of a sphere, the first author [Proc. R. Soc. Edinb., Sect. A 118, No. 1/2, 5-12 (1991; Zbl 0753.35039)]. In the paper under review the authors extend all the above results to the general case of a compact \(n\)-dimensional manifold with smooth boundary. They also generalize these results to arbitrary operators of Laplace type with Dirichlet boundary conditions and to arbitrary smooth initial conditions.

53C20 Global Riemannian geometry, including pinching
58J35 Heat and other parabolic equation methods for PDEs on manifolds
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
Full Text: DOI