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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.

MSC:
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
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