Heat Flow Out of a Compact Manifold

@article{Berg2013HeatFO,
  title={Heat Flow Out of a Compact Manifold},
  author={Michiel van den Berg and Peter Gilkey},
  journal={The Journal of Geometric Analysis},
  year={2013},
  volume={25},
  pages={1576-1601}
}
  • M. Berg, P. Gilkey
  • Published 26 June 2013
  • Mathematics
  • The Journal of Geometric Analysis
We discuss the heat content asymptotics associated with the heat flow out of a smooth compact manifold in a larger compact Riemannian manifold. Although there are no boundary conditions, the corresponding heat content asymptotics involve terms localized on the boundary. The classical pseudo-differential calculus is used to establish the existence of the complete asymptotic series, and methods of invariance theory are used to determine the first few terms in the asymptotic series in terms of… 
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References

SHOWING 1-10 OF 16 REFERENCES
Neumann heat content asymptotics with singular initial temperature and singular specific heat
We study the asymptotic behavior of the heat content on a compact Riemannian manifold with boundary and with singular specific heat and singular initial temperature distributions imposing Robin
Heat content asymptotics with singular initial temperature distributions
Heat content and Hardy inequality for complete Riemannian manifolds
Short-time heat flow and functions of bounded variation in RN
— We prove a characterisation of sets with finite perimeter and BV functions in terms of the short time behaviour of the heat semigroup in RN . For sets with smooth boundary a more precise result is
A Semigroup Version of the Isoperimetric Inequality
AbstractIn this article we extend the heat semigroup version of the isoperimetric inequality in Rn established by M. Ledoux. To this purpose, the notion of the perimeter of a Caccioppoli set
Invariance Theory Heat Equation and Atiyah Singer Index Theorem
Pseudo-Differential Operators Introduction Fourier Transform and Sobolev Spaces Pseudo-Differential Operators on Rm Pseudo-Differential Operators on Manifolds Index of Fredholm Operators Elliptic
Stochastic analysis on manifolds
Introduction Stochastic differential equations and diffusions Basic stochastic differential geometry Brownian motion on manifolds Brownian motion and heat kernel Short-time asymptotics Further
Short-time heat flow and functions of bounded variation in \mathbf{R}^N
On prouve une caracterisation des ensembles avec perimetre fini et des fonctions a variation bornee en termes du comportement du semi-groupe de la chaleur dans R N au voisinage de t = 0. On prouve
Topics in pseudo-differential operators
The subject of pseudo-differential operators has sprung up in the last few years out of the earlier work of Giraud, Mihlin, and Calderon and Zygmund, and is still in the process of development. There
Isoperimetric Inequalities: Differential Geometric and Analytic Perspectives
Part I. Introduction: 1. The isoperimetric problem 2. The isoperimetric inequality in the plane 3. Preliminaries 4. Bibliographic notes Part II. Differential Geometric Methods: 1. The C2 uniqueness
...
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