Abstract
We establish that heat diffusion with periodic conductivity is governed by two scales. The small time diffusion is described by the geodesic distance but the large time behaviour is dictated by the distance associated with an homogenized system obtained by a suitable averaging process. Our methods are quite general and apply to diffusion on a stratified Lie group.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bensoussan, A., Lions, J. L. and Papanicolaou, G. C.Asymptotic analysis for periodic structures. Studies in Mathematics and its Applications 5. North-Holland, 1978.
Davies, E. B.Heat kernels and spectral theory. Cambridge Tracts in Mathematics92. Cambridge University Press, Cambridge etc., 1989.
Davies, E. B. Explicit constants for Gausssian upper bounds on heat kernels.Amer. Journ. Math.109, 319–333 (1987).
Davies, E. B. Heat kernels in one dimension.The Quarterly Journal of Mathematics, Oxford 2nd Series 44(175), 283–299 (1993).
Fabes, E. B. and Stroock, D. W. A new proof of Moser’s parabolic Harnack inequality using the old ideas of Nash.Arch. Rat. Mech. and Anal. 96, 327–338 (1986).
Folland, G. B. and Stein, E. M.Hardy spaces on homogeneous groups. Mathematical Notes 28. Princeton University Press, Princeton, 1982.
Helgason, S.Differential geometry, Lie groups and symmetric spaces. Academic Press, New York, 1978.
Lehner, J.A short course in automorphic functions. R. W. Holt Publishing Inc., New York, 1966.
Nash, J. Continuity of solutions of parabolic and elliptic equations.Amer. J. Math. 80, 931–954 (1958).
Norris, J. R. Heat kernel bounds and homogenization of elliptic operators.Bulletin of the London Mathematical Society 26, 75–87 (1994).
Robinson, D. W.Elliptic operators and Lie groups. Oxford Mathematical Monographs. Oxford University Press, Oxford etc., 1991.
Saloff-Coste, L. and Stroock, D. W. Opérateurs uniformement sous-elliptiques sur les groupes de Lie.J. Funct. Anal. 98 (1991), 97–121.
Varopoulos, N., Saloff-Coste, L. and Coulhon, T.Analysis and geometry on groups. Cambridge University Press, Cambridge, 1992.
Zhikov, V. V., Kozlov, S. M., Oleinik, O. A. and Kha T’en Ngoan. Averaging and G-convergence of differential operators.Russ. Math. Surveys 34, 69–147 (1979).