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Journal of the London Mathematical Society 2001 63(3):735-753; doi:10.1017/S0024610701002149
© 2001 by London Mathematical Society
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© The London Mathematical Society

On Second-Order Almost-Periodic Elliptic Operators

N. Dungey, A. F. M. ter Elst and Derek W. Robinson

Centre for Mathematics and its Applications, School of Mathematical Sciences, Australian National University Canberra, ACT 0200, Australia
Department of Mathematics and Computing Science, Eindhoven University of Technology PO Box 513, 5600 MB Eindhoven, Netherlands

Received 27 January 2000. Revision received 25 September 2000.

The paper considers second-order, strongly elliptic, operators H with complex almost-periodic coefficients in divergence form on Rd. First, it is proved that the corresponding heat kernel is Hölder continuous and Gaussian bounds are derived with the correct small and large time asymptotic behaviour on the kernel and its Hölder derivatives. Secondly, it is established that the kernel has a variety of properties of almost-periodicity. Thirdly, it is demonstrated that the kernel of the homogenization Formula of H is the leading term in the asymptotic expansion of t ↦ Kt.


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