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Mathematische Zeitschrift

, Volume 238, Issue 3, pp 569–637 | Cite as

On second-order periodic elliptic operators in divergence form

  • A.F.M. ter Elst
  • Derek W. Robinson
  • Adam Sikora

Abstract.

We consider second-order, strongly elliptic, operators with complex coefficients in divergence form on \({\bf R}^d\). We assume that the coefficients are all periodic with a common period. If the coefficients are continuous we derive Gaussian bounds, with the correct small and large time asymptotic behaviour, on the heat kernel and all its Hölder derivatives. Moreover, we show that the first-order Riesz transforms are bounded on the \(L_p\)-spaces with \(p\in\langle1,\infty\rangle\). Secondly if the coefficients are Hölder continuous we prove that the first-order derivatives of the kernel satisfy good Gaussian bounds. Then we establish that the second-order derivatives exist and satisfy good bounds if, and only if, the coefficients are divergence-free or if, and only if, the second-order Riesz transforms are bounded. Finally if the third-order derivatives exist with good bounds then the coefficients must be constant.

Mathematics Subject Classification (2000): 35K05, 35B27, 35B40, 47A58 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • A.F.M. ter Elst
    • 1
  • Derek W. Robinson
    • 2
  • Adam Sikora
    • 2
  1. 1.Department of Mathematics and Computing Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The NetherlandsNL
  2. 2.Centre for Mathematics and its Applications, School of Mathematical Sciences, Australian National University, Canberra, ACT 0200, AustraliaAU

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