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On the Liouville Property for Divergence Form Operators

Published online by Cambridge University Press:  20 November 2018

Martin T. Barlow
Martin T. Barlow*
Affiliation:
Department of Mathematics University of British Columbia Vancouver, British Columbia V6T 1Z2
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Abstract

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In this paper we construct a bounded strictly positive function $\sigma $ such that the Liouville property fails for the divergence form operator $L\,=\,\nabla ({{\sigma }^{2}}\nabla )$. Since in addition $\Delta \sigma /\sigma $ is bounded, this example also gives a negative answer to a problem of Berestycki, Caffarelli and Nirenberg concerning linear Schrödinger operators.

Keywords

31C0560H1035J10
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 50 , Issue 3 , 01 June 1998 , pp. 487 - 496
Copyright
Copyright © Canadian Mathematical Society 1998

References

[Bas] Bass, R.F., Diffusions and Elliptic Operators. Springer, New York, 1997.Google Scholar
[BCN] Berestycki, H., Cafarelli, L. and Nirenberg, L., Further qualitative properties for elliptic equations in unbounded domains. preprint.Google Scholar
[GG] Ghoussoub, N. and Gui, C., On a conjecture of De Giorgi and some related problems. Math. Ann., to appear.Google Scholar
[IW] Ikeda, N. and Watanabe, S., Stochastic Differential Equations and Diffusion processes. North-Holland, Kodansha, 1981.Google Scholar
[P] Pitman, J., One dimensional Brownian motion and the three-dimensional Bessel process. J. Appl. Probab. 7(1975), 511–526.Google Scholar
[RW] Rogers, L.C.G. and Williams, D., Diffusions, Markov Processes, and Martingales, Volume I. (2nd edition), Wiley, 1994.Google Scholar