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On the Liouville property for divergence form operators |
Canad. J. Math. 50(1998), 487-496
Published:1998-06-01
Printed: Jun 1998
Printed: Jun 1998
- Martin T. Barlow
Abstract
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 Schr\"odinger operators.
| MSC Classifications: |
31C05 - Harmonic, subharmonic, superharmonic functions 60H10 - Stochastic ordinary differential equations [See also 34F05] 35J10 - Schrodinger operator [See also 35Pxx] |