## Positivity and strong ellipticity

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- by A. F. M. ter Elst, Derek W. Robinson and Yueping Zhu PDF
- Proc. Amer. Math. Soc.
**134**(2006), 707-714 Request permission

## Abstract:

We consider partial differential operators $H=-\operatorname {div} (C\nabla )$ in divergence form on $\mathbf {R}^d$ with a positive-semidefinite, symmetric, matrix $C$ of real $L_\infty$-coefficients, and establish that $H$ is strongly elliptic if and only if the associated semigroup kernel satisfies local lower bounds, or, if and only if the kernel satisfies Gaussian upper and lower bounds.## References

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## Additional Information

**A. F. M. ter Elst**- Affiliation: Centre for Mathematics and its Applications, Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia
- Address at time of publication: Department of Mathematics and Computing Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
- MR Author ID: 63185
- Email: terelst@win.tue.nl
**Derek W. Robinson**- Affiliation: Centre for Mathematics and its Applications, Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia
- MR Author ID: 191422
- Email: Derek.Robinson@anu.edu.au
**Yueping Zhu**- Affiliation: Centre for Mathematics and its Applications, Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia
- Address at time of publication: Department of Mathematics, Nantong University, Nantong, 226007, Jiangsu Province, People’s Republic of China
- Email: zhuyueping@ntu.edu.cn
- Received by editor(s): September 30, 2004
- Published electronically: September 28, 2005
- Additional Notes: This work was carried out while the first author was visiting the Centre for Mathematics and its Applications at the Australian National University. He thanks the Australian Research Council for its support and the CMA for its hospitality. The third author was an ARC Research Associate for the duration of the collaboration
- Communicated by: Andreas Seeger
- © Copyright 2005 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**134**(2006), 707-714 - MSC (2000): Primary 35Jxx
- DOI: https://doi.org/10.1090/S0002-9939-05-08180-3
- MathSciNet review: 2180888