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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Positivity and strong ellipticity


Authors: A. F. M. ter Elst, Derek W. Robinson and Yueping Zhu
Journal: Proc. Amer. Math. Soc. 134 (2006), 707-714
MSC (2000): Primary 35Jxx
Published electronically: September 28, 2005
MathSciNet review: 2180888
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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.


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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
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
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

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08180-3
PII: S 0002-9939(05)08180-3
Keywords: Elliptic operator, semigroup, kernel, upper bounds, lower bounds
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
Article copyright: © Copyright 2005 American Mathematical Society