Positivity and strong ellipticity

ter Elst, A. F. M.; Robinson, Derek; Zhu, Yueping

doi:10.1090/S0002-9939-05-08180-3

Positivity and strong ellipticity
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by A. F. M. ter Elst, Derek W. Robinson and Yueping Zhu

Proc. Amer. Math. Soc. 134 (2006), 707-714

DOI: https://doi.org/10.1090/S0002-9939-05-08180-3

Published electronically: September 28, 2005

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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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