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July  2009, 8(4): 1159-1201. doi: 10.3934/cpaa.2009.8.1159

On a class of hypoelliptic operators with unbounded coefficients in $R^N$

Bálint Farkas 1, and  Luca Lorenzi 2,

1. 

Technische Universität Darmstadt, Fachbereich Mathematik, AG Analysis, Schloßgartenstraße 7, D-64289, Darmstadt, Germany
2. 

Dipartimento di Matematica, Universitá degli Studi di Parma, Viale G. Usberti 85/A, 43100 Parma

Received  June 2008 Revised  December 2008 Published  March 2009

We consider second order linear partial differential operators $A$ on $R^N$ which are not assumed to be uniformly elliptic and whose coefficients in the second order part may grow quadratically, while the drift part has essentially linear growth. Instead of uniform ellipticity, we require a much weaker hypothesis of uniform hypoellipticity, which in an equivalent formulation connects the behaviour of the diffusion and the drift coefficients, by requiring that a Kalman-type condition is satisfied for them. By refining Bernstein's method we prove the existence of a semigroup {$T(t)$} of bounded linear operators (in the space of bounded and continuous functions) associated to the operator $A$. We also show uniform estimates for the spatial derivatives of the semigroup {$T(t)$} in (an)isotropic spaces of (Hölder-) continuous functions. As a consequence, we obtain Hölder estimates for the solutions of some elliptic and parabolic problems associated to the operator $A$.
Keywords: anisotropic Hölder estimates., distributional solutions to elliptic and parabolic problems, uniform estimates, Degenerate elliptic operators with unbounded coefficients in $R^N$.
Mathematics Subject Classification: Primary: 35K65; Secondary: 35J70, 35B65, 35K1.
Citation: Bálint Farkas, Luca Lorenzi. On a class of hypoelliptic operators with unbounded coefficients in $R^N$. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1159-1201. doi: 10.3934/cpaa.2009.8.1159
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