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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Regularity properties of solutions of a class of elliptic-parabolic nonlinear Levi type equations


Authors: G. Citti and A. Montanari
Journal: Trans. Amer. Math. Soc. 354 (2002), 2819-2848
MSC (2000): Primary 35J70, 35K65; Secondary 22E30
Published electronically: February 14, 2002
MathSciNet review: 1895205
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Abstract: In this paper we prove the smoothness of solutions of a class of elliptic-parabolic nonlinear Levi type equations, represented as a sum of squares plus a vector field. By means of a freezing method the study of the operator is reduced to the analysis of a family $L_{\xi_0}$ of left invariant operators on a free nilpotent Lie group. The fundamental solution $\Gamma_{\xi_0}$ of the operator $L_{\xi_0}$ is used as a parametrix of the fundamental solution of the Levi operator, and provides an explicit representation formula for the solution of the given equation. Differentiating this formula and applying a bootstrap method, we prove that the solution is $C^\infty$.


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Additional Information

G. Citti
Affiliation: Dipartimento di Matematica, Universita di Bologna, Piazza di Porta S. Donato 5, 40127, Bologna, Italy
Email: citti@dm.unibo.it

A. Montanari
Affiliation: Dipartimento di Matematica, Universita di Bologna, Piazza di Porta S. Donato 5, 40127, Bologna, Italy
Email: montanar@dm.unibo.it

DOI: http://dx.doi.org/10.1090/S0002-9947-02-02928-8
PII: S 0002-9947(02)02928-8
Keywords: Levi equation, elliptic-parabolic nonlinear equation, freezing method, Lie groups, fundamental solution, regularity properties
Received by editor(s): May 3, 2000
Received by editor(s) in revised form: August 8, 2001
Published electronically: February 14, 2002
Additional Notes: Investigation supported by University of Bologna, founds for selected research topics.
Article copyright: © Copyright 2002 American Mathematical Society