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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Regularity properties of solutions of a class of elliptic-parabolic nonlinear Levi type equations
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by G. Citti and A. Montanari PDF
Trans. Amer. Math. Soc. 354 (2002), 2819-2848 Request permission

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
  • 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.
  • © Copyright 2002 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 354 (2002), 2819-2848
  • MSC (2000): Primary 35J70, 35K65; Secondary 22E30
  • DOI: https://doi.org/10.1090/S0002-9947-02-02928-8
  • MathSciNet review: 1895205