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

DOI:
https://doi.org/10.1090/S0002-9947-02-02928-8

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 of left invariant operators on a free nilpotent Lie group. The fundamental solution of the operator 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 .

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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:
https://doi.org/10.1090/S0002-9947-02-02928-8

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