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

Citti, G.; Montanari, A.

doi:10.1090/S0002-9947-02-02928-8

Regularity properties of solutions of a class of elliptic-parabolic nonlinear Levi type equations
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by G. Citti and A. Montanari

Trans. Amer. Math. Soc. 354 (2002), 2819-2848

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

Published electronically: February 14, 2002

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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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