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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 $L_{\xi_0}$ 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 $C^\infty$.


References [Enhancements On Off] (What's this?)

  • [BG] E. Bedford, B. Gaveau, Envelopes of holomorphy of certain 2-spheres in $C^2$, Amer. J. Math., 105, 1983, 975-1009. MR 84k:32016
  • [BK] E. Bedford, W. Klingenberg, On the envelopes of holomorphy of a -spheres in $C^2$, Journal of the A.M.S. , 4, 3, 1991, 623-646.MR 92j:32034
  • [CS] E. M. Chirka, Shcherbina, Pseudoconvexity of rigid domains and foliation of hulls of graphs, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 21, 1995, 707-735.MR 98f:32016
  • [C] G. Citti, $C^\infty$ regularity of solutions of a quasilinear equation related to the Levi operator, Ann. Scuola Norm. Sup. Pisa, Cl. Sci. (4), 23, 3, (1996), 483-529. MR 98b:35072
  • [Csem] G. Citti, Regolarità di grafici non Levi-piatti, Seminario di Analisi, Dip. Mat. Univ. Bologna (A.A. 1996/97), Tecnoprint, Bologna, 186-195.
  • [CM1] G. Citti, A. Montanari, Analytic estimates of solutions of the Levi equation, J. Differential Equations, 173, (2001), 356-389.
  • [CM2] G. Citti, A. Montanari, $C^\infty$ regularity of solutions of an equation of Levi's type in $R^{2n+1}$, Annali di Matematica Pura Appl. (4), 180, (2001), 27-58 CMP 2001:11
  • [CPP] G. Citti, A. Pascucci, S. Polidoro, On the regularity of solutions to a nonlinear ultraparabolic equation in mathematical finance, Diff. Int. Eq, 14 (2001), 701-738.
  • [F] G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Arkiv Mat., 13, 1975, 161-207.MR 58:13215
  • [FS] G. B. Folland, E. M. Stein, Estimates for the $\bar\partial_b$ Complex and Analysis on the Heisenberg Group, Comm. Pure Appl. Math. 20, 1974, 429 - 522.MR 51:3719
  • [FL] B. Franchi, E. Lanconelli, Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 10, 1983, 523-541. MR 85k:35094
  • [H] L. Hörmander Hypoelliptic second order differential equations Acta Math., 119, 1967, 147-171. MR 36:5526
  • [HK] G. Huisken, W. Klingenberg, Flow of real hypersurfaces by the trace of the Levi form, Mathematical Research Letters 6 (1999), 645-662. MR 2001f:53141
  • [K] W. Klingenberg, Real hypersurfaces in Kähler manifolds preprint.
  • [M] A. Montanari, Real hypersurfaces evolving by Levi curvature: smooth regularity of solutions to the parabolic Levi equation, to appear in Comm. Partial Differential Equations 26 (9&10) (2001) 1633-1664.
  • [NSW] A. Nagel, E. M. Stein, S. Wainger Balls and metrics defined by vector fields I: Basic properties. Acta Math. 155, 1985, 103-147. MR 86k:46049
  • [RS] L. Rothschild, E. M. Stein Hypoelliptic differential operators and nilpotent groups Acta Math. 137, 1977, 247-320.MR 55:9171
  • [Sc] N. V. Shchcerbina, On the polynomial hull of a graph. Indiana Univ. Math J., 42, 1993, 477-503. MR 95e:32017
  • [ST1] Z. Slodkowski, G. Tomassini, Weak solutions for the Levi equation and Envelope of Holomorphy, J. Funct. Anal, 101, no. 4, 1991, 392-407. MR 93c:32018
  • [ST2] Z. Slodkowski, G. Tomassini, Evolution of subsets in ${\mathbb C}^2$ and a parabolic problem for the Levi equation, Ann. Scuola Norm. Sup. Pisa, Cl. Sci. (4), 25, (1997), 757-784. MR 2000c:32013
  • [ST3] Z. Slodkowski, G. Tomassini, Evolution of special subsets of ${\mathbb C}^2$, Adv. Math. 152, (2000), no. 2, 336-358. MR 2001i:32019
  • [ST4] Z. Slodkowski, G. Tomassini, Evolution of a graph by Levi form, Gulliver, Robert (ed.) et al., Differential geometric methods in the control of partial differential equations. (Boulder, CO, 1999), Providence, RI: American Mathematical Society (AMS). Contemp. Math. 268, 373-382 (2000). CMP 2001:07

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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

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