On the smoothness of viscosity solutions of the prescribed Levi-curvature equation

Giovanna Citti; Ermanno Lanconelli; Annamaria Montanari

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1999)

  • Volume: 10, Issue: 2, page 61-68
  • ISSN: 1120-6330

Abstract

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In this paper a C ∞ -regularity result for the strong viscosity solutions to the prescribed Levi-curvature equation is announced. As an application, starting from a result by Z. Slodkowski and G. Tomassini, the C ∞ -solvability of the Dirichlet problem related to the same equation is showed.

How to cite

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Citti, Giovanna, Lanconelli, Ermanno, and Montanari, Annamaria. "On the smoothness of viscosity solutions of the prescribed Levi-curvature equation." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 10.2 (1999): 61-68. <http://eudml.org/doc/252347>.

@article{Citti1999,
abstract = {In this paper a \( C^\{\infty\} \)-regularity result for the strong viscosity solutions to the prescribed Levi-curvature equation is announced. As an application, starting from a result by Z. Slodkowski and G. Tomassini, the \( C^\{\infty\} \)-solvability of the Dirichlet problem related to the same equation is showed.},
author = {Citti, Giovanna, Lanconelli, Ermanno, Montanari, Annamaria},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Levi equation; Viscosity solutions; Non-linear vector fields; $ C^\{\infty \} $ C ∞ -regularity; Boundary value problem; -solvability; Dirichlet problem},
language = {eng},
month = {6},
number = {2},
pages = {61-68},
publisher = {Accademia Nazionale dei Lincei},
title = {On the smoothness of viscosity solutions of the prescribed Levi-curvature equation},
url = {http://eudml.org/doc/252347},
volume = {10},
year = {1999},
}

TY - JOUR
AU - Citti, Giovanna
AU - Lanconelli, Ermanno
AU - Montanari, Annamaria
TI - On the smoothness of viscosity solutions of the prescribed Levi-curvature equation
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1999/6//
PB - Accademia Nazionale dei Lincei
VL - 10
IS - 2
SP - 61
EP - 68
AB - In this paper a \( C^{\infty} \)-regularity result for the strong viscosity solutions to the prescribed Levi-curvature equation is announced. As an application, starting from a result by Z. Slodkowski and G. Tomassini, the \( C^{\infty} \)-solvability of the Dirichlet problem related to the same equation is showed.
LA - eng
KW - Levi equation; Viscosity solutions; Non-linear vector fields; $ C^{\infty } $ C ∞ -regularity; Boundary value problem; -solvability; Dirichlet problem
UR - http://eudml.org/doc/252347
ER -

References

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  1. Bedford, E. - Gaveau, B., Envelopes of holomorphy of certain 2-spheres in C 2 . Am. J. of Math., 105, 1983, 975-1009. Zbl0535.32008MR708370DOI10.2307/2374301
  2. Citti, G., C ∞ regularity of solutions of the Levi equation. Analyse non Linéaire, Annales de l’I.H.P, 15, 1998, 517-534. Zbl0921.35033MR1632929DOI10.1016/S0294-1449(98)80033-4
  3. Cabré, X. - Caffarelli, L., Fully nonlinear elliptic equations. Amer. Math. Society, Providence1995. Zbl0834.35002MR1351007
  4. Crandall, M. G. - Ishii, H. - Lions, P. L., User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math Soc., (N.S.), 27, 1992, 1-67. Zbl0755.35015MR1118699DOI10.1090/S0273-0979-1992-00266-5
  5. Citti, G. - Montanari, A., Strong solutions for the Levi curvature equation. Advances in Differential Equations, to appear. Zbl1211.35112MR1734545
  6. Citti, G. - Montanari, A., C ∞ regularity of solutions of an equation of Levi’s type in R 2 ⁢ n + 1 . Preprint. Zbl1030.35019MR1848050DOI10.1007/s10231-001-8196-z
  7. Folland, G. B. - Stein, E. M., Estimates for the ∂ ¯ b complex and analysis on the Heisenberg group. Comm. Pure Appl. Math., 20, 1974, 429-522. Zbl0293.35012MR367477
  8. Gilgarg, D. - Trudinger, N. S., Elliptic partial differential equations of second order. Springer-Verlag, Berlin-Heidelberg-New York-Tokio1983. Zbl0361.35003MR737190
  9. Range, R. M., Holomorphic functions and integral representations in several complex variables. Springer-Verlag, Berlin-Heidelberg-New York-Tokio1986. Zbl0591.32002MR847923
  10. Slodkowski, Z. - Tomassini, G., Weak solutions for the Levi equation and envelope of holomorphy. J. Funct. Anal, 101, 4, 1991, 392-407. Zbl0744.35015MR1136942DOI10.1016/0022-1236(91)90164-Z

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