Smooth regularity for solutions of the Levi Monge-Ampère equation

Francesca Lascialfari; Annamaria Montanari

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2001)

  • Volume: 12, Issue: 2, page 115-123
  • ISSN: 1120-6330

Abstract

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We present a smooth regularity result for strictly Levi convex solutions to the Levi Monge-Ampère equation. It is a fully nonlinear PDE which is degenerate elliptic. Hence elliptic techniques fail in this situation and we build a new theory in order to treat this new topic. Our technique is inspired to those introduced in [3] and [8] for the study of degenerate elliptic quasilinear PDE’s related to the Levi mean curvature equation. When the right hand side has the meaning of total curvature of a real hypersurface in C n + 1 , the Levi Monge-Ampère equation arises in the study of envelopes of holomorphy and has important applications in the theory of holomorphic functions of several complex variables.

How to cite

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Lascialfari, Francesca, and Montanari, Annamaria. "Smooth regularity for solutions of the Levi Monge-Ampère equation." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 12.2 (2001): 115-123. <http://eudml.org/doc/252425>.

@article{Lascialfari2001,
abstract = {We present a smooth regularity result for strictly Levi convex solutions to the Levi Monge-Ampère equation. It is a fully nonlinear PDE which is degenerate elliptic. Hence elliptic techniques fail in this situation and we build a new theory in order to treat this new topic. Our technique is inspired to those introduced in [3] and [8] for the study of degenerate elliptic quasilinear PDE’s related to the Levi mean curvature equation. When the right hand side has the meaning of total curvature of a real hypersurface in $\mathbb\{C\}^\{n+1\}$, the Levi Monge-Ampère equation arises in the study of envelopes of holomorphy and has important applications in the theory of holomorphic functions of several complex variables.},
author = {Lascialfari, Francesca, Montanari, Annamaria},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Levi Monge-Ampère equation; Fully nonlinear degenerate elliptic PDE; Non-linear vector fields; Schauder-type estimate; Smooth regularity of strictly Levi convex solutions; fully nonlinear degenerate elliptic PDE; nonlinear vector fields; smooth regularity of strictly Levi convex solutions},
language = {eng},
month = {6},
number = {2},
pages = {115-123},
publisher = {Accademia Nazionale dei Lincei},
title = {Smooth regularity for solutions of the Levi Monge-Ampère equation},
url = {http://eudml.org/doc/252425},
volume = {12},
year = {2001},
}

TY - JOUR
AU - Lascialfari, Francesca
AU - Montanari, Annamaria
TI - Smooth regularity for solutions of the Levi Monge-Ampère equation
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2001/6//
PB - Accademia Nazionale dei Lincei
VL - 12
IS - 2
SP - 115
EP - 123
AB - We present a smooth regularity result for strictly Levi convex solutions to the Levi Monge-Ampère equation. It is a fully nonlinear PDE which is degenerate elliptic. Hence elliptic techniques fail in this situation and we build a new theory in order to treat this new topic. Our technique is inspired to those introduced in [3] and [8] for the study of degenerate elliptic quasilinear PDE’s related to the Levi mean curvature equation. When the right hand side has the meaning of total curvature of a real hypersurface in $\mathbb{C}^{n+1}$, the Levi Monge-Ampère equation arises in the study of envelopes of holomorphy and has important applications in the theory of holomorphic functions of several complex variables.
LA - eng
KW - Levi Monge-Ampère equation; Fully nonlinear degenerate elliptic PDE; Non-linear vector fields; Schauder-type estimate; Smooth regularity of strictly Levi convex solutions; fully nonlinear degenerate elliptic PDE; nonlinear vector fields; smooth regularity of strictly Levi convex solutions
UR - http://eudml.org/doc/252425
ER -

References

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  3. Citti, G., C ∞ regularity of solutions of a quasilinear equation related to the Levi operator. Ann. Scuola Norm. Sup. di Pisa Cl. Sci., s. 4, v. XXIII, 1996, 483-529. Zbl0872.35018MR1440031
  4. Citti, G., C ∞ regularity of solutions of the Levi equation. Ann. Inst. H. Poincaré, Anal. non Linéaire, 15, 4, 1998, 517-534. Zbl0921.35033MR1632929DOI10.1016/S0294-1449(98)80033-4
  5. Citti, G. - Lanconelli, E. - Montanari, A., On the smoothness of viscosity solutions of the prescribed Levi-curvature equation. Rend. Mat. Acc. Lincei, s. 9, v. 10, 1999, 61-68. Zbl1010.35024MR1768190
  6. Citti, G. - Montanari, A., Strong solutions for the Levi curvature equation. Adv. in Diff. Eq., v. 5, 1-3, 2000, 323-342. Zbl1211.35112MR1734545
  7. Citti, G. - Montanari, A., Regularity properties of Levi flat graphs. C.R. Acad. Sci. Paris, t. 329, s. 1, 1999, 1049-1054. Zbl0940.35053MR1735882DOI10.1016/S0764-4442(00)88472-4
  8. Citti, G. - Montanari, A., C ∞ regularity of solutions of an equation of Levi’s type in R 2 ⁢ n + 1 . Ann. Mat. Pura Appl., to appear. Zbl1030.35019MR1848050DOI10.1007/s10231-001-8196-z
  9. Citti, G. - Montanari, A., Regularity properties of solutions of a class of elliptic-parabolic nonlinear Levi type equations. Preprint. Zbl1008.35030MR1895205DOI10.1090/S0002-9947-02-02928-8
  10. Gilgarg, D. - Trudinger, N.S., Elliptic partial differential equations of second order. Grundlehrer der Math. Wiss., vol. 224, Springer-Verlag, New York1977. Zbl0361.35003MR473443
  11. Nagel, A. - Stein, E.M. - Wainger, S., Balls and metrics defined by vector fields I: basic properties. Acta Math., 155, 1985, 103-147. Zbl0578.32044MR793239DOI10.1007/BF02392539
  12. Slodkowski, Z. - Tomassini, G., The Levi equation in higher dimension and relationships to the envelope of holomorphy. American Journal of Mathematics, 116, 1994, 479-499. Zbl0802.35050MR1269612DOI10.2307/2374937
  13. Slodkowski, Z. - Tomassini, G., Weak solutions for the Levi equation and Envelope of Holomorphy. J. Funct. Anal., 101, n. 4, 1991, 392-407. Zbl0744.35015MR1136942DOI10.1016/0022-1236(91)90164-Z
  14. Tomassini, G., Geometric Properties of Solutions of the Levi equation. Ann. Mat. Pura Appl., 4, 152, 1988, 331-344. Zbl0681.35017MR980986DOI10.1007/BF01766155
  15. Xu, C.J., Regularity for Quasilinear Second-Order Subelliptic Equations. Comm. Pure and Appl. Math., 45, 1992, 77-96. Zbl0827.35023MR1135924DOI10.1002/cpa.3160450104

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