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September  2012, 11(5): 1911-1922. doi: 10.3934/cpaa.2012.11.1911

On the characteristic curvature operator

Vittorio Martino 1,

1. 

Department of Mathematics, Rutgers University, Hill Center for the Mathematical Sciences, 110 Frelinghuysen Rd., Piscataway 08854-8019 NJ, United States

Received  May 2011 Revised  September 2011 Published  March 2012

We introduce the Characteristic Curvature as the curvature of the trajectories of the Hamiltonian vector field with respect to the normal direction to the isoenergetic surfaces; by using the Second Fundamental Form we relate it to the Classical and Levi Mean Curvature. Then we prove existence and uniqueness of viscosity solutions for the related Dirichlet problem and we show the Lipschitz regularity of the solutions under suitable hypotheses. At the end we show that neither Strong Comparison Principle nor Hopf Lemma hold for the Characteristic Curvature Operator.
Keywords: viscosity solutions, characteristic direction for hypersurfaces., Quasilinear degenerate elliptic PDEs.
Mathematics Subject Classification: Primary: 35J70, 35J25; Secondary: 49L2.
Citation: Vittorio Martino. On the characteristic curvature operator. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1911-1922. doi: 10.3934/cpaa.2012.11.1911
References:
[1]

E. Bedford and B. Gaveau, Hypersurfaces with bounded Levi form, Indiana University Journal, 27 (1978), 867-873. doi: 10.1512/iumj.1978.27.27058.  Google Scholar

[2]

A. Bogges, "CR Manifolds and the Tangential Cauchy-Riemann Complex," Studies in Advanced Mathematics, 1991.  Google Scholar

[3]

M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order Partial differential equations, Bull. Amer. Soc., 27 (1992), 1-67.  Google Scholar

[4]

F. Da Lio and A. Montanari, Existence and uniqueness of Lipschitz continuous graphs with prescribed Levi curvature, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 1-28.  Google Scholar

[5]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," second edition, Springer-Verlag, 1983.  Google Scholar

[6]

H. Hofer and E. Zehnder, "Symplectic Invariants and Hamiltonian Dynamics," Birkhäuser Verlag, Basel, 1994  Google Scholar

[7]

J. G. Hounie and E. Lanconelli, An Alexander type theorem for Reinhardt domains of $\mathbbC^2$. Recent progress on some problems in several complex variables and partial differential equations, Contemp. Math., Amer. Math. Soc., Providence, RI, 400 (2006), 129-146.  Google Scholar

[8]

J. G. Hounie and E. Lanconelli, A sphere theorem for a class of Reinhardt domains with constant Levi curvature, Forum Mathematicum, 20 (2008), 571-586. doi: 10.1515/FORUM.2008.029.  Google Scholar

[9]

H. Ishii and P. L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. Differential Equations, 83 (1990), 26-78. doi: 10.1016/0022-0396(90)90068-Z.  Google Scholar

[10]

E. Lanconelli and A. Montanari, Pseudoconvex fully nonlinear partial differential operators. Strong comparison theorems, J. Differential Equations, 202 (2004), 306-331. doi: 10.1016/j.jde.2004.03.017.  Google Scholar

[11]

V. Martino, A symmetry result on Reinhardt domains, Differential and Integral Equations, 24 (2011), 495-504.  Google Scholar

[12]

V. Martino and A. Montanari, Graphs with prescribed the trace of the Levi form, Ann. Univ. Ferrara Sez. VII Sci. Mat., 52 (2006), 371-382. doi: 10.1007/s11565-006-0027-0.  Google Scholar

[13]

Z. Slodkowski and G. Tomassini, Weak solutions for the Levi equation and envelope of holomorphy, J. Funct. Anal., 101 (1991), 392-407. doi: 10.1016/0022-1236(91)90164-Z.  Google Scholar

[14]

Z. Slodkowski, G. Tomassini, The Levi equation in higher dimensions and relationships to the envelope of holomorphy, Amer. J. Math., 116 (1994), 479-499. doi: 10.2307/2374937.  Google Scholar

show all references

References:
[1]

E. Bedford and B. Gaveau, Hypersurfaces with bounded Levi form, Indiana University Journal, 27 (1978), 867-873. doi: 10.1512/iumj.1978.27.27058.  Google Scholar

[2]

A. Bogges, "CR Manifolds and the Tangential Cauchy-Riemann Complex," Studies in Advanced Mathematics, 1991.  Google Scholar

[3]

M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order Partial differential equations, Bull. Amer. Soc., 27 (1992), 1-67.  Google Scholar

[4]

F. Da Lio and A. Montanari, Existence and uniqueness of Lipschitz continuous graphs with prescribed Levi curvature, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 1-28.  Google Scholar

[5]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," second edition, Springer-Verlag, 1983.  Google Scholar

[6]

H. Hofer and E. Zehnder, "Symplectic Invariants and Hamiltonian Dynamics," Birkhäuser Verlag, Basel, 1994  Google Scholar

[7]

J. G. Hounie and E. Lanconelli, An Alexander type theorem for Reinhardt domains of $\mathbbC^2$. Recent progress on some problems in several complex variables and partial differential equations, Contemp. Math., Amer. Math. Soc., Providence, RI, 400 (2006), 129-146.  Google Scholar

[8]

J. G. Hounie and E. Lanconelli, A sphere theorem for a class of Reinhardt domains with constant Levi curvature, Forum Mathematicum, 20 (2008), 571-586. doi: 10.1515/FORUM.2008.029.  Google Scholar

[9]

H. Ishii and P. L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. Differential Equations, 83 (1990), 26-78. doi: 10.1016/0022-0396(90)90068-Z.  Google Scholar

[10]

E. Lanconelli and A. Montanari, Pseudoconvex fully nonlinear partial differential operators. Strong comparison theorems, J. Differential Equations, 202 (2004), 306-331. doi: 10.1016/j.jde.2004.03.017.  Google Scholar

[11]

V. Martino, A symmetry result on Reinhardt domains, Differential and Integral Equations, 24 (2011), 495-504.  Google Scholar

[12]

V. Martino and A. Montanari, Graphs with prescribed the trace of the Levi form, Ann. Univ. Ferrara Sez. VII Sci. Mat., 52 (2006), 371-382. doi: 10.1007/s11565-006-0027-0.  Google Scholar

[13]

Z. Slodkowski and G. Tomassini, Weak solutions for the Levi equation and envelope of holomorphy, J. Funct. Anal., 101 (1991), 392-407. doi: 10.1016/0022-1236(91)90164-Z.  Google Scholar

[14]

Z. Slodkowski, G. Tomassini, The Levi equation in higher dimensions and relationships to the envelope of holomorphy, Amer. J. Math., 116 (1994), 479-499. doi: 10.2307/2374937.  Google Scholar

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