# American Institue of Mathematical Sciences

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

## On the characteristic curvature operator

 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.
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. doi: 10.1512/iumj.1978.27.27058. [2] A. Bogges, "CR Manifolds and the Tangential Cauchy-Riemann Complex,", Studies in Advanced Mathematics, (1991). [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. [4] F. Da Lio and A. Montanari, Existence and uniqueness of Lipschitz continuous graphs with prescribed Levi curvature,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 23 (2006), 1. [5] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", second edition, (1983). [6] H. Hofer and E. Zehnder, "Symplectic Invariants and Hamiltonian Dynamics,", Birkh\, (1994). [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., 400 (2006), 129. [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. doi: 10.1515/FORUM.2008.029. [9] H. Ishii and P. L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations,, J. Differential Equations, 83 (1990), 26. doi: 10.1016/0022-0396(90)90068-Z. [10] E. Lanconelli and A. Montanari, Pseudoconvex fully nonlinear partial differential operators. Strong comparison theorems,, J. Differential Equations, 202 (2004), 306. doi: 10.1016/j.jde.2004.03.017. [11] V. Martino, A symmetry result on Reinhardt domains,, Differential and Integral Equations, 24 (2011), 495. [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. doi: 10.1007/s11565-006-0027-0. [13] Z. Slodkowski and G. Tomassini, Weak solutions for the Levi equation and envelope of holomorphy,, J. Funct. Anal., 101 (1991), 392. doi: 10.1016/0022-1236(91)90164-Z. [14] Z. Slodkowski, G. Tomassini, The Levi equation in higher dimensions and relationships to the envelope of holomorphy,, Amer. J. Math., 116 (1994), 479. doi: 10.2307/2374937.

show all references

##### References:
 [1] E. Bedford and B. Gaveau, Hypersurfaces with bounded Levi form,, Indiana University Journal, 27 (1978), 867. doi: 10.1512/iumj.1978.27.27058. [2] A. Bogges, "CR Manifolds and the Tangential Cauchy-Riemann Complex,", Studies in Advanced Mathematics, (1991). [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. [4] F. Da Lio and A. Montanari, Existence and uniqueness of Lipschitz continuous graphs with prescribed Levi curvature,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 23 (2006), 1. [5] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", second edition, (1983). [6] H. Hofer and E. Zehnder, "Symplectic Invariants and Hamiltonian Dynamics,", Birkh\, (1994). [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., 400 (2006), 129. [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. doi: 10.1515/FORUM.2008.029. [9] H. Ishii and P. L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations,, J. Differential Equations, 83 (1990), 26. doi: 10.1016/0022-0396(90)90068-Z. [10] E. Lanconelli and A. Montanari, Pseudoconvex fully nonlinear partial differential operators. Strong comparison theorems,, J. Differential Equations, 202 (2004), 306. doi: 10.1016/j.jde.2004.03.017. [11] V. Martino, A symmetry result on Reinhardt domains,, Differential and Integral Equations, 24 (2011), 495. [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. doi: 10.1007/s11565-006-0027-0. [13] Z. Slodkowski and G. Tomassini, Weak solutions for the Levi equation and envelope of holomorphy,, J. Funct. Anal., 101 (1991), 392. doi: 10.1016/0022-1236(91)90164-Z. [14] Z. Slodkowski, G. Tomassini, The Levi equation in higher dimensions and relationships to the envelope of holomorphy,, Amer. J. Math., 116 (1994), 479. doi: 10.2307/2374937.
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