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Annamaria Montanari


Department of Mathematics

Academic discipline: MAT/05 Mathematical Analysis


With Levi-Monge-Ampère equations (LMA equations) we denote a class  of fully nonlinear PDE's which can be viewed as the pseudoconvex  counterpart of the usual real curvatures equations (mean curvature equation, Gauss  curvature equations, etc..) and of the Monge-Ampère equations [ML].  They are NONELLIPTIC at any point and they are subelliptic when computed on  functions which are pseudoconex in a suitable generalized sense. From  this crucial property several results have been already obtained. Our aim is to first prove optimal regularity of viscosity  solutions for the stationary problem, then to study the flow by Levi curvature to get new information about symmetry problems.

Let M be a real hypersurface in C^{n+1}, boundary of a domain D. 
The Levi form at a point p of M is a Hermitian form on the tangent 
space. Since this space has
complex n, the Levi form has n real eigenvalues. Given a 
generalized symmetric function ( in the sense of 
Caffarelli-Nirenberg-Spruck [CaNiSp]) it is
natural to call S- Levi curvature of M at p, the function S computed 
on the eigenvalues of the Levi form. If M is the graph of a function u, and one requires that 
M has a given
S-curvature, one obtains a second order PDO which is fully 
nonlinear and NOT elliptic at any point. We call this 
Levi-Monge-Ampère operator (LMA
operator, in short). The LMA class was implicitly introduced by 
Bedford and Gaveau [BeGa] and by Tomassini [To], in studying pseudoconvex domains and 
envelopes of holomrphy
in C^{n+1}. The LMA class have been precisely formalized by Montanari 
and Lanconelli [MoLa]. In [MoLa] it is proved that the natural 
linearizations of the LMA
operators are NOT elliptic at any point. However, they become 
subelliptic when computed on the S-pseudoconcex functions. Indeed, they 
can be written as "sums of
squares" of NON linear vector fields ( i.e. nonlinear first order 
PDE).On  S-pseudoconvex functions vector  fields  directions
are the only ellipticity
directions for the LMA operator. However, the missing ellipticity direction can 
be recovered by commutation.
This is the crucial property on which are based most of the results in 
literature on the LMA operators. We now summarize the state of the arts on this topic. The following results have already been proved.
(a). The Dirichlet problem for the LMA operators has been studied by Da Lio and Montanari[DaMo]. They prove existence and 
uniqueness of a Lipschitz
continuous viscosity solution under quite optimal assumptions. The 
problem of the optimal regularity of these solutions is quite 
completely open. Only in the lowest
dimensional case a result by Citti-Lanconelli-Montanari [CiLaMo] gives 
an almost complete solution of the problem. Indeed, in [CLM] it is proved 
that all the locally
Lipschitz viscosity solutions to the Levi equation are of class 
C^{infty}if the given curvature is smooth and strictly positive. The 
problem, in higher dimension,
is very hard, and cannot be faced by using the usual known 
techniques for the fully nonlinear elliptic equations. (b) Strong Comparison Principle, in every 
dimension [MoLa]. (c) Spherical symmetry of the bounded Reinhardt 
domains of C^2 with
constant Levi curvature [HoLa]. In 
[MaMo2] some integral formulas, and isoperimetric inequalities for 
compact domains,
involving the Levi curvatures are proved. The same paper contains some 
spherical symmetry result for starlike domains with constant Levi 
curvatures. In [MaMo3] by using differential geometry tecniques, we prove that if the unit characteristic direction  is a geodesic, then it is an eigenvector of the second fundamental form and the relative eigenvalue is constant. As an application we get a symmetry result, of Alexandrov type, for compact hypersurfaces with positive constant Levi mean curvature. In [MoMo] it is proved that the Levi-umbelical surfaces (i.e. with all the eigenvalues of the Levi form equal and constant) are spheres, if compact, cylinders if not bounded.  (d) Gradient interior estimates for the Levi-mean curvature equation [MaMo1]. The optimal regularity of 
this solutions could be
studied with the techniques introduced in [CiLaMo] for the Levi 
equation in the lowest dimension.
We therefore plan to first study regularity 
properties of the solutions to fully nonlinear PDE's modeled on Levi equations We then plan to study the motion by Levi curvature and  to get from this study new information on symmetry properties.   We plan to study some notions of curvatures associated with pseudoconvexity and the Levi form the way the classical Gauss and Mean curvatures are related to the convexity and to the Hessian matrix.  In particular, given a prescribed non negative function k, the Levi Monge Ampère equation for the graph of a real value function u is det L = k(x,u)(1+|Du|^2)^ {(n+1)/2}, where L is the Levi form of the graph u and Du is the Euclidean gradient of u. More generally, we shall consider elementary symmetric functions in the eigenvalues of the Levi form L. These curvature equations contain information about the geometric feature of a closed hypersurface and the curvature operator lead to a new class of second order fully nonlinear equations whose characteristic form, when computed on generalized pseudoconvex functions, is nonnegative definite with  kernel of dimension one. Thus, the equations are not elliptic at any point. However, they have the following redeeming feature: the missing ellipticity direction can be recovered by suitable commutation relations. We shall use this property to study existence, uniqueness and regularity of viscosity solutions of the Dirichlet problem for graphs with prescribed Levi curvature. To attack the problem of local regularity of Lipschitz continuous viscosity solutions in THE FIRST YEAR  we plan to study the regularity theory for three related problems:  [1.] The Levi Monge Ampère equation in cylindrical   coordinates. In this case we can prove that the prescribed Levi curvature equation   is a quasilinear degenerate elliptic equation in two variables,   whose prototype is the Grushin operator.  [2.] The subelliptic Monge Ampère equation, with respect to   left invariant vector fields  [3.] Elementary symmetric function in the eigenvalues of real matrices of the type   A(x,Du)D^2uA^T(x,Du), where D^2u is the real Hessian, Du is the Euclidean gradient of u,  and whose classical solutions are good candidates to approximate viscosity solutions of the previous PDE's. The problem of existence and uniqueness of  viscosity for the Dirichlet problem will be jointly studied with Francesca Da Lio, University of Padova. To attack the problem of regularity for these equations we plan to use the regularity theory developed in [CaCa] and in [CaNiSp]. In THE SECOND YEAR we plan to study the related diffusion problems. First of all we plan to study the motion by the trace of the Levi form. A beautiful paper by Huisken and Klingerberg [HK] asserts that if the initial hypersurface is strictly pseudoconvex and C^{2,\alpha} then for small time the evolution by the trace of the Levi form is C^{2,\alpha}. Moreover, by [Mo] the evolution is smooth for small time. We plan to study what happen to Levi flat sides of the initial hypersurface by the Levi curvature flow. We also plan to study  existence, uniqueness and regularity for the motion by elementary symmetric functions of the Levi Form and to apply these results to get isoperimetric estimates for the initial problems and characterization properties  for closed hypersurfaces with some constant Levi curvatures. References  [CaCa] Caffarelli, Luis A.; X Cabré, Xavier Fully nonlinear elliptic 
equations. AMS , Providence, RI, 1995.
[CaNiSp] L. Caffarelli, L.Nirenberg,J.Spruck, Comm.Pure Appl. Math. (1985)
 [CiLaMo] G.Citti,E.Lanconelli,A.Montatori, Smoothness of Lipschtiz 
continuous graphs with nonvanisching Levi curvature, Acta Math., 188, 
(2002) 87-128
[DaMo] F. Da Lio, A. Montanari: Existence and Uniqueness of Lipschitz 
Continuous Graphs, Ann. Inst. H. Poincaré Anal. Non Linèaire 23 no. 1 
(2006) , 1-28.

[HK] G. Huisken; W. Klingenberg. Flow of real hypersurfaces by the trace of the Levi form. Math. Res. Lett. 6 (1999), no. 5-6, 645--661.  [HoLa] J.Hounie.E. Lanconelli, An Alexander type Theorem for Reinhardt 
domains of C^2, Contemporary Math, 400 (2006), 129-14

[MaMo1] V. Martino, A. Montanari: Local Lipschitz continuity of graphs 
with prescribed Levi urature, NoDEA, Nonlinear Differential Equations and Applications, 14, 2007,377-390 [MaMo2]. V. Martino, A. Montanari: Integral formulas for a class of 
curvature PDE's and applications., Forum Mathematicum, Issue 22:2 (2010), 255—267 [MaMo3] V. Martino, A. Montanari, On the characteristic direction of real hypersurfaces in CN+1 and a symmetry result, to appear in Advances in Geometry [Mo] A. Montanari, Real hypersurfaces evolving by Levi curvature: smooth regularity of solutions to the parabolic Levi equation,   Comm. Partial Differential Equations 26 (2001), no. 9-10, 1633--1664
[MoLa] A. Montanari, E. Lanconelli, Pseudoconvex Fully Nonlinear 
Partiall Differential Operators. J.Diff. Eq. 202(2004) 306-333.
 [MoMo] R. Monti D.Morbidelli., J. Reine Angew. Math. 603, (2007), 113-131.

 [To] G. Tomassini, Ann. Mat. Pura Appl.(1988)

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