Keywords:
Fully nonlinear degenerate elliptic PDE's
Subelliptic PDE's
SubRiemannian structures
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)