Two Classes Of Fully Nonlinear Elliptic Equations On Complex Manifolds

Constructing some special geometric structures,such as Einstein metric,is an impor-tant problem in differential geometry.Those problems are reduced to certain analysis prob-lems.Fully nonlinear elliptic equation is an important method to construct special geometric structures.In this paper,using the Bernstein method,we study two classes of(nondegenerate or degenerate)fully nonlinear elliptic equations with first order derivatives terms on complex manifolds and give a priori C2-estimates of admissible solutions of equations.Moreover,we solve Dirichlet problems in special cases.There are two parts in this paper as follows.In the first part,we consider a class of fully nonlinear elliptic equations containing gradient term(?)u on compact Hermitian manifolds.By means of Bernstein method,barrier functions construction and maximum principle,we obtain a priori interior C2-estimates,global C2-estimates and boundary estimates for second derivatives of admissible solutions.Moreover,these estimates can be applied to degenerate equations.Here ? + s>0(s = ± 1)in equations plays a crucial role in the estimates.Our main results are as follows:1)Under the assumption of the equations satisfying some proper growth rate conditions,we derive interior gradient estimates and global gradient estimates of admissible solutions.2)We derive interior second estimates and global second estimates of admissible solutions.In our proof of second order estimates,we use an argument used by G.Szekelyhidi[1]to control nontrivial torsion terms.3)Using the distance function to the boundary,we construct desired barrier functions to derive boundary estimates for second derivatives.4)Based on a priori C2-estimates as above,we give real Hessian estimates of admissible solution via Bernstein method.5)We solve Dirichlet problems of some special equations by continuity method and degree argument under the assumption of existence of admissible subsolutions.In the second part,we consider a class of fully nonlinear elliptic equations on Kahler cones,we obtain a priori C2-estimates for admissible solutions to the Dirichlet problems under the assumptions that the existence of admissible subsolution.Moreover,the estimates can be applied to degenerate fully nonlinear elliptic equations.The C2-estimates heavily depend on the admissible subsolutions.The reason is that the subsolutions play important roles in constructing test functions and barrier functions.Our main results are as follows:1)Assume that Dirichlet problems satisfy proper conditions,we prove that all admissible solutions are basic.The basic property of admissible solution plays an important role in overcoming the difficulties arising from(?)u/(?)r contained in equations.Constructing a new test function,we establish global second order estimates of admissible solutions.2)Using admissible subsolutions and the distance functions to the boundary,we construct the desired barrier functions to derive boundary estimates for second order derivatives.In particular,these estimates can be applied to degenerate elliptic equations.3)We prove gradient estimates by means of Blow up argument.4)If there exist admissible subsolutions of Dirichlet problems,by means of degree argument and continuity method,we prove the existence of admissible solutions for the Dirichlet prob-lems of non-degenerate or degenerate elliptic equations.Moreover,we study the regularity of the admissible solutions.In particular,we discover the sufficient and necessary condition of the existence of admissible solution of the Dirichlet problem of non-degenerate equation.