The Convexity Of A Fully Nonlinear Elliptic Operator And Related Eigenvalue Problem

The eigenvalue and convexity problems are two important subjects in the study of fully nonlinear elliptic equations.The former is a natural generalization of the ei-genvalue problem for the Laplacian operator,while the latter one connects geometric properties to analysis inequalities.This thesis considers a special nonlinear eigenvalue problem.We first establish the a prior estimate and use the super-sub solution method to get an existence and uniqueness result for the problem.Then we prove a constant rank theorem and use it to get some strict convexity property of the solution when the problem is defined on a strictly convex domain.The thesis is divided into three chapters.Chapter 1:We first give an introduction to the backgrounds of the nonlinear elliptic eigenvalue problem and convexity problem,then we state some preliminary knowledge and our main results;Chapter 2:Establish the a prior estimates for a class of degenerate equation and use the super-sub solution method to get the existence result for our eigenvalue problem and use the contradiction argument to obtain the uniqueness result;Chapter 3:Establish the constant rank theorem for the eigenvalue problem and use the constant rank theorem and deformation technique to get the strict convexity result.