On the Regularity and Rigidity Theorems and Problems for the Solutions of Some Class of the Degenerate Elliptic Partial Differential Equations

This is a short survey paper based on the topics of the author’s research along with his collaborators. Main topics presented in the paper are on the regularity and rigidity theorems for the solutions of the elliptic differential equations. In particular, the author poses several open problems in these topics for further study. The paper contains six sections. Regularity theorems for the elliptic equations of the nondivergence form with rough coefficients are presented in Section 1. In Section 2, we introduce and summarize some recent developments on the rigidity problems and theorems for the solution of some linear degenerate elliptic equations. A typical example is the rigidity problem for the solution of the Dirichlet problem of the Laplace-Beltrami operator on the unit ball in C with Bergman metric. In Section 3, the rigidity theorems and problems for harmonic maps between two complete non-compact Kähler manifolds are discussed. In Section 4, we summarize the approximation formula for the potential function of the Kähler-Einstein metric. In Section 5, we summarize some rigidity theorems for the degenerate Monge-Ampère equations as well as some characterization theorems for some strictly pseudoconvex pseudo-Hermitian manifolds. Finally, in Section 6, we summarize some recent results on the bottom of the spectrum of the Laplace-Beltrami operators on Kähler manifolds. 1. Boundary value problem for uniform elliptic PDEs Let Ω be a bounded domain in IR, and let A(x) = [aij(x)] be an n × n symmetric matrix-valued function on Ω. Let