Regularity Estimation Of Nonlinear Elliptic Equations With Feature Degeneration And Its Application In Geometry

In this dissertation, we mainly discuss second order elliptic partial differen-tial equations with characteristic degenerate boundary. Our main results concern with the regularities of Monge-Ampere equations, the compactness of Alexandrov-Nirenberg surfaces and L∞-estimates of some semi-linear elliptic equations.In the first part, by using Fourier transformation we construct a resolvent op-erator for a special modeling second order elliptic equation. Meanwhile, we estab-lish IP and Holder estimates of the resolvent operator by means of the anisotropic Calderon-Zygmund decomposition and the technique of oscillatory integral.In the second part, we study the regularities of Monge-Ampere equations with characteristic degenerate boundary. We transfer Monge-Ampere equation to some quasi-linear elliptic equation in a neighborhood of the boundary by Ampere trans-formation. By the a priori estimates of the resolvent operator established in Chapter2, in a2-D and strictly convex domain, we can prove that any C3-convex solution of Monge-Ampere equations with boundary characteristic degenerate is C∞-smooth up to the boundary and the estimates of high order derivatives only depend on the C2-norms and the modulus of continuity of second order derivatives.In the third part, we investigate the compactness of Alexandrov-Nirenberg sur-faces. We get the L∞-estimates of the second fundamental form L, M, N by maximal principle. Then, by De Giogi iteration and blow up analysis we achieve the uniform Holder estimates and Lipschitz estimates respectively near the boundary. Prom the a priori estimates established in Chapter2, we finally get the compactness of Alexandrov-Nirenberg surfaces.Finally, we study the a priori bounds of some semi-linear elliptic equations with boundary characteristic degenerate in a bounded domain. By blow up analysis, we only need to investigate the existence of non-negative solutions of some semi-linear elliptic equation in the whole space or in the half space. We obtain Liouville theorem for subcritical case and classify the solutions for critical case in the half space. By Liouville theorem, we get the a priori bounds.