Degenerate Elliptic Monge-amp¨¨re Equation Existence Of Solutions And Regular

This thesis mainly focuses on elliptic Monge-Amp(?)re equations degenerating on the boundary in strictly convex domains of Euclidean Spaces.Existences and regularities of solutions to Dirichlet problems in the domain of different shapes in two or higher dimensions are discussed.The present thesis also studies the existences and regularities of some eigenvalue problems and some general problems of degenerate Monge-Amp(?)re equations.In the first part of the thesis,the author uses some ODE and analysis techniques to prove that in the ball-shaped domains,a large kind of eigenvalue problems of Monge-Amp(?)re equations possesses symmetric nontrivial C²(?) solutions. Furthermore,the solutions are proved to be smooth up to the boundaries.In the second part,the author applies the "Legendre transformation" and "Higher dimensional partial Legendre transformation" to change Monge-Amp(?)re equations into quasilinear equations.Then by the decomposition of solution,and their a priori estimates of resolvent operators,the author proves that the C² solutions of degenerate elliptic Monge-Amp(?)re equations with finite order degeneracies on the boundary are essentially smooth up to the boundary in two dimensions,and in higher dimensions,any C² solutions of Monge-Amp(?)re equations with order-one boundary degeneracies are C^{∞}up to the boundaries,and any C^2,α solutions with order-two boundary degeneracies are C^{∞}too.Finally,the paper studies perturbations on Monge-Amp(?)re equations and proves a subtle Sobolev type a priori estimate of the modified linearized equation. Applying the Nash-Moser iteration technique,the author proved the solvabilities of Dirichlet problems for degenerate elliptic Monge-Amp(?)re equations in two dimensions after small perturbations.