The Oblique Boundary Value Problem For Monge-Ampère Type Equations

The Monge-Ampère equation was pioneerly studied in the 19th century by two French Mathematicians Gaspard Monge[1]and Andre-Marie Ampère[2],who origi-nally investigated the equations with two variables.In the 20th century,the further research of this equation and its development are closely related to two kinds of geo-metric problems:Minkowski problem[3]and Wely problem[4].Monge-Ampère type equations,originated from the study on the optimal transportation problems,are more widely applied than Monge-Ampère equation.They play a significant role in the study of a lot of geometry problems,such as affine geometry,geometrical optics and con-formal geometry,etc.Monge-Ampère type equations are very important members of fully nonlinear partial differential equations and they attract more and more attentions because of the wide applications in fluid mechanics,statistical physics,digital image processing,statistics,meteorology and so on.Thus deeper and further investigations on Monge-Ampère type equations not only provide various solutions to the problem-s described above,but also offer us the possibilities to complete the theories on fully nonlinear partial differential equations.In recent years,the main tools to study the ex-istence and regularity of solutions for the Monge-Ampère type equations are continuity method and weak solution theory.When we deal with Neumann boundary value problem,there is lack of the infor-mation for the tangential direction,thus,the gradient estimates and second order deriva-tives estimates for the tangential direction on the boundary are difficult.In this thesis,the problems to deal with are to get the corresponding gradient estimates and second order derivatives estimates for the tangential direction via the analysis of boundary and constructing barrier function.By constructing barrier function,using the property of function at the maximum value point and the maximum principle,we obtain the inner gradient estimates,boundary gradient estimates and near the boundary gradient esti-mates of Monge-Ampère type equation det[D~2u-A(x,u)]= B(x,u)with oblique boundary condition in n dimension,and then we get the global estimates of the equa-tions.Furthermore,using eigenvalue relation for two dimensional derivatives matrix in two dimension,we study two dimensional Monge-Ampère type equations with Neu-mann boundary value problem.By constructing barrier function,we reduce the global estimates to those on the boundary,and then we divide the boundary estimates into tangential direction,normal direction,non-tangential and non-normal direction.This dissertation includes four sections.In section 1,we present the context of Monge-Ampère equations and Monge-Ampère type equations.In section 2,we intro-duce some notations,the maximum principle and the continuity method.In section 3,we prove the gradient estimates of Monge-Ampère type equations with Neumann boundary value problem and the gradient estimates of Monge-Ampère type equations with oblique boundary value problem.In section 4,we give the second order derivatives estimates of Monge-Ampère type equations with two dimensions.