The Classical Monge-Ampere Equation

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The classical Monge-Ampere equation $detD2u(x) = f(x)$ is one of the most important partial dierential equations. This equation is closely related to the theory of optimal transportation and dierential geometry. It was studied by many famous mathematicians including Nirenberg, Yau, Caarelli, and etc. A very beautiful existence and regularity theory has been developed. The tools and methods used to study this equation are extremely useful in studying fully nonlinear partial dierential equations. In this project we will start from the basic denition of the generalized solutions. We will learn how to use the Perron's method to prove the existence of generalized solutions. Then, most parts of the project will be devoted to understanding Caarelli's $C1;, C2$ and $W2;p$ estimates of this equation when the right hand side $f$ is bounded away from $0$ and $1$.

Updated:  24 June 2017/Responsible Officer:  Director/Page Contact:  School Manager