We introduce the background and techniques required to approach the theory of fully nonlinear second order PDEs with a view towards the study of the Monge-Ampère equation. In particular, we discuss the Alexandrov-Bakelman-Pucci (ABP) technique, which allows us to obtain analogues of the maximum principle and of a prior estimates for nonlinear PDEs.

We introduce the background and techniques required to approach the theory of fully nonlinear second order PDEs with a view towards the study of the Monge-Ampère equation. In particular, we discuss the Alexandrov-Bakelman-Pucci (ABP) technique, which allows us to obtain analogues of the maximum principle and of a prior estimates for nonlinear PDEs.

We introduce the background and techniques required to approach the theory of fully nonlinear second order PDEs with a view towards the study of the Monge-Ampère equation. In particular, we discuss the Alexandrov-Bakelman-Pucci (ABP) technique, which allows us to obtain analogues of the maximum principle and of a prior estimates for nonlinear PDEs.

Within this talk, we restrict ourselves to the fully nonlinear uniformly elliptic equation of the form \(F(D^2 u)=0\), which is invariant under translation. We will introduce a useful approximation of the viscosity solution, called Jenson Approximation, which effectively captures the translation invariance of our PDE. Using Jenson Approximation, we establish a variation of Jenson’s uniqueness theorem for Dirichlet problem, which concerns the difference of two viscosity solutions. And again, due to the translation invariance of the PDE, we can apply the same idea to study the difference quotient of a viscosity solution, leading to interior \(C^{1,\alpha}\) estimate. If time permits, we will also cover the application to concave \(F\), in which we basically apply the same idea to study second order difference quotient (hence second derivatives) of a solution. (Notice: the additional concavity condition is crucial for discussing second order difference quotient.)

In this talk, we restrict ourselves to the fully nonlinear uniformly elliptic equation of the form \(F(D^2 u)=0\), with the additional condition that \(F\) is concave as a function on symmetric matrices. In particular, we will make use of weak Harnack inequality and local maximum principle in chapter 4 of Caffarelli-Cabre as ingredients, demonstrating their application in proving the Evans-Krylov theorem and interior \(C^{2,\alpha}\) regularity for concave equations. During the proof, some results implied by Jensen approximation we discussed last time (Chapter 5 of Caffarelli-Cabre) would also be used, leveraging translation invariance of the PDE we are interested in.

In this talk, we continue our discussion of the concave equation \(F(D^2u)=0\), focusing on its Dirichlet Problem. We first note that the standard Schauder estimate enables us to demonstrate that interior/global \(C^{2,\alpha}\) solutions are indeed interior/global smooth. (No concavity is necessary in this "bootstrap" process.) Next, we establish the \(C^{2,\alpha}\) estimate up to the boundary, progressing step by step (\(C^0\), \(C^1\), \(C^2\), \(C^{2,\alpha}\)). The main technique at each step is selecting appropriate test functions and applying the Maximum Principle. Finally, we use everything we have discussed (uniqueness and regularity) along with the method of continuity to solve the Dirichlet problem.

We will provide a general introduction to the theory of Alexandrov solutions for the Monge-Ampère equation. The underlying observation is that the Monge-Ampère operator possesses a hidden divergence structure, which enables the formulation of a notion of weak solutions using integral formulas. Following this, we will discuss the Alexandrov maximum principle and employ it to demonstrate the existence and uniqueness of solutions to the Dirichlet problem within a strictly convex domain. 

We will discuss the regularity of a convex solution of the Monge-Ampère equation with locally bounded RHS in 2D. It will be either locally \(C^1\) or affine on a line segment. We will then discuss the strict convexity of a solution in 2D with RHS locally bounded from below. 

We will first discuss some examples about Alexandrov solutions that are not classical. These examples were first constructed by Pogorelov in dimension at least 3, and later generalized by Caffarelli. We will then discuss how to obtain a priori \(C^2\) estimate if the boundary data is nice, using a deep estimate due to Pogorelov. 

We will start discussing Caffarelli's results regarding the interior regularity of Alexandrov solutions to the Monge-Ampère equation. Unlike the smooth case, where regularity is usually obtained by differentiating the equations, the analysis of weak solutions are based on the geometry of "sections". We will prove some general properties of sections for strictly convex solutions and then discuss Caffarelli's interior \(C^{1,\alpha}) regularity.  

We study the properties of the sections of the Monge-Ampère equation by utilizing the affine invariance.