Organizers: Raphael Tsiamis, Jingbo Wan, Yipeng Wang
Time: Tuesday 4:30 pm  6:00 pm
Location: Room 528 at Columbia Math Department
This seminar provides a thorough introduction to the regularity theory of fully nonlinear elliptic equations. We initiate with a concise review of the KrylovSafonov Harnack inequality and the EvansKrylov Theorem, which give Hölder continuity and \(C^{2, \alpha}\) regularity to viscosity solutions of uniformly convex elliptic equations. This foundational understanding sets the stage for a deeper examination of Caffarelli's influential work in 1990, specifically his groundbreaking advances in the interior \(C^{1,\alpha}\) and \(W^{2,p}\) regularity theory of the MongeAmpère equation. If time permits, we will discuss further applications in optimal transport and other fully nonlinear equations that appear naturally in differential geometry.
References:
Date 
Speaker 
Title and abstract 

Jan 16th 

Organizational Meeting 
Jan 23rd 
Raphael Tsiamis 
Introduction to fully nonlinear PDEs and the AlexandrovBakelmanPucci (ABP) estimate Show/hide AbstractsWe introduce the background and techniques required to approach the theory of fully nonlinear second order PDEs with a view towards the study of the MongeAmpère equation. In particular, we discuss the AlexandrovBakelmanPucci (ABP) technique, which allows us to obtain analogues of the maximum principle and of a prior estimates for nonlinear PDEs.

Jan 30th 
Raphael Tsiamis 
Introduction to fully nonlinear PDEs and the AlexandrovBakelmanPucci (ABP) estimate: Part II Show/hide AbstractsWe introduce the background and techniques required to approach the theory of fully nonlinear second order PDEs with a view towards the study of the MongeAmpère equation. In particular, we discuss the AlexandrovBakelmanPucci (ABP) technique, which allows us to obtain analogues of the maximum principle and of a prior estimates for nonlinear PDEs.

Feb 6th 
Raphael Tsiamis 
Introduction to fully nonlinear PDEs and the AlexandrovBakelmanPucci (ABP) estimate: Part III Show/hide AbstractsWe introduce the background and techniques required to approach the theory of fully nonlinear second order PDEs with a view towards the study of the MongeAmpère equation. In particular, we discuss the AlexandrovBakelmanPucci (ABP) technique, which allows us to obtain analogues of the maximum principle and of a prior estimates for nonlinear PDEs.

Feb 13th 
Jingbo Wan 
Jenson Approximation and uniqueness of solutions Show/hide AbstractsWithin 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.)

Feb 20th 
No seminar (Oberwolfach) 

Feb 27th 
Jingbo Wan 
EvansKrylov and interior \(C^{2,\alpha}\) regularity for concave equationss Show/hide AbstractsIn 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 CaffarelliCabre as ingredients, demonstrating their application in proving the EvansKrylov 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 CaffarelliCabre) would also be used, leveraging translation invariance of the PDE we are interested in.

Mar 5th 
No Seminar 
Show/hide Abstracts

Mar 12th 
No seminar (Spring break) 

Mar 19th 
Jingbo Wan 
The Dirichlet Problem for Concave Equations Show/hide AbstractsIn 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.

Mar 26th 
Yipeng Wang 
Weak Solutions of the MongeAmpère Equation Show/hide AbstractsWe will provide a general introduction to the theory of Alexandrov solutions for the MongeAmpère equation. The underlying observation is that the MongeAmpère 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.

Apr 2nd 
Chilin Zhang 
The \(C^{1}\) regularity and convexity of the MongeAmpère equations in 2D. Show/hide AbstractsWe will discuss the regularity of a convex solution of the MongeAmpère 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.

Apr 9th 
Yipeng Wang 
Smooth Solutions of the MongeAmpère Equations Show/hide AbstractsWe 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.

Apr 16th 
Yipeng Wang> 
Interior Regularity of Weak Solutions Show/hide AbstractsWe will start discussing Caffarelli's results regarding the interior regularity of Alexandrov solutions to the MongeAmpère 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.

Apr 23rd 
Chilin Zhang> 
Sections of the MongeAmpère Equation Show/hide AbstractsWe study the properties of the sections of the MongeAmpère equation by utilizing the affine invariance.
