Student Geometric & Analysis Seminar (Spring 2024): Regularity Theory of the Monge-Ampère Equation

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 non-linear elliptic equations. We initiate with a concise review of the Krylov-Safonov Harnack inequality and the Evans-Krylov 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 Monge-Ampère equation. If time permits, we will discuss further applications in optimal transport and other fully non-linear equations that appear naturally in differential geometry.  

References: 


Title and Abstract (Spring 2024)

Date

Speaker

Title and abstract

Jan 16th


Organizational Meeting

Jan 23rd

Raphael Tsiamis

Introduction to fully nonlinear PDEs and the Alexandrov-Bakelman-Pucci (ABP) estimate 

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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.

 

 


Jan 30th

Raphael Tsiamis

Introduction to fully nonlinear PDEs and the Alexandrov-Bakelman-Pucci (ABP) estimate: Part II 

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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.

 

 


Feb 6th

Raphael Tsiamis

Introduction to fully nonlinear PDEs and the Alexandrov-Bakelman-Pucci (ABP) estimate: Part III 

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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.

 

 


Feb 13th

Jingbo Wan

Jenson Approximation and uniqueness of solutions 

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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.)

 

 


Feb 20th

No seminar (Oberwolfach)

Feb 27th

Jingbo Wan

Evans-Krylov and interior \(C^{2,\alpha}\) regularity for concave equationss 

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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.

 

 


Mar 5th


 

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Mar 12th

No seminar (Spring break)

Mar 19th


 

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Mar 19th


 

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Apr 2nd


 

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Apr 9th


 

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Apr 16th


 

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