## Evans-Krylov and interior $C^{2,lpha}$ regularity for concave equationss

Talk, [Student Geometric & Analysis Seminar (Spring 2024)](https://math.columbia.edu/~ypharry/seminar/ma), Columbia University

In this talk, we restrict ourselves to the fully nonlinear uniformly elliptic equation of the form $F(D^2u)=0$ 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. References: [Caffarelli-Cabre Fully Nonlinear Elliptic Equations]