High-order estimates for fully nonlinear second order equations under weak concavity assumptions

In 1982 L.C. Evans and N.V. Krylov proved interior a priori second derivatives estimates in Hölder spaces for fully nonlinear second  order uniformly elliptic equations under the main assumption that the  operator is either concave or convex in the Hessian variable. Since  then, a remarkable question in the theory is to determine which hypotheses on the operator in between convexity/concavity and no assumptions ensure that solutions to general second  order fully nonlinear equations are classical. In this direction, N. Nadirashvili and  S. Vladut exhibited counterexamples in dimension higher than or equal  to 5 to show that the sole uniform ellipticity is in general not  enough to reach classical regularity. Despite this progress, the  minimal assumptions guaranteeing classical regularity are unknown, and  the above question has remained largely open.
After a review of the regularity theory for fully nonlinear  equations, I will discuss some advances on the Evans-Krylov theory and show how to prove interior  C^{2,alpha} and C^{1,1} regularity for elliptic and parabolic problems  under the assumption that the operator is quasi-concave/convex. I will also consider interior estimates for functionals that are concave/convex or “close to a hyperplane” at  infinity as well as C^{2,alpha} parabolic regularity in dimension 3 or under Cordes assumptions on the ellipticity. The approach is based on  linearization arguments and Bernstein methods.  I will conclude with some consequences about the Calderón-Zygmund regularity of solutions to (fully nonlinear) second order Hamilton-Jacobi-Bellman/Isaacs equations with power-growth gradient terms and discuss the relation with a conjecture posed by P.-L. Lions.