Higher regularity of solutions to fully nonlinear elliptic equations

We establish higher regularity properties of solutions to fully nonlinear elliptic equations at interior critical points. The key novelty of our estimates lies in the fact that they yield smoothness properties that go beyond the inherent regularity limitations dictated by the heterogeneity of the problem. We explore various scenarios, revealing a plethora of improved regularity estimates. Notably, depending on the model's parameters, we establish estimates that transcend the natural regularity regime of the model, from $C^{0,\alpha_0}$ to $C^{1,\alpha_1}$ and further to $C^{2,\alpha_2}$, with the potential for even higher estimates.