Higher regularity of solutions to fully nonlinear elliptic equations

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