From Qualitative Smoothness to Uniform Regularity Estimates for Elliptic Equations

The starting point of this talk is a qualitative-to-quantitative question: when does the mere fact that every solution is smooth force a uniform estimate? We discuss this question for pure second-order fully nonlinear elliptic equations and for degenerate equations such as the p-Laplacian. This is joint work with Eduardo Teixeira (Oklahoma State University) and José Miguel Urbano (KAUST).