We discuss the very basics of fully nonlinear elliptic equations, in close connection with regularity results. The latter include the celebrated Krylov-Safonov and Evans-Krylov theorems and the fundamental developments in Caffarelli’s theory. We also put forward more recent advances, such as the smoothness of flat solutions and the partial regularity result. In face of this panorama, we present some of our recent contributions to the theory. It covers fractional regularity estimates, the use of the Harnack approach and the connection with free boundary problems. We conclude with a discussion of a few open questions and their main challenges.

Given a Gabor orthonormal basis of $L^2(\mathbb{R})$\[\mathcal{G}(g,T,S):=\big\{ g(x-t) e^{2\pi is x}: g\in L^2(\mathbb{R}), \,t\in T,\, s\in S\big\},\]we study periodicity properties of the translation and modulation sets $T$ and $S$. In particular, we show that if the window function $g$ is compactly supported, then $T$ and $S$ must be periodic sets, i.e., of the form\[T = a\mathbb{Z}+ \{t_1,\ldots,t_n\}, \qquad S = b\mathbb{Z} + \{s_1,\ldots,s_m\}.\]To achieve this, we first obtain a result of independent interest: if the system $\mathcal{G}(g,T,S)$ is an orthonormal basis of $L^2(\mathbb{R})$, then both $|g|^2$ and $|\widehat{g}|^2$ tile $\mathbb{R}$ by translations (when translated along the sets $T$ and $S$, respectively), and moreover,\[\sum_{t\in T} |g(x-t)|^2=D(T), \qquad \sum_{s\in S} |\widehat{g}(x-s)|^2=D(S), \qquad \text{a.e. }x\in \mathbb{R},\]where $D(\Lambda)$ denotes the uniform density of a set $\Lambda\subset \mathbb{R}$.

Partial results towards the Liu-Wang conjecture are also obtained.

This talked in based on a joint work with Nir Lev (Bar-Ilan University, Israel)