A glimpse at partial differential equations: on the existence and regularity of solutions

The analysis of partial differential equations is a prominent area of  research in both pure and applied mathematics. It has attracted the  attention of several researchers, working on various topics, with important spillovers across mathematical disciplines and  beyond. In this talk, we discuss two fundamental questions arising in this field, namely: the existence of solutions and their regularity in certain functional spaces. Through simple examples, we aim at entering  the main challenges, as well as a number of techniques to by-pass them. Our model-problems include mean-field games systems, fully nonlinear elliptic equations and a preliminary analysis of free boundary problems.