The Monge-Ampère equation det D2u = 1 is a fully nonlinear elliptic PDE that arises in several physical and geometric contexts. In the first lecture we will discuss weak solutions and the solvability of the Dirichlet problem. In the second lecture we will discuss the interior regularity of weak solutions, which is closely related to the classical solvability of the Dirichlet problem.
The aim of the lectures is to introduce a recent line of research that is still largely unexplored. We will describe the ambient of metric graphs, functions and functional spaces on graphs and the Schrödinger equation on these ramified structures. The main problem that we will discuss is the search for solutions of minimal energy (ground states) with fixed mass. Through a series of examples we will identify assumptions of a topological nature that guarantee or prevent the existence of grond states. The techniques are those of the Calculus of Variations: lower semicontinuity, coercivity, rearrangements, Gagliardo-Nirenberg inequalities, level estimates an so on.