Free discontinuity and free boundary problems with Robin conditions

In this talk I will present my recent works on free discontinuity and free boundary problems involving Robin or other similar boundary conditions. I will start with a scalar case that involves both free boundary and free discontinuity problems with obstacle constraints, whose motivation was originally to establish a quantitative version of the Saint-Venant inequality (dealing with the maximization of torsional rigidity among sets of fixed area), as well as a similar problem involved in thermal insulation. Then I will show some recent results on the regularity of a free boundary problem using an epiperimetric inequality method. Finally, I will discuss current works on vectorial free discontinuity problems in fluids mechanics, namely a theoretical approach to the drag minimization of an object immersed in a creeping flow with Navier conditions.