Monotonicity Formulas and Fractional Obstacle Problems

Obstacle problems play a ubiquitous role in the applied sciences, with applications ranging from linear elasticity to fluid dynamics, from temperature control to financial mathematics. In this talk we will show how seemingly different phenomena can be expressed in terms of a problem of obstacle type for the parabolic fractional heat operator. We will also discuss some recent results concerning the regularity of the solution and of its free boundary. In particular, we will highlight the pervasive role played by some families of monotonicity formulas.