ERC Starting Grant Project (ERC-2019-STG): *VAREG - Variational approach to the regularity of free boundaries*

**Bozhidar Velichkov**, **Full Professor** at the **Department of Mathematics** as of 1 June 2020, is the Principal Investigator of the **Variational approach to the regularity of free boundaries** (VAREG) project, awarded a prestigious **ERC Starting Grant **by the European Research Council worth €1,330,325.

Bozhidar Velichkov has a glittering academic track record: born in **1985**, after graduating from the University of Pisa and the Scuola Normale Superiore he worked as Assistant Professor at the Université de Grenoble Alpes. In 2019 he became associate professor at the Federico II University of Naples, and in the same year he was awarded the ERC Grant, with which he arrived at the University of Pisa in 2020, thanks also to the **University direct call incentive measure** for ERC grant winners.

The Professor explains his project dedicated to the variational** approach to the regularity of free boundaries as follows:** “A partial differential equation (PDE) is an equation that involves the partial derivatives (derivatives with respect to the different variables) of an unknown function. PDEs are used to describe important physical phenomena such as wave and heat propagation. They are present in various fields such as quantum mechanics, relativity, electrodynamics, aerodynamics and fluid mechanics, and also form the basis of different mathematical models in biology, economics, and medicine. A PDE problem has three main “ingredients”: 1. the differential equation within a given set (the domain of the equation); 2. the form of the domain; 3. the boundary conditions (of the domain). For example, using the heat equation, one can describe the temperature inside a material starting only from the data on its outer surface (the boundary conditions); in particular, to determine the temperature inside the material it is not enough to know that it is described by the heat equation, but it is also necessary to know the shape of the material (a very thin object heats up faster than a bulky object) and the conditions at the boundary (for example, room temperature).

Free boundary problems are a particular class of PDE in which **the boundary conditions determine the shape of the object itself**. A typical example is a melting block of ice. At any moment, knowing the shape of the ice and using the heat equation (the PDE), we can accurately determine the evolution of the temperature in its interior. On the other hand, when the temperature reaches zero degrees Celsius, the ice melts; so it is the temperature itself (the solution of the PDE) that determines the shape of the domain (the block of ice).

The ERC VAREG project is dedicated to the development of new theoretical methods for the study of free boundary problems, in particular, to **mathematical techniques that make it possible to determine the geometric structure of the boundary**. The project was born from the discovery of new relationships between the quantity of energy located in a specific area of the surface, and the geometry of the border itself. In particular, these new relationships allow us to study the formation of singularities on the border and their geometric structure."

More details are available on **the VAREG project website**.

**Prof. Velichkov**'s web page can be found at **this link**.