### Variational approach to the regularity of the free boundaries

Project Type: ERC-STG - Starting Grant

Funded by: European Research Council

Period: Jun 01, 2020 – May 31, 2025

Budget: € 1330325

Website: http://www.velichkov.it/vareg.html

Principal Investigator: Bozhidar Velichkov (Università di Pisa)

##### Participants

Giulia Bevilacqua (Università di Pisa), Roberto Ognibene (Università di Pisa), Carlo Gasparetto (Università di Pisa), Luca Benatti (Università di Pisa), Giorgio Tortone (Università di Pisa), Lorenzo Ferreri (SNS, Pisa)

##### Description

Free boundary problems, which involve a partial differential equation on a domain whose boundary is free, naturally arise in many different models in physics, engineering and economy. The EU-funded VAREG project plans to study free boundary problems from a purely theoretical point of view. The focus will be on the regularity of the free boundaries arising in the context of variational minimisation problems such as the one-phase, the two-phase and the vectorial Bernoulli problems as well as the obstacle and thin-obstacle problems. The project will develop new techniques for analysing the fine structure of the free boundaries, especially around singularities. Many tools and methods developed can find application in other problems and domains, including area-minimising surfaces, harmonic maps, free-discontinuity problems and parabolic and non-local free boundary problems.