In this project, we study free boundary problems from a purely theoretical point of view. The focus is on the regularity of the free boundaries arising in the context of variational minimization problems as, for instance, the one-phase, the two-phase and the vectorial Bernoulli problems; the obstacle and the thin-obstacle problems. The aim is to develop new techniques for the analysis of the fine structure of the free boundaries, especially around singularities. Many tools and methods developed in this context can find application to other problems and domains, including shape optimization problems, area-minimizing surfaces, harmonic maps, free discontinuity problems, parabolic and non-local free boundary problems.

Keywords: free boundary regularity, the one-phase Bernoulli problem, Alt-Caffarelli,
the two-phase Bernoulli problem, Alt-Caffarelli-Friedman, the vectorial Bernoulli problem, the obstacle problem, the Signorini problem, epiperimetric inequality,
logarithmic epiperimetric inequality, monotonicity formulas.