A free boundary problem is a boundary value problem
that involves partial differential equation on a domain whose boundary is *free*,
that is, it is not a priori known and depends on the solution of the PDE itself.
These problems naturally arise in many different models in Physics, Engineering and Economy.
A typical example is a block of melting ice; in this case, the free boundary is the surface
of the ice, the PDE is the heat equation and its solution (the state function) is the temperature distribution.

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.

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.

• Lecture notes on the regularity of the one-phase free boundaries: pdf .

the 2D case, applications; singularities.

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Title: * Variational approach to the regularity of the free boundaries *

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Primary coordinator: * Bozhidar Velichkov *

Host institution: * Università degli Studi di Napoli Federico II *

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Acknowledge as: *"This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 853404)"*

Laboratoire Jean Kuntzmann (Université Grenoble Alpes), where the PI was Maître de Conférences from 2014 to 2019.

It is a pleasure to acknowledge LJK, UGA and the projects ANR CoMeDiC and GeoSpec, for supporting the PI's research in this period,

and the team of Fostering Science for the support during the preparation of the proposal.