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.
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.
References:
• Lecture notes on the regularity of the one-phase free boundaries: pdf .
• The vectorial Bernoulli problem: MTV, CSY, KL, KL2, MTV2.
