Project description
A variational principle to the regularity of free boundary problems
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.
Objective
The focus of this project is the regularity theory of free boundary problems. This is a fascinating topic, which combines methods from Analysis and Geometry, and has numerous applications to a large variety of problems in Physics, Engineering and Economy, which involve partial differential equations on domains whose boundary is free, that is, it is not a priori known. Typical examples are the Stefan problem, describing the evolution of a block of melting ice, and the American stock options. Since the shape of the boundary is free, it is a deep and usually extremely difficult question to study its fine structure. The regularity theory is precisely the art of deducing the local structure of the free boundary just by looking at a global energy-minimization property of the state function. In this project I aim to develop new techniques to study the regularity of the free boundaries and to give a precise description of the structure of the free boundaries around singular points. I will introduce a new variational method for the analysis of the free boundaries, aiming to solve several major open questions related to the classical one-phase, two-phase and the vectorial Bernoulli problems, the obstacle and thin-obstacle problems, which are the most important models both from a theoretical and applicative point of view. The techniques that I will develop in this project will have an impact on several domains, including the minimal surfaces, harmonic maps, free discontinuity problems, parabolic and non-local free boundary problems.
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Funding Scheme
ERC-STG - Starting GrantHost institution
56126 Pisa
Italy