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Regularity of free interfaces in parabolic transmission problems arising from the jump of conductivity

Regularity of free interfaces in parabolic transmission problems arising from the jump of conductivity

Free boundary problems describe several natural phenomena that arise in diverse fields such as physics, biology, and engineering. The analysis of free boundaries has been a central topic in PDEs over the past fifty years, and it is still a very active research topic with numerous open problems. In this talk, we will introduce a two-phase parabolic free boundary problem motivated by the jump of conductivity in composite materials that undergo a phase transition. Our main goal will be to establish strong regularity properties of the free boundary, following the classical strategy:

I. Flat free boundaries are $C^{1,\alpha}$;

II. $C^{1, \alpha}$ implies smooth.

In the first part, we will focus on the main ideas and techniques developed by Daniela De Silva for elliptic Bernoulli problems, and we will discuss how to adapt these methods to our parabolic context. In the second part, we will introduce the Hodograph transform as a tool to achieve higher regularity of the free boundary. This is a joint work with Dennis Kriventsov.

## Department of Mathematics and Statistics