Luis Caffarelli (UTexas, Austin)

The Interaction of Free Boundary with a Diffusion on a Plane

Donatella Danielli (ASU, Phoenix)

A penalized boundary obstacle problem for the bi-Laplacian

In this talk we are concerned with a two-penalty boundary obstacle problem for the bi-Laplace operator in the upper unit ball. This problem arises in connection with unilateral phenomena for flat elastic plates. It can also be seen as an obstacle-type problem for the fractional Laplacian $(−\Delta)^{3/2}$. Our goals are to establish the well-posedness and the optimal regularity of the solution, and to study the structure of the free boundary. The proofs are based on monotonicity formulas of Almgren- and Monneau-type. This is joint work with Alaa Haj Ali.

Alessio Figalli (ETH, Zurich)

The singular set in the Stefan problem

The Stefan problem describes phase transitions such as ice melting to water, and it is among the most classical free boundary problems. It is well known that the free boundary consists of a smooth part (the regular part) and singular points. In this talk, I will describe a recent result with Ros-Oton and Serra, where we analyze the singular set in the Stefan problem and prove a series of fine results on its structure.

Alfio Quarteroni (Politecnico, Milano)

The Beat of Math

Mathematical models based on first principles allow the description of the blood motion in the human circulatory system, as well as the interaction between electrical, mechanical, and fluid-dynamical processes occurring in the heart. This is a classical environment where multi-physics and multi-scale processes must be addressed.
Appropriate systems of nonlinear differential equations (either ordinary and partial) and efficient numerical strategies must be devised to allow for the analysis of both heart function and dysfunction, and the simulation, control and optimization of therapy and surgery.
This presentation will address some of these issues and a few representative applications of clinical interest.
Acknowledgment: The work presented in this talk is part of the project iHEART that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 740132).