Abstract:
We introduce non-local fractional diffusion and evolution processes modeling diverse structures in the formation of free boundary problems and diffusion phenomena arising, for instance, from the evolution of energy-partition methods, involving a regularizing mechanism such as local and fractional free energies and convex-like potentials.

The applications are abundant in natural sciences and engineering structures, stochastic processes in financial modeling and beyond.

This lecture was held by Luis Caffarelli at The University of Oslo, May 24, 2023 and was part of the Abel Prize Lectures in connection with the Abel Prize Week celebrations.

Program for the Abel lectures 2023:
1. Luis A. Caffarelli: "Non-linear and non-local surface structure problems and some of its applications"
2. Sylvia Serfaty: "From diffusions to fluid equations: the question of regularity"
3. Alessio Figalli: "From elastic membranes to ice melting"
4. Luis Silvestre: "Fully nonlinear elliptic equations and applications"