Obstacle Problem

Schedule: To be announced

Summary

This seminar focuses on the modern regularity theory via blow-up and compactness methods and the main goal is to study three state of the art papers. For the integro-differential obstacle problem, Caffarelli, Ros-Oton and Serra extended the regularity theory of free boundaries for the fractional Laplacian to a more general class of integro-differential operators. In the fully nonlinear setting, Savin and Yu developed fine estimates on the singular set of the obstacle problem for fully non-linear operators. Finally, in a very recent paper by Figalli, Ros-Oton, Serra we will see consider nonlocal obstacle problems with subcritical scaling due to a drift term, their result extend to the critical case and include the parabolic equation as well. A common feature of these three results is that they develop the regularity theory of the free boundary without the necessity of a monotonicity formula.

We recommend having a look at the surveys below.

Papers

• Caffarelli, Ros-Oton, Serra
  Obstacle problems for integro-differential operators: regularity of solutions and free boundaries
  Inventiones 2017.
• Savin, Yu
  Regularity of the singular set in the fully nonlinear obstacle problem
  JEMS 2019.
• Figalli, Ros-Oton, Serra
  Regularity theory for nonlocal obstacle problems with critical and subcritical scaling
  2024.

Surveys

• Ros-Oton
  Obstacle problems and free boundaries: an overview.
• Yu
  A brief survey on the obstacle problem.
• Figalli
  Free boundary regularity in obstacle problems.

Prerequisite

• Caffarelli
  The obstacle problem revisited
  Journal of Fourier Analysis and Applications 1998.

Selected references