##### Department of Mathematics,

University of California San Diego

****************************

### Math 258 - Differential Geometry

## Xavier Fernandez-Real Girona

#### EPFL

## The non-regular part of the free boundary for the fractional obstacle problem

##### Abstract:

The fractional obstacle problem in $\mathbb R^n$ with obstacle $\varphi\in C^\infty(\mathbb{R}^n)$ can be written as \[ \min\{(-\Delta)^s u , u-\varphi\} = 0,\quad\textrm{in }\quad\mathbb{R}^n. \] The set $\{u = \varphi\} \subset \mathbb{R}^n$ is called the contact set, and its boundary is the free boundary, an unknown of the problem. The free boundary for the fractional obstacle problem can be divided between two subsets: regular points (around which the free boundary is smooth, and is $n-1$ dimensional) and degenerate points. The set of degenerate points, even for smooth obstacles, can be very large (for example, with infinite $\mathcal{H}^{n-1}$ measure). In a joint work with X. Ros-Oton we show, however, that generically solutions to the fractional obstacle problem have a lower dimensional degenerate set. That is, for almost every solution (in an appropriate sense), the set of degenerate points is lower dimensional.

Hosts: Lei Ni and Luca Spolaor

### October 28, 2020

### 11:00 AM

### Zoom ID: 960 7952 5041

****************************