# List All Events

Nonlinear Analysis

## Minimizers for the thin one-phase free boundary problem

#### Yannick Sire, John Hopkins University

Location:  Room 705
Date & time: Wednesday, 22 January 2020 at 1:40PM - 3:00PM

Abstract: We consider the thin one-phase free boundary problem,
associated to minimizing a weighted Dirichlet energy of the function in
the half-space plus the area of the positivity set of that function
restricted to the boundary. I will provide a rather complete picture of
the (partial ) regularity of the free boundary, providing content and
structure estimates on the singular set of the free boundary when it
exists. All of these results hold for the full range of the relevant
weight related to an anomalous diffusion on the boundary. The approach
does not follow the standard one introduced in the seminal work of Alt
and Caffarelli. Instead, the nonlocal nature of the distributional
measure associated to a minimizer necessitates arguments which are less
reliant on the underlying PDE. This opens several directions of research
that I will try to describe.