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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.

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