Boundary unique continuation of Dini domains

Seminar: 
Geometric Analysis and Application
Event time: 
Friday, October 1, 2021 - 2:00pm
Speaker: 
Zihui Zhao
Speaker affiliation: 
University of Chicago
Event description: 

Abstract:

Let u be a harmonic function in a domain \Omega \subset \mathbb{R}^d. It is known that in the interior, the singular set \mathcal{S}(u) = \{u=|\nabla u|=0 \} is (d-2)-dimensional, and moreover \mathcal{S}(u) is (d-2)-rectifiable and its Minkowski content is bounded (depending on the frequency of u). We prove the analogue near the boundary for C^1-Dini domains: If the harmonic function u vanishes on an open subset E of the boundary, then near E the singular set \mathcal{S}(u) \cap \overline{\Omega} is (d-2)-rectifiable and has bounded Minkowski content. Dini domain is the optimal domain for which \nabla u is continuous towards the boundary, and in particular every C^{1,\alpha} domain is Dini. The main difficulty is the lack of the monotonicity formula for the frequency function near the boundary of a Dini domain. This is joint work with Carlos Kenig

Special note: 
Email organizers jiewon.park@yale.edu or lu.wang@yale.edu for Zoom info