The Putnam seminar meets every Wednesday from 4 to 5:30 in LOM 214.  As always, everyone is warmly welcomed to come to hang out, learn more cool math, and meet folks.  The seminar is casual, and folks can come and go as they like.  See Pat Devlin’s webpage (and/or contact him) for more information.  Folks can sign up for the mailing list here: https://forms.gle/nYPx72KVJxJcgLha8

This is the fourth lecture in the series. Details can be found here: Algebra and Geometry lecture series (yale.edu)

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