Friday, October 1, 2021
Time  Items 

All day 

2:00pm 
10/01/2021  2:00pm 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 (d2)dimensional, and moreover \mathcal{S}(u) is (d2)rectifiable and its Minkowski content is bounded (depending on the frequency of u). We prove the analogue near the boundary for C^1Dini 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 (d2)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 Location: 