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Boundary unique continuation of Dini domains

Zihui Zhao (University of Chicago)
Tuesday, December 1, 2020 - 1:30pm to 3:30pm
Zoom (link will be distributed via email; if you'd like to attend please email Silvia at ghinassi@uw.edu)

Let \$u\$ be a harmonic function in \$\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 at the boundary for 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 monotonicity formula for boundary points of a Dini domain. This is joint work with Carlos Kenig.

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