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Unique Continuation at the Boundary

Ignacio Tejeda, University of Washington
Thursday, May 18, 2023 - 12:30pm to 1:20pm
PDL C-401

Unique continuation at the boundary is one of a number of problems that seek to understand nodal or singular sets of solutions to elliptic and parabolic equations. The question it poses is the following: given a nonconstant solution that vanishes continuously on an open subset of the boundary of the domain, what can be said about the size of the subset where the normal derivative vanishes as well? In this talk we will discuss the recent progress that has been made in the particular case of the Laplace equation, under various assumptions on the regularity of the domain. We will pay special attention to  the techniques that have been used thus far, most notably the use of Almgren's frequency function, and to the challenges that arise when weaker assumptions on the regularity of the domain are made.