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Learning Seminar on PDE and Applications

Quantitative boundary unique continuation II

Zihui Zhao, University of Chicago

Location:  Hill Center Room 705
Date & time: Wednesday, 21 September 2022 at 9:30AM - 11:30AM

Abstract: In the first part, I have talked about recent progress in quantitative boundary unique continuation, and in particular, the work of X. Tolsa and my joint work with C. Kenig estimating the size of the singular set for harmonic functions near the boundary. In the second part, I will discuss some key ideas in their proofs. Both proofs rely on the monotonicity of Almgren's frequency function (which is a quantifier of the growth rate of a harmonic function), to different extents. However, the former proof is a harmonic analysis approach inspired by the work of Logunov on Yau's conjecture for Laplace eigenfunctions, and the latter uses tools from geometric measure theory inspired by the work of Naber-Valtorta on geometric variational problems.

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