Unique continuation at the boundary for harmonic functions in C¹ domains and Lipschitz domains with small constant

2020/04/22 by Tolsa, Xavier · 4 citations
#31B05 31B20 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2004.10721

Abstract

Let Ω⊂\mathbb Rⁿ be a C¹ domain, or more generally, a Lipschitz domain with small local Lipschitz constant. In this paper it is shown that if u is a function harmonic in Ω and continuous in Ω which vanishes in a relatively open subset Σ⊂∂Ω and moreover the normal derivative ∂_νu vanishes in a subset of Σ with positive surface measure, then u is identically 0.

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• Quantitative unique continuation for Neumann problem in planar C^1,α domains
• Elliptic unique continuation below the Lipschitz threshold
• Edge asymptotics for weighted Robin equations on cylinders with applications to spectral fractional Laplacians
• Smooth Failure of Boundary Unique Continuation for Harmonic Functions

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