Constant frequency and the higher regularity of branch sets

2014/10/27 by Brian Krummel, Krummel, Brian
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #math.AP #math.DG #msc:35J47

paper · pdf · doi:10.48550/arxiv.1410.7339

arxiv created 2014/10/27 · arxiv updated 2014/10/28

Abstract

We consider a two-valued function u that is either Dirichlet energy minimizing, C^1,μ harmonic, or in C^1,μ with an area-stationary graph such that Almgren's frequency (restricted to the singular set) is continuous at a singular point Y₀. As a corollary of recent work of Wickramasekera and the author, if the frequency of u at Y₀ equals 1/2+k for some integer k ≥ 0, then the singular set of u is a C^1,τ submanifold and we have estimates on the asymptotic behavior of u at singular points. Using a nontrivial modification of the argument of Wickramasekera and author, we show that the frequency of u at Y₀ cannot equal an integer and therefore must equal 1/2+k for some integer k ≥ 0. We then use the asymptotic behavior of u and partial Legendre-type transformations based on those of Kinderlehrer, Nirenberg, and Spruck to show that the singular set in this case is in fact real analytic.

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