2021/11/24 by Krummel, Brian, Wickramasekera, Neshan · 1 citation
#35J47 #49Q05 #53A10 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2111.12246
We study (higher order) asymptotic behaviour near branch points of stationary n-dimensional two-valued C1, μ graphs in an open subset of \mathbb Rn+m. Specifically, if M is the graph of a two-valued C1, μ function u on an open subset Ω⊂ \mathbb Rn taking values in the space of un-ordered pairs of points in \mathbb Rm, and if the integral varifold V = (M, θ), where the multiplicity function θ : M → \1, 2\ is such that θ=2 on the set where the two values of u agree and θ=1 otherwise, is stationary in Ω× \mathbb Rm with respect to the mass functional, we show that at \mathcal Hn-2-a.e. point Z along its branch locus u decays asymptotically, modulo its single valued average, to a unique non-zero two-valued cylindrical harmonic tangent function φ(Z) which is homogeneous of some degree ≥ 3/2. As a corollary, we obtain that the branch locus of u is countably (n-2)-rectifiable, and near points Z where the degree of homogeneity of φ(Z) is equal to 3/2, the branch locus is an embedded real analytic submanifold of dimension n-2. These results, combined with the recent works \citeM and \citeMW, imply a stratification theorem for the (relatively open) set of density < 3 points of a stationary codimension 1 integral n-varifold with stable regular part and no triple junction singularities.