2019/10/31 by Jeffrey S. Case, Yi Wang, Case, Jeffrey S. +1
Mathematics · #35J66 #53C21 #58E11 #58J32 #Analysis of PDEs (math.AP) #Analytic and geometric function theory #Ball (mathematics) #Boundary (topology) #Differential Geometry (math.DG) #Dimension (graph theory) #Euclidean geometry #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry #Mathematical analysis #Mathematics #Nonlinear Partial Differential Equations #Nonlinear system #Physics #Pure mathematics #Sobolev inequality #Sobolev space #Symmetrization #TRACE (psycholinguistics) #Unit sphere #math.AP #math.DG #msc:35J66 #msc:53C21 #msc:58E11 #msc:58J32
paper · pdf · doi:10.48550/arxiv.1910.14232
25 pages
arxiv created 2019/10/31 · openalex publication_date 2019/10/31 · arxiv updated 2019/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We classify local minimizers of ∫σ2+\oint H2 among all conformally flat metrics in the Euclidean (n+1)-ball, 4≤ n≤ 5, for which the boundary has unit volume, subject to an ellipticity assumption. We also classify local minimizers of the analogous functional in the critical dimension n+1=4. If minimizers exist, this implies a fully nonlinear sharp Sobolev trace inequality. Our proof is an adaptation of the Frank--Lieb proof of the sharp Sobolev inequality, and in particular does not rely on symmetrization or Obata-type arguments.