2012/10/29 by A. G. Ramm, А. Г. Рамм, Ramm, A. G.
Mathematics · #31B20 #35J05 #Algebraic and Geometric Analysis #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #math.AP #math.FA #msc:31B20 #msc:35J05
paper · pdf · doi:10.48550/arxiv.1210.7670
arxiv created 2012/10/29 · openalex publication_date 2012/10/29 · arxiv updated 2012/10/30 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
Let f ∈ Lloc1 (\Rn)∩ S', where S' is the Schwartz class of distributions, and ∫σ(D) f(x) dx = 0 ∀ σ∈ G, (*) where D⊂ \Rn is a bounded domain, the closure D of which is diffeomorphic to a closed ball. Then the complement of D is connected and path connected. Here G denotes the group of all rigid motions in \Rn. This group consists of all translations and rotations. It is conjectured that if f≠ 0 and (*) holds, then D is a ball. Other conjectures, equivalent to the above one, are formulated and discussed. Several new short proofs are given for the earlier proved results.