2013/04/08 by A. G. Ramm, А. Г. Рамм, Ramm, A. G.
Mathematics · #31B20 #35J05 #Algebraic Geometry and Number Theory #Analysis of PDEs (math.AP) #FOS: Mathematics #History and Theory of Mathematics #Nonlinear Partial Differential Equations #math.AP #msc:31B20 #msc:35J05
paper · pdf · doi:10.48550/arxiv.1304.2297
openalex publication_date 2013/04/08 · arxiv created 2013/04/13 · arxiv updated 2013/04/16 · openalex created_date 2025/10/10 · 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, and S is its boundary. Then the comp is connected and path connected. By G the group of all rigid motions of \Rn is denoted. This group consists of all translations and rotations. A proof of the following theorem is given. Theorem 1. \it Assume that n=2, f\not≡ 0, and (*) holds. Then D is a ball. Corollary. \it If the problem (∇2+k2)u=0 in D, uN|S=0, u|S=const≠ 0 has a solution, then D is a ball. Here N is the outer unit normal to S.