Proof of the Schiffer's conjecture

2017/06/09 by A. G. Ramm, А. Г. Рамм, Ramm, A. G.
Mathematics · #35J05 #35R30 #Analysis of PDEs (math.AP) #Analytic and geometric function theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #math.AP #msc:35J05 #msc:35R30

paper · pdf · doi:10.48550/arxiv.1706.03032

the argument is not complete

openalex publication_date 2017/06/09 · arxiv created 2018/02/09 · arxiv updated 2018/02/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The following conjecture has been known for many decades as Schiffer's symmetry problem (or Schiffer's conjecture): Assume that Δu+k²u=0 in D, u|_S=0, u_N|_S=1, where D⊂ ℝ³ is a bounded, connected, C²-smooth domain, S is its boundary, N is a unit normal to S pointing out of D, k²>0 is a constant. Then S is a sphere. In this paper the above conjecture is proved. It is also proved that the relation ∫_Se^{ikβ⋅ s}ds=0, ∀ β∈ S² implies that S is a sphere.

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