Solution to the Pompeiu problem and the related symmetry problem

2016/06/20 by A. G. Ramm, Ramm, A. G.
Mathematics · #35J05 #35J25 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35J05 #msc:35J25

paper · pdf · doi:10.48550/arxiv.1606.05976

arxiv created 2016/08/15 · arxiv updated 2016/08/16

Abstract

Assume that D⊂ ℝ³ is a bounded domain with C¹-smooth boundary. Our result is: \bf Theorem 1. \em If D has P-property, then D is a ball. Four equivalent formulations of the Pompeiu problem are discussed. A domain D has P-property if there exists an f≠ 0, f∈ L¹_loc(ℝ³) such that ∫_Df(gx+y)dx=0 for all y∈ ℝ³ and all g∈ SO(2), where SO(2) is the rotation group. The result obtained concerning the related symmetry problem is: \bf Theorem 2. \em If (∇² +k²)u=0 in D, u|_S=1, u_N|_S=0, and k>0 is a constant, then D is a ball.

Related

• The Pompeiu problem
• A solution to the Pompeiu problem
• Isoparametric foliations and the Pompeiu problem
• A symmetry problem
• Symmetry problems in harmonic analysis