2019/04/21 by Ramm, Alexander G.
#35J05 #35R30 #42B05 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1904.11363
Symmetry problems in harmonic analysis are formulated and solved. One of these problems is equivalent to the refined Schiffer's conjecture which was recently proved by the author. Let k=const>0 be fixed, S2 be the unit sphere in ℝ3, D be a connected bounded domain with C2-smooth boundary S, j0(r) be the spherical Bessel function. The harmonic analysis symmetry problems are stated in the following theorems: \bf Theorem A. \em Assume that ∫S eikβ⋅ sds=0 for all β∈ S2. Then S is a sphere of radius a, where j0(ka)=0. \bf Theorem B. \em Assume that ∫D eikβ⋅ xdx=0 for all β∈ S2. Then D is a ball.