On the harmonicity of a function with a Bôcher-Koebe type condition

Abstract

Let B_R be an open ball of radius R in R^n with the center at zero, B_(0,R)=B_R\{0}, and a function f be harmonic in B_(0,R). If f has zero residue at the point x=0, then the flow of its gradient through any sphere lying in B_(0,R), is zero. In this paper, the reverse phenomenon is studied for the case when only spheres of one or two fixed radii r_1 and r_2. are allowed. A description of the class H_r (B_(0,R) )={f∈C^∞ (B_(0,R) ):∫_(S_r (x))▒〖∂x/∂"n" dω=0 ∀x∈B_(R-r)\S_r 〗}, was found, where r∈(0,R/2), S_r (x)={y∈R^n: |y-x|=r}, S_r=S_r (0). It is proved that if r_1/r_2 is not a ratio of the zeros of the Bessel function J_(n/2) and f∈(H_(r_1 ) ∩ H_(r_2 ) )(B_(0,R) ), then the function f is harmonic in B_(0,R) and "Res"(f,0)=0. This result cannot be significantly improved. Namely, if r_1/r_2=α/β, where J_(n/2) (α)=J_(n/2) (β)=0, or R<r_1+r_2, then there exists a function f∈C^∞ (B_R ) non-harmonic in B_(0,R) and such that ∫_(S_(r_j ) (x))▒〖∂x/∂"n" dω=0, x∈B_(R-r_j ),j=〗 {1;2}. In addition, the condition f∈C^∞ (B_(0,R) ) cannot be replaced, generally speaking, by the requirement f∈C^S (B_R ) for an arbitrary fixed s∈N.