On the harmonicity of a function with a B\^{o}cher--Koebe type condition

N. P. Volchkova ; V. V. Volchkov

Vestnik rossijskih universitetov. Matematika, Tome 29 (2024) no. 146, pp. 125-137

Résumé

Let $B_R$ be an open ball of radius $R$ in $\mathbb{R}^n$ with the center at zero, $B_{0,R}=B_R\backslash \{0\},$ 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 $r_1$ и $r_2$ are allowed. A description of the class \begin{equation*} \mathfrak{H}_r(B_{0,R})=\bigg\{f\in C^{\infty}(B_{0,R}): \int_{S_{r}(x)} \frac{\partial f}{\partial \mathbf{n}}\, d\omega =0\quad \forall x\in B_{R-r}\backslash S_{r}\bigg\} \end{equation*} was found, where $r\in (0,R/2),$ $S_r(x)=\{y\in \mathbb{R}^n: |y-x|=r\},$ $S_r=S_r(0).$ It is proved that if $r_1/r_2$ is not a ratio of the zeros of the Bessel function $J_{n/2}$ and $f\in(\mathfrak{H}_{r_1}\cap\mathfrak{H}_{r_2})(B_{0,R}),$ then the function $f$ is harmonic in $B_{0,R}$ and ${\mathrm{Res}}\, (f,0)=0.$ This result cannot be significantly improved. Namely, if $r_1/r_2 =\alpha/\beta,$ where $J_{n/2}(\alpha)=J_{n/2}(\beta)=0,$ or $R r_1+r_2,$ then there exists a function $f\in C^{\infty}(B_{R})$ non-harmonic in $B_{0,R}$ and such that \begin{equation*} \int_{S_{r_j}(x)} \frac{\partial f}{\partial \mathbf{n}}\, d\omega =0,\quad x\in B_{R-r_j},\quad j\in \{1;2\}. \end{equation*} In addition, the condition $f\in C^{\infty}(B_{0,R})$ cannot be replaced, generally speaking, by the requirement $f\in C^{s}(B_{R})$ for an arbitrary fixed $s\in \mathbb{N}.$

Détail

Keywords: harmonic functions, Bôcher–Koebe condition, spherical harmonics
Mots-clés : Pompeiu sets