On the harmonicity of a function with a Bôcher–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

Bibliographie

[1] M. Bôcher, “On harmonic functions in two dimensions”, Proc. Amer. Acad. Arts and Sci., 41 (1906), 557–583

[2] P. Koebe, “Herleitung der partiellen differentialgleichungen der potentialfunktion aus deren integraleigenschaft”, Sitzungsber. Berlin. Math. Gessellschaft, 5 (1906), 39–42

[3] G. C. Evans, “Problems of potential theory”, Proc. Natl. Acad. Sci. USA, 7 (1921), 89–98 | DOI

[4] G. E. Raynor, “On the integro-differential equation of the Bôcher type in three space”, Bull. Amer. Math. Soc., 32 (1926), 654–658 | DOI | MR

[5] J. J. Gergen, “Note on a theorem of Bôcher and Koebe”, Bull. Amer. Math. Soc., 37 (1931), 591–596 | DOI | MR | Zbl

[6] S. Saks, “Note on defining properties of harmonic functions”, Bull. Amer. Math. Soc., 38 (1932), 380–382 | DOI | MR | Zbl

[7] E. F. Beckenbach, “Concerning the definition of harmonic functions”, Bull. Amer. Math. Soc., 51 (1945), 240–245 | DOI | MR | Zbl

[8] N. Kuznetsov, “Mean value properties of harmonic functions and related topics (a Survey)”, J. Math. Sci., 242 (2019), 177–199 | DOI | MR | Zbl

[9] I. Netuka, J. Veselý, “Mean value property and harmonic functions”, Classical and Modern Potential Theory and Applications. NATO ASI Series, 430 (1994), 359–398 | MR | Zbl

[10] K. A. Berenstein, D. Struppa, “Kompleksnyi analiz i uravneniya v svertkakh”, Kompleksnyi analiz – mnogie peremennye – 5, Itogi nauki i tekhn. Ser. Sovrem. probl. mat. Fundam. napravleniya, 54, 1989, 5–111 ; C. A. Berenstein, D. C. Struppa, “Complex analysis and convolution equations”, Encyclopaedia of Mathematical Sciences, v. 54, Several complex variables $\mathrm{V}.$ Complex analysis in partial differential equations and mathematical physics, ed. G. M. Khenkin, Springer, Berlin, Heidelberg, 1993, 1–108 | MR

[11] L. Zalcman, “A bibliographic survey of the Pompeiu problem”, Approximation by Solutions of Partial Differential Equations, Nato Science Series C: Mathematical and Physical Sciences, 365, eds. B. Fuglede, M. Goldstein, W. Haussmann, W. K. Hayman, L. Rogge, Kluwer Academic Publishers, Dordrecht, 1992, 185–194 | MR

[12] L. Zalcman, “Supplementary bibliography to "A bibliographic survey of the Pompeiu problem”, Contemporary Mathematics, v. 278, Radon Transform and Tomography, eds. E. T. Quinto, L. Ehrenpreis, A. Faridani, F. Gonzalez, E. Grinberg, AMS, 2001, 69–74 | DOI | MR | Zbl

[13] V. V. Volchkov, Integral Geometry and Convolution Equations, Kluwer Academic Publishers, Dordrecht, 2003 | MR | Zbl

[14] V. V. Volchkov, Vit. V. Volchkov, Offbeat Integral Geometry on Symmetric Spaces, Birkhäuser, Basel, 2013 | MR | Zbl

[15] L. Zalcman, “Analyticity and the Pompeiu problem”, Arch. Rat. Anal. Mech., 47 (1972), 237–254 | MR | Zbl

[16] L. Brown, B. M. Schreiber, B. A. Taylor, “Spectral synthesis and the Pompeiu problem”, Ann. Inst. Fourier, 23 (1973), 125–154 | DOI | MR | Zbl

[17] J. D. Smith, “Harmonic analysis of scalar and vector fields in $\mathbb R^n$”, Proc. Cambridge Philos. Soc., 72 (1972), 403–416 | DOI | MR | Zbl

[18] C. A. Berenstein, R. Gay, “A local version of the two-circles theorem”, Israel J. Math., 55 (1986), 267–288 | DOI | MR | Zbl

[19] V. V. Volchkov, “Teoremy o dvukh radiusakh na ogranichennykh oblastyakh evklidovykh prostranstv”, Diff. uravn., 30:10 (1994), 1719–1724 | MR | Zbl

[20] V. V. Volchkov, “Okonchatelnyi variant lokalnoi teoremy o dvukh radiusakh”, Matem. sb., 186:6 (1995), 15–34 | MR | Zbl

[21] S. Axler, P. Bourdon, W. Ramey, Harmonic Function Theory, Springer, New York, 1992 | MR | Zbl

[22] I. Stein, G. Veis, Vvedenie v garmonicheskii analiz na evklidovykh prostranstvakh, Mir, M., 1974; E. M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton–New Jersey, 1971 | MR | Zbl

[23] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii, Nauka, M., 1974; A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher Transcendental Functions, McGraw-Hill, New York, 1953 | MR

[24] N. Ya. Vilenkin, Spetsialnye funktsii i teoriya predstavlenii grupp, Nauka, M., 1991 ; N. Ja. Vilenkin, Special Functions and the Theory of Group Representations, AMS, RI, 1968 | MR | MR | Zbl

[25] V. V. Volchkov, “Reshenie problemy nositelya dlya nekotorykh klassov funktsii”, Matem. sb., 188:9 (1997), 13–30 | DOI | MR | Zbl