Abstract
We establish a mean value property for the functions which is satisfied to Laplace–Bessel equation. Also results involving generalized divergence theorem and the second Green’s identities relating the bulk with the boundary of a region on which differential Bessel operators act we obtained.
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REFERENCES
F. John, Plane Waves and Spherical Means (Springer, New York, 1981).
I. A. Kipriyanov, Singular Elliptic Boundary Value Problems (Nauka, Moscow, 1997) [in Russian].
A. Weinstein, ‘‘Discontinuos integrals and generalized theory of potential,’’ Trans. Am. Math. Soc. 63, 342–354 (1948).
A. Weinstein, ‘‘Generalized axially symmetric potential theory,’’ Bull. Am. Math. Soc. 59, 20–38 (1953).
I. A. Kipriyanov and V. V. Katrakhov, ‘‘On a boundary-value problem for elliptic equations of the second order at the sphere domain,’’ Dokl. Akad. Nauk SSSR 313, 545–548 (1990).
I. A. Kipriyanov and V. I. Kononenko, ‘‘Fundamental solutions of B-elliptic equations,’’ Differ. Uravn. 3, 114–129 (1967).
V. V. Katrakhov and S. M. Sitnik, ‘‘The transmutation method and boundary-value problems for singular elliptic equations,’’ Sovrem. Mat. Fundam. Napravl. 64, 211–426 (2018).
L. N. Lyakhov, ‘‘Fundamental solutions of singular differential equations with a Bessel \(D_{B}\) operator,’’ Proc. Steklov Inst. Math. 278, 139–151 (2012).
L. N. Lyakhov and A. V. Ryzhkov, ‘‘Solutions of the B-polyharmonic equation,’’ Differ. Equat. 36, 1507–1511 (2000).
Ya. I. Zhitomirskii, ‘‘Cauchy’s problem for systems of linear partial differential equations with differential operators of Bessel type,’’ Mat. Sb. (N.S.) 36 (78), 299–310 (1955).
E. L. Shishkina and S. M. Sitnik, Transmutations, Singular and Fractional Differential Equations with Applications to Mathematical Physics (Elsevier, Amsterdam, 2020).
B. M. Levitan, ‘‘Expansion in Fourier series and integrals with Bessel functions,’’ Usp. Mat. Nauk 6 (2), 102–143 (1951).
L. N. Lyakhov, I. P. Polovinkin, and E. L. Shishkina, ‘‘On a Kipriyanov problem for a singular ultrahyperbolic equation,’’ Differ. Equat. 50, 513–525 (2014).
L. N. Lyakhov, I. P. Polovinkin, and E. L. Shishkina, ‘‘Formulas for the solution of the Cauchy problem for a singular wave equation with Bessel time operator,’’ Dokl. Math. 90, 737–742 (2014).
L. N. Lyakhov and E. L. Shishkina, ‘‘Weighted mixed spherical means and singular ultrahyperbolic equation,’’ Analysis 36 (2), 65–70 (2016).
E. L. Shishkina and S. M. Sitnik, ‘‘General form of the Euler–Poisson–Darboux equation and application of the transmutation method,’’ Electron. J. Differ. Equat. 177, 1–20 (2017).
S. M. Sitnik and E. L. Shishkina, Transmutation Method for Differential Equations with Bessel Operators (Fizmatlit, Moscow, 2019) [in Russian].
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Shishkina, E.L. Mean-Value Theorem for B-Harmonic Functions. Lobachevskii J Math 43, 1401–1407 (2022). https://doi.org/10.1134/S1995080222090232
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DOI: https://doi.org/10.1134/S1995080222090232