Abstract
A mean-value formula for a linear partial differential hyperbolic equation with an operator splitting into first-order factors is obtained. This formula can be interpreted as an extension of the Ásgeirsson principle.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Benzoni-Gavage and D. Serre, Multi-Dimensional Hyperbolic Partial Differential Equations: First-Order Systems and Applications, Oxford Univ. Press(2006).
A. V. Bitsadze and A. M. Nakhushev, “On the theory of mixed-type equations in multidimentional domains,” Differ. Uravn., 10, No. 12, 2184–2191 (1974).
R. Courant and D. Hilbert, Methods of Mathematical Physics, Interscience, New York (1953).
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin–Heidelberg–New York–Tokyo (1983).
L. Hörmander, The Analysis of Linear Differential Operators, Springer-Verlag, Heidelberg (1983).
V. A. Il’in, “Fourier series in fundamental systems of functions of the Beltrami operator,” Differ. Uravn., 5, No. 11, 1940–1978 (1969).
V. A. Il’in, “On some properties of regular solutions of the Helmholtz equation in a planar domain,” Mat. Zametki, 15, No. 6, 885–890 (1974).
V. A. Il’in and E. I. Moiseev, “A mean-value formula for the associated functions of the Laplace operator,” Differ. Uravn., 17, No. 10, 1908–1910 (1981).
F. John, Plane Waves and Spherical Means, Interscience, New York–London (1955).
A. N. Kolmogorov, “On the notion of mean,” in: Selected Works of A. N. Kolmogorov. Vol. I: Mathematics and Mechanics, Springer (1991), pp. 144–146.
V. Z. Meshkov and I. P. Polovinkin, “Mean-value properties of solutions of linear partial differential equations,” J. Math. Sci., 160, No. 1, 45–52 (2009).
V. Z. Meshkov, I. P. Polovinkin, M. V. Polovinkina, Yu. D. Ermakova, and S. A. Rabeeakh, “A mean-value formula for a two-dimentional linear hyperbolic equation,” Vestn. Voronezh. Univ. Ser. Fiz. Mat., No. 4, 121–126 (2016).
E. I. Moiseev, “Asymptotic mean-value formula for regular solutions of differential equations,” Differ. Uravn., 16, No. 5, 827–844 (1980).
E. I. Moiseev, “A mean-value formula for a harmonic function in a circular sector,” Dokl. Ross. Akad. Nauk, 432, No. 5, 592–593 (2010).
S. M. Sitnik, “Generalization of the Cauchy–Bunyakovsky inequalities by method of averages and their applications,” in: Fundamental Methematics [in Russian] (L. A. Minin, I. Ya. Novikov, V. A. Rodin, and S. M. Sitnik, eds.), Voronezh (2005), pp. 3–42.
L. Zalcman, “Mean values and differential equations,” Isr. J. Math., 14, 339–352 (1973).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 172, Proceedings of the Voronezh Winter Mathematical School “Modern Methods of Function Theory and Related Problems,” Voronezh, January 28 – February 2, 2019. Part 3, 2019.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Polovinkina, M.V. Mean-Value Formula for a Hyperbolic Equation with a Factorizable Operator. J Math Sci 268, 124–129 (2022). https://doi.org/10.1007/s10958-022-06184-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-022-06184-1
Keywords and phrases
- accompanying distribution
- mean-value formula
- factorization of a differential operator
- inclusion-exclusion formula