Mean value properties of solutions to the m-dimensional Helmholtz and modified Helmholtz equations are considered. An elementary derivation of these properties is given. It involves the Euler–Poisson–Darboux equation. Despite the similar form of these properties for both equations, their consequences distinguish essentially. The restricted mean value property of harmonic functions is amended so that a function satisfying this property in a bounded domain of a special class solves the modified Helmholtz equation in this domain.
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Translated from Problemy Matematicheskogo Analiza 111, 2021, pp. 109-118.
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Kuznetsov, N. Mean Value Properties of Solutions to the Helmholtz and Modified Helmholtz Equations. J Math Sci 257, 673–683 (2021). https://doi.org/10.1007/s10958-021-05509-w
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DOI: https://doi.org/10.1007/s10958-021-05509-w