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Theorems on ball mean values for solutions of the Helmholtz equation on unbounded domains. (English. Russian original) Zbl 1246.35071

Izv. Math. 76, No. 2, 365-374 (2012); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 76, No. 2, 161-170 (2012).
In this article, a geometric description of the set of solutions of the Helmholtz equation on unbounded domains is obtained. The author extends the spherical mean theorem for solutions of the Helmholtz equation for half-spaces.

MSC:

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
43A90 Harmonic analysis and spherical functions
58C35 Integration on manifolds; measures on manifolds
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
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