On spherical inversions of polyharmonic functions
Author:
Allen T. Chwang
Journal:
Quart. Appl. Math. 44 (1987), 793-799
MSC:
Primary 31B30
DOI:
https://doi.org/10.1090/qam/872829
MathSciNet review:
872829
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Abstract: A general spherical inversion theorem for polyharmonic functions of order $m$ has been obtained, and it reduces to the Kelvin transformation for $m = 1$. For biharmonic functions $\left ( {m = 2} \right )$, the present theorem has been applied to generate an explicit solution which satisfies the prescribed homogeneous boundary conditions on the surface of a sphere.
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C. S. Yih, Fluid mechanics, McGraw-Hill, Inc., pp. 96–97 (1969)
E. Almansi, Sull’integrazione dell’ equazione differenziale ${\Delta ^{2n}} = 0$, Ann. Mat. Pura Appl. (3) 2, 1–51 (1899)
J. C. Burns, A generalization of Milne-Thomson’s circle theorem, J. Math. Phys. Sci. 7, 373–382 (1973)
S. F. J. Butler, A note on Stokes’s stream function for motion with a spherical boundary, Proc. Cambridge Philos. Soc. 49, 169–174 (1953)
W. D. Collins, A note on Stokes’s stream function for the slow steady motion of viscous fluid before plane and spherical boundaries, Mathematika 1, 125–130 (1954)
W. D. Collins, Note on a sphere theorem for the axisymmetric Stokes flow of a viscous fluid, Mathematika 5, 118–121 (1958)
R. Courant and D. Hilbert, Methods of mathematical physics, Vol. 2, Interscience, pp. 242–243 and 286–290 (1962)
O. D. Kellogg, Foundations of potential theory, J. Springer, Berlin, pp. 231–233 and 251–253, 1929
G. S. S. Ludford, J. Martinek, and G. C. K. Yeh, The sphere theorem in potential theory, Proc. Cambridge Philos. Soc. 51, 389–393 (1955)
L. M. Milne-Thomson, Hydrodynamical images, Proc. Cambridge Philos. Soc. 36, 246–247 (1940)
P. Weiss, On hydrodynamical images—arbitrary irrotational flow disturbed by a sphere, Proc. Cambridge Philos. Soc. 40, 259–261 (1944)
G. C. K. Yeh, J. Martinek, and G. S. S. Ludford, A general sphere theorem for hydrodynamics, heat, magnetism, and electrostatics, Z. Angew. Math. Mech. 36, 111–116 (1956)
C. S. Yih, Solutions of the hyper-Bessel equation, Quart. Appl. Math. 13, 462–463 (1956)
C. S. Yih, Fluid mechanics, McGraw-Hill, Inc., pp. 96–97 (1969)
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Article copyright:
© Copyright 1987
American Mathematical Society