Abstract
All solutions of a continuously perturbed Laplace-Beltrami equation in its angular coordinates in the open unit ball Bn+2 ⊂ ℝn+2, n ≥ 1, are characterized. Moreover, it is shown that such pertubations yield distributional boundary values which are different from, but algebraically and topologically equivalent to, the hyperfunctions of Lions & Magenes. This is different from the case of radially perturbed Laplace- Beltrami operators (cf. [7]) where one has stability of distributional boundary values under such pertubations.
This work was supported by Sandia National Laboratories. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy (DOE) under contract DE-ACo4-94AL85000.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
R. E. Bellman, Stability Theory of Differential Equations, McGraw - Hill, New York, 1953.
R. Courant and D. Hilbert, Methoden der Mathematischen Physik I. Springer Verlag, Heidelberg, 1968.
M. S. P. Eastham, The spectral theory of periodic differential equations, McGraw - Hill, New York, 1973.
E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics, Chelsea Publishing Company, New York, 1955.
G. Johnson, Harmonic functions on the unit disk I, Illinois J. Math. 12 (1968), 143–154.
J. Kelingos and P. R. Massopust, A characterization of solutions to a perturbed Laplace equation II, Rocky Mountain J. Math. 24, No. 2 (1994), 549–562.
J. Kelingos and P. R. Massopust, A characterization of solutions to a radially perturbed Laplace equation in the unit n-ball J. Math. Anal. and Appl. 199 (1996), 728–747.
G. Köthe, Topological Vector Spaces, Springer Verlag, New York, 1969.
J. L. Lions and E. Magenes, Problèmes aux limites non homogènes, VII Ann. Mat. Pura Appl. 63 (1963), 201–224.
M. Sato, Theory of Hyperfunctions I and II, J. Fac. Sci. Univ. Tokyo 8 (1959–60), 139–193, 387–437.
P. O. Staples and J. Kelingos, A characterization of solutions to a perturbed Laplace equation, Illinois J. Math. 22 (1978), 208–216.
Editor information
Editors and Affiliations
Additional information
Dedicated to the Memory of John Kelingos
Rights and permissions
Copyright information
© 1999 Birkhäuser Boston
About this chapter
Cite this chapter
Massopust, P.R. (1999). On angularly perturbed Laplace equations in the unit ball and their distributional boundary values. In: Bray, W.O., Stanojević, Č.V. (eds) Analysis of Divergence. Applied and Numerical Harmonic Analysis. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-2236-1_21
Download citation
DOI: https://doi.org/10.1007/978-1-4612-2236-1_21
Publisher Name: Birkhäuser Boston
Print ISBN: 978-1-4612-7467-4
Online ISBN: 978-1-4612-2236-1
eBook Packages: Springer Book Archive