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Cauchy’s Problem for Harmonic Functions with Entire Data on a Sphere

  • Dmitry Khavinson (a1)
Abstract

We give an elementary potential-theoretic proof of a theorem of G. Johnsson: all solutions of Cauchy’s problems for the Laplace equations with an entire data on a sphere extend harmonically to the whole space RN except, perhaps, for the center of the sphere.

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References
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[ABR] Axler, S., Bourdon, P. and Ramey, W., Harmonic Function Theory, Springer-Verlag, 1992.
[BS] Bony, J. M. and Schapira, P., Existence et prolongement des solutions holomorphes des équations aux dérivées partielles, Invent. Math. 17 (1972), 95105.
[J] Johnsson, G., The Cauchy problem in Cn for linear second order partial differential equations with data on a quadric surface, Trans.Amer.Math. Soc. 344 (1994), 148.
[KS1] Khavinson, D. and Shapiro, H. S., The Schwarz potential in Rn and Cauchy's problem for the Laplace equation, Research Report TRITA-MAT-1989-36, Royal Institute of Technology, Stockholm, 1989.
[KS2] Khavinson, D. and Shapiro, H. S., Dirichlet's problem when the data is an entire function, Bull. London Math. Soc. 24 (1992), 456468.
[L] Leray, J., Uniformisation de la solution des problème linéaire analytique de Cauchy, prés de la variété qui porte les données de Cauchy, Bull. Soc. Math. France 85 (1957), 389429.
[S] Schaeffer, A. D., Inequalities of A. Markoff and S. Bernstein for polynomials and related functions, Bull. Amer. Math. Soc. 47 (1941), 565579.
[SS] Sternin, B. Yu. and Shatalov, V. E., Differential Equations on Complex Manifolds, Kluwer, 1994.
[Z] Zerner, M., Domains d’holomorphie des fonctions vérifiant une équation aux dérivées partielles, C. R. Acad. Sci. Paris, Sér. I Math. 272 (1971), 16461648.
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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