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A criterion for the existence of biharmonic Green's functions

Published online by Cambridge University Press:  09 April 2009

Leo Sario
Leo Sario
Affiliation:
521 Georgina Avenue Santa Monica California, 90402, U.S.A.
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Abstract

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The biharmonic Green's function of a simply supported plate is generalized to Riemannian manifolds and shown to exist if and only if the harmonic measure of the ideal boundary is square integrable.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 21 , Issue 2 , March 1976 , pp. 155 - 165
Copyright
Copyright © Australian Mathematical Society 1976

References

L. Chung (ot appear), ‘Manifolds carrying bounded quasiharmonic but no bounded harmonic functions’ Math. Scand.Google Scholar
L. Chung (to appear), ‘Asymptotic behavior and bihamonic degeneracy’.Google Scholar
Chung, L. and Sario, L. (to appear), ‘Harmonic and quasiharmonic degeneracy of Riemannian manifolds’ Tôkoku Math. J.Google Scholar
Chung, L., Sario, L. and Wang, C. (1973), ‘Riemannian manifolds with bounded Dirichlet finite polyharmonic functions’, Ann. Scuola Norm. Sup. Pisa. 27, 1–6.Google Scholar
Chung, L. and Wang, C. (to appear), ‘Quasiharmonic Lp functions on Riemannian manifolds’ Ann Scuols Norm. Sup. Pisa.Google Scholar
Hada, D., Sario, L. and Wang, C. (1975), ‘Dirichlet finite biharmonic functions on the Poincaré N-ball’, J. Reine Angew. Math. 272, 92–101.Google Scholar
Hada, D., Sario, L. and Wang, C. (1974), ‘N-manifolds carrying bounded but no Dirichlet finite harmonic functions’, Nagoya Math. J. 54, 1–6.CrossRefGoogle Scholar
Hada, D., Sario, L. and Wang, C. (to appear), ‘Bounded biharmonic functions on the Poincaré N-ball’, Kōdai Math. Sem. Rep.Google Scholar
Haupt, O. (1913), ‘Über das asymptotische Verhalten der Lösungen gewisser linearer gewöhnlicher Differentialgleichungen’, Matk. Z. 48, 289–292.CrossRefGoogle Scholar
Hille, E. (1952), ‘Behavior of solutions of linear second order differential equations’, Ark. Mat. 2, 25–41.CrossRefGoogle Scholar
Kwon, Y. K., Sario, L. and Walsh, B. (1971), ‘Behavior of biharmonic functions on Wiener's and Royden's compactifications’, Ann. Inst. Fourier (Grenoble) 21, 217–226.CrossRefGoogle Scholar
Mirsky, N., Sario, L. and Wang, C. (1973), ‘Bounded polyharmonic functions and the dimension of the manifold’, J. Math. Kyoto Univ. 13, 529–535.Google Scholar
Mirsky, N., Sario, L. and Wang, C. (1974), ‘Parabolicity and existence of bounded or Dirichlet finite polyharmonic functions’, Rend. 1st. Mat. Univ. Trieste 6, 1–9.Google Scholar
Nakai, M. and Sario, L. (1967), ‘Completeness and function-theoretic degeneracy of Riemannian spaces’, Proc. Nat. Acad. Sci. 57, 29–31.CrossRefGoogle ScholarPubMed
Nakai, M. and Sario, L. (1971), ‘Biharmonic classification of Riemannian manifolds’, Bull. Amer. Math. Soc. 77, 432–436.CrossRefGoogle Scholar
Nakai, M. and Sario, L. (1972), ‘Quasiharmonic classification of Riemannian manifolds’, Proc. Amer. Math. Soc. 31, 165–169.CrossRefGoogle Scholar
Nakai, M. and Sario, L. (1971), ‘Dirichlet finite biharmonic functions with Dirichlet finite Laplacians’, Math. Z. 122, 203–216.CrossRefGoogle Scholar
Nakai, M. and Sario, L., (1971), ‘A property of biharmonic functions with Dirichlet finite Laplacians’, Math. Scand. 29, 307–316.CrossRefGoogle Scholar
Nakai, M. and Sario, L. (1973), ‘Existence of Dirichlet finite biharmonic functions’, Ann. Acad. Sci. Fenn. A. I. 532, 1–33.Google Scholar
Nakai, M. and Sario, L. (1973), ‘Existence of bounded biharmonic functions’, J. Reine Angew. Math. 259, 147–156.Google Scholar
Nakai, M. and Sario, L. (1972), ‘Existence of bounded Dirichlet finite biharmonic functions’, Ann. Acad. Sci. Fenn. A. I. 505, 1–12.Google Scholar
Nakai, M. and Sario, L. (1972), ‘Biharmonic functions on Riemannian manifolds’, Continuum Mechanics and Related Problems of Analysis (Nauka, Moskow, 1972).Google Scholar
Ralston, J. and Sario, L. (to appear), ‘A relation between biharmonic Green's functions of simply supported and clamped bodies’.Google Scholar
Sario, L. (1968–1970), ‘Biharmonic and quasiharmonic functions on Riemannian manifolds’ Duplicated lecture notes (University of California, Los Angeles, 1968–1970).Google Scholar
Sario, L. (1974), ‘Quasiharmonic degeneracy of Riemannian N-manifolds’, Kōdai Math. Sem. Rep. 26, 53–57.Google Scholar
Sario, L. (1974), ‘Completeness and existence of bounded biharmonic functions on a Riemannian manifold’, Ann. Inst. Fourier (Grenoble) 24, 311–317.CrossRefGoogle Scholar
Sario, L. (1974), ‘Biharmonic measure’, Ann. Acad. Sci. Fenn. A. I. 587, 1–18.Google Scholar
Sario, L. (to appear), ‘Biharmonic Green's functions and harmonic degeneracy’, J. Math. Kyoto Univ..Google Scholar
Sario, L. and Nakai, M. (1970), Classification Theory of Riemann Surfaces (Springer Verlag, 1970).CrossRefGoogle Scholar
Sario, L. and Wang, C. (1972), ‘The class of (pq)-biharmonic functions’, Pacific J. Math. 41, 799–808.CrossRefGoogle Scholar
Sario, L. and Wang, C. (1974), ‘Counterexamples in the biharmonic classification of Riemannian 2-manifolds’, Pacific J. Math. 50, 159–162.CrossRefGoogle Scholar
Sario, L. and Wang, C. (1972), ‘Generators of the space of bounded biharmonic functions’, Math. Z. 127, 273–280.CrossRefGoogle Scholar
Sario, L. and Wang, C. (1973), ‘Quasiharmonic functions on the Poincaré N-ball’, Rend Mat. (4) 6, 1–14.Google Scholar
Sario, L. and Wang, C. (1974), ‘Riemannian manifolds of dimension N ≧ 4 without bounded biharmonic funtions’, J. London Math. Soc.(2) 7, 635–644.CrossRefGoogle Scholar
Sario, L. and Wang, C. (1973), ‘Existence of Dirichlet finite biharmonic functions on the Poincaré 3-ball’, Pacific J. Math. 48, 267–274.CrossRefGoogle Scholar
Sario, L. and Wang, C. (1974), ‘Negative quasiharmonic functions’, Tôhoku Math. J. 26, 85–93.Google Scholar
Sario, L. and Wang, C. (1973), ‘Radial quasiharmonic functions’, Pacific J. Math. 46, 515–522.CrossRefGoogle Scholar
Sario, L. and Wang, C. (1972), ‘Parabolicity and existence of bounded biharmonic functions’, Comm. Math. Helv. 47, 341–347.CrossRefGoogle Scholar
Sario, L. and Wang, C. (1973), ‘Positive harmonic functions and biharmonic degeneracy’, Bull Amer. Math. Soc. 79, 182–187.CrossRefGoogle Scholar
Sario, L. and Wang, C. (1974), ‘Parabolicity and existence of Dirichlet finite biharmonic functions’, J. London Math. Soc. 8, 145–148.CrossRefGoogle Scholar
Sario, L. and Wang, C. (1973), ‘Harmonic and biharmonic degeneracy’, Kōdai Math. Sem. Rep. 25, 392–396.Google Scholar
Sario, L. and Wang, C. (1975), ‘Harmonic Lp -functions on Riemannian manifolds’, Kōdai Math. Sem. Rep. 26, 204–209.Google Scholar
Sario, L., Wang, C. and Range, M. (1971), Biharmonic projection and decomposition’, Ann. Acad. Sci. Fenn. A. I. 494, 1–14.Google Scholar
Wang, C. (1974), ‘Biharmonic Green's function and quasiharmonic degeneracy’, Math. Scand. 35, 38–42.CrossRefGoogle Scholar
Wang, C. (to appear), ‘Biharmonic Green's function and biharmonic degeneracy’, Math. Scand.Google Scholar
Wang, C. and Sario, L. (1972), ‘Polyharmonic classification of Riemannian manifolds’, J. Math. Kyoto Univ. 12, 129–140.Google Scholar