Hostname: page-component-cb9f654ff-rkzlw Total loading time: 0 Render date: 2025-08-25T13:07:32.455Z Has data issue: false hasContentIssue false
  • English
  • Français

A relation between biharmonic Green’s functions of simply supported and clamped bodies

Published online by Cambridge University Press:  22 January 2016

James Ralston  and
Leo Sario
James Ralston
Affiliation:
University of California, Los Angeles
Leo Sario
Affiliation:
University of California, Los Angeles
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The deflection, under a point load, of a thin elastic plate clamped at the edges is the biharmonic Green’s function β with the boundary data β = ∂β/∂n = 0. If the boundary of the region is reasonably smooth, the construction of β offers no difficulty. In contrast, nothing is known about the existence of β in the general case. The purpose of our study is to give a sufficient condition for the existence of β on a given Riemannian manifold of arbitrary dimension and to construct β. Our results will have, apart from their physical meaning in elasticity, consequences in the biharmonic classification theory of Riemannian manifolds.

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 61 , July 1976 , pp. 59 - 71
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1976

References

[1] Agmon, S., Lectures on Elliptic Boundary Value Problems, Van Nostrand, Princeton, 1965, 291 pp.Google Scholar
[2] Chung, L., Manifolds carrying bounded quasiharmonic but no bounded harmonic functions Math. Scand. (to appear).Google Scholar
[3] Chung, L., Asymptotic behavior and biharmonic degeneracy (to appear).Google Scholar
[4] Chung, L. and Sario, L., Harmonic and quasiharmonic degeneracy of Riemannian manifolds, Tôhoku Math. J. 27, 487–496.Google Scholar
[5] Chung, L. and Sario, L., Harmonic Lp functions and quasiharmonic degeneracy, J. Indian Math. Soc. (to appear).Google Scholar
[6] Chung, L., Sario, L. and Wang, C., Riemannian manifolds with bounded Dirichlet finite polyharmonic functions, Ann. Scuola Norm. Sup. Pisa 27 (1973), 1–6.Google Scholar
[7] Chung, L., Sario, L. and Wang, C., Quasiharmonic Lp functions on Riemannian manifolds, Ann. Scuola Norm. Sup. Pisa 2 (1974), 469–478.Google Scholar
[8] Duffin, R. J., On a question of Hadamard concerning super-biharmonic functions, J. Math. Physics 27 (1949), 253–258.Google Scholar
[9] Duffin, R. J., Some problems of mathematics and science, Bull. Amer. Math. Soc. 80 (1974), 1053–1070.Google Scholar
[10] Garabedian, P. R., A partial differential equation arising in conformal mapping, Pacific J. Math. 1 (1951), 485–524.Google Scholar
[11] Hada, D., Sario, L. and Wang, C., Dirichlet finite biharmonic functions on the Poincaré N-ball, J. Reine Angew. Math. 272 (1975), 92–101.Google Scholar
[12] Hada, D., Sario, L. and Wang, C., N-manifolds carrying bounded but no Dirichlet finite harmonic functions, Math. J. 54 (1974), 1–6.Google Scholar
[13] Hada, D., Sario, L. and Wang, C., Bounded biharmonic functions on the Poincaré N-ball, Ködai Math. Sem. Rep. 26 (1975), 327–342.Google Scholar
[14] Hadamard, J., Mémoire sur le problème d’analyse relatif à l’équilibre des plaques élastiques encastrées, Mémoires présentés par divers savants étrangers à l’Académie des Sciences, 33 (1908), 515–629.Google Scholar
[15] Haupt, O., Über das asympototische Verhalten der Lösungen gewisser linearer gewöhnlicher Differentialgleichungen, Math. Z, 48 (1913), 289–292.Google Scholar
[16] Hille, E., Behavior of solutions of linear second order differential equations, Ark. Mat. 2(1952), 25–41.Google Scholar
[17] Hörmander, L., Linear Partial Differential Operators, Springer, New York, 1969, 285 pp.CrossRefGoogle Scholar
[18] Kwon, Y. K., Sario, L. and Walsh, B., Behavior of biharmonic functions on Wiener’s and Royden’s compactifications, Ann. Inst. Fourier (Grenoble) 21 (1971), 217–226.Google Scholar
[19] Loewner, C., On generalization of solutions of the biharmonic equation in the plane by conformal mappings, Pacific J. Math. 3 (1953), 417–436.Google Scholar
[20] Mirsky, N., Sario, L. and Wang, C., Bounded polyharmonic functions and the dimension of the manifold, J. Math. Kyoto Univ. 13 (1973), 529–535.Google Scholar
[21] Mirsky, N., Sario, L. and Wang, C., Parabolicity and existence of bounded or Dirichlet finite polyharmonic functions, Rend. 1st. Mat. Univ. Trieste 6 (1974), 1–9.Google Scholar
[22] Nakai, M. and Sario, L., Completeness and function-theoretic degeneracy of Riemannian spaces, Proc. Nat. Acad. Sci. 57 (1967), 29–31.Google Scholar
[23] Nakai, M. and Sario, L., Biharmonic classification of Riemannian manifolds, Bull. Amer. Math. Soc. 77 (1971), 432–436.Google Scholar
[24] Nakai, M. and Sario, L., Quasiharmonic classification of Riemannian manifolds, Proc. Amer. Math. Soc. 31 (1972), 165–169.Google Scholar
[25] Nakai, M. and Sario, L., Dirichlet finite biharmonic functions with Dirichlet finite Laplacians, Math. Z. 122 (1971), 203–216.Google Scholar
[26] Nakai, M. and Sario, L., A property of biharmonic functions with Dirichlet finite Laplacians, Math. Scand. 29 (1971), 307–316.Google Scholar
[27] Nakai, M. and Sario, L., Existence of Dirichlet finite biharmonic functions, Ann. Acad. Sci. Fenn. A. I. 532 (1973), 1–33.Google Scholar
[28] Nakai, M. and Sario, L., Existence of bounded biharmonic functions, J. Reine Angew. Math. 259 (1973), 147–156.Google Scholar
[29] Nakai, M. and Sario, L., Existence of bounded Dirichlet finite biharmonic functions, Ann. Acad. Sci. Fenn. A. I. 505 (1972), 1–12.Google Scholar
[30] Nakai, M. and Sario, L., Biharmonic functions on Riemannian manifolds, Continuum Mechanics and Related Problems of Analysis, Nauka, Moscow, 1972, 329–335.Google Scholar
[31] Sario, L., Biharmonic and quasiharmonic functions on Riemannian manifolds, Duplicated lecture notes 1968–70, University of California, Los Angeles.Google Scholar
[32] Sario, L., Quasiharmonic degeneracy of Riemannian N-manifolds, Ködai Math. Sem. Rep. 26 (1974), 53–57.Google Scholar
[33] Sario, L., Completeness and existence of bounded biharmonic functions on a Riemannian manifold, Ann. Inst. Fourier (Grenoble) 24 (1974), 311–317.CrossRefGoogle Scholar
[34] Sario, L., A criterion for the existence of biharmonic Green’s functions, Research Announcement, Bull. Amer. Math. Soc. 80 (1974), 1183–1184. To appear in extenso in J. Austral. Math. Soc.CrossRefGoogle Scholar
[35] Sario, L., Biharmonic measure, Ann. Acad. Sci. Fenn. A. I. 587 (1974), 1–18.Google Scholar
[36] Sario, L., Biharmonic Green’s functions and harmonic degeneracy, J. Math. Kyoto Univ. 15 (1975), 351–362.Google Scholar
[37] Sario, L. and Nakai, M., Classification Theory of Riemann Surfaces, Springer-Verlag, 1970, 446 pp.Google Scholar
[38] Sario, L. and Wang, C., The class of (p, q)-biharmonic functions, Pacific J. Math. 41 (1972), 799–808.Google Scholar
[39] Sario, L. and Wang, C., Counterexamples in the biharmonic classification of Riemannian N-manifolds, Pacific J. Math. 50 (1974), 159–162.Google Scholar
[40] Sario, L. and Wang, C., Generators of the space of bounded biharmonic functions, Math. Z. 127 (1972), 273–280.Google Scholar
[41] Sario, L. and Wang, C., Quasiharmonic functions on the Poincaré N-ball, Rend. Mat. (4) 6 (1973), 1–14.Google Scholar
[42] Sario, L. and Wang, C., Riemannian manifolds of dimension N>4 without bounded biharmonic functions, J. London Math. Soc. (2) 7 (1974), 635–644.Google Scholar
[43] Sario, L. and Wang, C., Existence of Dirichlet finite biharmonic functions on the Poincaré N-ball, Pacific J. Math. 48 (1973), 267–274.Google Scholar
[44] Sario, L. and Wang, C., Negative quasiharmonic functions, Tôhoku Math. J. 26 (1974), 85–93.Google Scholar
[45] Sario, L. and Wang, C., Radial quasiharmonic functions, Pacific J. Math. 46 (1973), 515–522.Google Scholar
[46] Sario, L. and Wang, C., Parabolicity and existence of bounded biharmonic functions, Comm. Math. Helv. 47 (1972), 341–347.Google Scholar
[47] Sario, L. and Wang, C., Positive harmonic functions and biharmonic degeneracy, Bull. Amer. Math. Soc. 79 (1973), 182–187.Google Scholar
[48] Sario, L. and Wang, C., Parabolicity and existence of Dirichlet finite biharmonic functions, J. London Math. Soc. 8 (1974), 145–148.Google Scholar
[49] Sario, L. and Wang, C., Harmonic and biharmonic degeneracy, Ködai Math. Sem. Rep. 25 (1973), 392–396.Google Scholar
[50] Sario, L. and Wang, C., Harmonic Lp-functions on Riemannian manifolds, Ködai Math. Sem. Rep. 26 (1975), 204–209.Google Scholar
[51] Sario, L., Wang, C. and Range, M., Biharmonic projection and decomposition, Ann. Acad. Sci. Fenn. A. I. 494 (1971), 1–14.Google Scholar
[52] Szegö, G., Remark on the preceding paper by Charles Loewner, Pacific J. Math. 3 (1953), 437–446.Google Scholar
[53] Wang, C., Biharmonic Green’s function and quasiharmonic degeneracy, Math. Scand. 35 (1974), 38–42.Google Scholar
[54] Wang, C., Biharmonic Green’s function and biharmonic degeneracy, Math. Scand. 37 (1975),122–128.Google Scholar
[55] Wang, C. and Sario, L., Polyharmonic classification of Riemannian manifolds, J. Math. Kyoto Univ. 12 (1972), 129–140.Google Scholar