On the completeness of the Papkovich potentials
Author:
Robert F. Millar
Journal:
Quart. Appl. Math. 41 (1984), 385-393
MSC:
Primary 73C05; Secondary 31B99
DOI:
https://doi.org/10.1090/qam/724050
MathSciNet review:
724050
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Abstract: The Papkovich representation for the elastostatic displacement vector in a domain $D$ is considered. The possibility of eliminating from this representation either the scalar potential $\chi$ or a rectangular component $\psi$ of the vector potential $\psi$ is examined. Earlier work is discussed and the connection is made with the oblique derivative problem of potential theory. A convexity requirement on the boundary of $D$ is shown to be necessary in general in order that $\chi$ or $\psi$ may be eliminated.. A result of Stippes for a domain with an internal cavity is generalized, and two new classes of domains are found for which $\chi$ may be eliminated.
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L. Hörmander, Pseudo-differential operators and non-elliptic boundary problems, Ann. of Math. 83, 129–209 (1966)
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R. A. Eubanks and E. Sternberg, On the completeness of the Boussinesq-Papkovich stress functions, J. Rat. Mech. and Anal. 5, 735–746 (1956)
I. S. Sokolnikoff, Mathematical theory of elasticity, second edition, McGraw-Hill Book Company, New York, 1956
M. E. Gurtin, The linear theory of elasticity, Encyclopedia of Physics, Volume VIa/2, Springer-Verlag, Berlin, Heidelberg, New York, 1972, pp. 1–295
M. Stippes, Completeness of the Papkovich potentials, Quart. Appl. Math. 26, 477–483 (1969)
T. Tran Cong and G. P. Steven, On the representation of elastic displacement fields in terms of three harmonic functions, J.Elasticity 9, 325–333 (1979)
I. N. Vekua, New methods for solving elliptic equations, North-Holland Publishing Company, Amsterdam, 1967
A. V. Bitsadze, Boundary value problems for second order elliptic equations, North-Holland Publishing Company, Amsterdam, 1968
N. I. Muskhelishvili, Singular integral equations, P. Noordhoff Ltd., Groningen, 1953
S. G. Mikhlin, Multidimensional singular integrals and integral equations, Pergamon Press, Oxford, 1965
C. Miranda, Partial differential equations of elliptic type, second revised edition, Springer-Verlag, Berlin, 1970
L. Hörmander, Pseudo-differential operators and non-elliptic boundary problems, Ann. of Math. 83, 129–209 (1966)
B. Winzell, A boundary value problem with an oblique derivative, Comm. Partial Differential Equations 6, 305–328 (1981)
S. Bochner and W. T. Martin, Several complex variables, Princeton University Press, Princeton, 1948
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© Copyright 1984
American Mathematical Society