On the positive square root of the fourth derivative operator
Author:
D. L. Russell
Journal:
Quart. Appl. Math. 46 (1988), 751-773
MSC:
Primary 47E05; Secondary 47A60, 70J99, 73D30, 73K12
DOI:
https://doi.org/10.1090/qam/973388
MathSciNet review:
973388
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Abstract: It is only for a special subset of the natural boundary conditions for the operator \[ Aw = \frac {{{d^4}w}}{{d{x^4}}}\] that its positive square root is the negative second derivative operator. In this paper we develop a procedure for parametric description of all natural boundary conditions, we show which ones admit ${A^{1/2}}$ in the form just noted, and we show that in the other cases \[ - {D^2}w \equiv - \frac {{{d^2}w}}{{d{x^2}}} = \left [ {I + P} \right ]{A^{1/2}}w\] where $P$ is a bounded, but in general not compact, operator on the Hilbert space ${L^2}\left [ {0, \pi } \right ]$. Possible applications to the theory of the partial differential equation \[ \rho \frac {{{\partial ^2}w}}{{\partial {t^2}}} - 2\gamma \frac {{{\partial ^3}w}}{{\partial t\partial {x^2}}} + EI\frac {{{\partial ^4}w}}{{\partial {x^4}}} = 0\] are indicated.
- F. V. Atkinson, Discrete and continuous boundary problems, Mathematics in Science and Engineering, Vol. 8, Academic Press, New York-London, 1964. MR 0176141
- G. Chen and D. L. Russell, A mathematical model for linear elastic systems with structural damping, Quart. Appl. Math. 39 (1981/82), no. 4, 433–454. MR 644099, DOI https://doi.org/10.1090/S0033-569X-1982-0644099-3
- Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
G. H. Hardy, J. E. Littlewood, and G. Pòlya, Inequalities, Cambridge University Press, 1934
- L. F. Ho and D. L. Russell, Admissible input elements for systems in Hilbert space and a Carleson measure criterion, SIAM J. Control Optim. 21 (1983), no. 4, 614–640. MR 704478, DOI https://doi.org/10.1137/0321037
- Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
S. Kaczmarz and H. Steinhaus, Theorie der Orthogonalreihen, Monografje Matematyczne, Tom VI: Warsaw, 1935
- R. S. Phillips, Dissipative hyperbolic systems, Trans. Amer. Math. Soc. 86 (1957), 109–173. MR 90748, DOI https://doi.org/10.1090/S0002-9947-1957-0090748-2
- S. C. Power, Hankel operators on Hilbert space, Research Notes in Mathematics, vol. 64, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. MR 666699
- William T. Reid, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0273082
S. A. Janczewski, Sur quelques problèmes aux limites pour des équations differentielles linéaires ordinaires du quatrième ordre, Comptes Rendus 184, 141–143 (1927) Further, related, work by the same author appears in Comptes Rendus 184, 261–263 (1927), Comptes Rendus 186, 287–289 (1928), and in Annals of Mathematics (2) 29, 521–542 (1927/28).
F. V. Atkinson, Discrete and Continuous Boundary Value Problems, Vol. 8 of Mathematics in Science and Engineering, Academic Press, New York, 1964
G. Chen and D. L. Russell, A mathematical model for linear elastic systems with structural damping. Quart. Appl. Math. 39, 433–454 (1982)
P. L. Duren, Theory of H$^{p}$ Spaces, Academic Press, New York, 1970
G. H. Hardy, J. E. Littlewood, and G. Pòlya, Inequalities, Cambridge University Press, 1934
L. F. Ho and D. L. Russell, Admissible input elements for systems in Hilbert space and a Carleson measure criterion, SIAM J. Control Optim. 21, 614–640 (1983)
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966
S. Kaczmarz and H. Steinhaus, Theorie der Orthogonalreihen, Monografje Matematyczne, Tom VI: Warsaw, 1935
R. S. Phillips, Dissipative hyperbolic systems, Trans. Amer. Math. Soc. 86, 109–173 (1957)
S. C. Power, Hankel Operators on Hilbert Space, Research Notes in Mathematics, Vol. 64. Pitman Advanced Publishing Program, Boston, London, Melbourne, 1982
W. T. Reid, Ordinary Differential Equations, John Wiley and Sons, New York. 1971 Reference added in proof. Dr. G. Leugering has very kindly pointed out earlier work related to our boundary value classification scheme.
S. A. Janczewski, Sur quelques problèmes aux limites pour des équations differentielles linéaires ordinaires du quatrième ordre, Comptes Rendus 184, 141–143 (1927) Further, related, work by the same author appears in Comptes Rendus 184, 261–263 (1927), Comptes Rendus 186, 287–289 (1928), and in Annals of Mathematics (2) 29, 521–542 (1927/28).
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Article copyright:
© Copyright 1988
American Mathematical Society