Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is only for a special subset of the natural boundary conditions for the operator

$\displaystyle Aw = \frac{{{d^4}w}}{{d{x^4}}}$

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

$\displaystyle - {D^2}w \equiv - \frac{{{d^2}w}}{{d{x^2}}} = \left[ {I + P} \right]{A^{1/2}}w$

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

$\displaystyle \rho \frac{{{\partial ^2}w}}{{\partial {t^2}}} - 2\gamma \frac{{{... ...}{{\partial t\partial {x^2}}} + EI\frac{{{\partial ^4}w}}{{\partial {x^4}}} = 0$

are indicated.

References [Enhancements On Off] (What's this?)

  • [1] F. V. Atkinson, Discrete and Continuous Boundary Value Problems, Vol. 8 of Mathematics in Science and Engineering, Academic Press, New York, 1964 MR 0176141
  • [2] G. Chen and D. L. Russell, A mathematical model for linear elastic systems with structural damping. Quart. Appl. Math. 39, 433-454 (1982) MR 644099
  • [3] P. L. Duren, Theory of H$ ^{p}$ Spaces, Academic Press, New York, 1970 MR 0268655
  • [4] G. H. Hardy, J. E. Littlewood, and G. Pòlya, Inequalities, Cambridge University Press, 1934
  • [5] 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) MR 719525
  • [6] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966 MR 0203473
  • [7] S. Kaczmarz and H. Steinhaus, Theorie der Orthogonalreihen, Monografje Matematyczne, Tom VI: Warsaw, 1935
  • [8] R. S. Phillips, Dissipative hyperbolic systems, Trans. Amer. Math. Soc. 86, 109-173 (1957) MR 0090748
  • [9] S. C. Power, Hankel Operators on Hilbert Space, Research Notes in Mathematics, Vol. 64. Pitman Advanced Publishing Program, Boston, London, Melbourne, 1982 MR 666699
  • [10] 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. MR 0273082
  • [11] 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).

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 47E05, 47A60, 70J99, 73D30, 73K12

Retrieve articles in all journals with MSC: 47E05, 47A60, 70J99, 73D30, 73K12

Additional Information

DOI: https://doi.org/10.1090/qam/973388
Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society