A study of an operator arising in the theory of circular plates
Herrmann, Leopold
Applications of Mathematics, Tome 33 (1988), p. 337-353 / Harvested from Czech Digital Mathematics Library
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The operator $L_0:D_{L_0}\subset H \rightarrow H$, $L_0u = \frac 1r \frac d {dr} \left\{r \frac d{dr}\left[\frac 1r \frac d{dr}\left(r \frac {du}{dr}\right)\right] \right\}$, $D_{L_0}= \{u \in C^4 ([0,R]), u'(0)=u''''(0)=0, u(R)=u'(R)=0\}$, $H=L_{2,r}(0,R)$ is shown to be essentially self-adjoint, positive definite with a compact resolvent. The conditions on $L_0$ (in fact, on a general symmetric operator) are given so as to justify the application of the Fourier method for solving the problems of the types $L_0u=g$ and $u_{tt}+L_0u=g$, respectively.

Publié le : 1988-01-01
Classification:  34B20,  35C10,  47B25,  47E05,  73K12,  74H45,  74K20 
@article{104315,
     author = {Leopold Herrmann},
     title = {A study of an operator arising in the theory of circular plates},
     journal = {Applications of Mathematics},
     volume = {33},
     year = {1988},
     pages = {337-353},
     zbl = {0658.73037},
     mrnumber = {0961312},
     language = {en},
     url = {http://dml.mathdoc.fr/item/104315}
}

Herrmann, Leopold. A study of an operator arising in the theory of circular plates. Applications of Mathematics, Tome 33 (1988) pp. 337-353. http://gdmltest.u-ga.fr/item/104315/

R. A. Adams Sobolev spaces, Academic Press 1975. (1975) | MR 0450957 | Zbl 0314.46030

B. M. Budak A. A. Samarskii A. N. Tikhonov A collection of problems on mathematical physics, International series of monographs in pure and applied mathematics 52, Pergamon Press, Oxford 1964. (Russian: Izd. Nauka, Moscow 1980, 3rd ed.). (1964) | MR 0592954

A. Erdélyi; At. Higher transcendental functions, Vol 2. McGraw Hill 1953. (1953)

W. G. Faris Self-adjoint operators, Lecture Notes in Mathematics 433, Springer-Verlag 1975. (1975) | MR 0467348 | Zbl 0317.47016

P. Hartman Ordinary differential equations, J. Wiley and Sons 1964. (1964) | MR 0171038 | Zbl 0125.32102

E. Jahnke F. Emde Tables of functions, Dover Publ. 1945 (4th ed.). (1945) | MR 0015900

W. Magnus F. Oberhettinger Formeln und Sätze für die speziellen Funktionen der mathematischen Physik, Springer-Verlag 1948 (2nd ed.). (1948) | MR 0025629

D. E. Mc Farland B. L. Smith W. D. Bernhart Analysis of plates, Spartan Books 1972. (1972)

S. G. Mikhlin Linear partial differential equations, (Russian.) Vysš. škola, Moscow 1977. (1977) | MR 0510535

S. Timoshenko D. H. Young W. Weaver, Jr. Vibrations problems in engineering, John Wiley and Sons 1974 (4th ed.). (Russian: Mashinostrojenije, Moscow 1985.) (1974)

S. Timoshenko S. Woinowski-Krieger Theory of plates and shells, McGraw Hill 1959. (1959)

O. Vejvoda; Al. Partial differential equations: time-periodic solutions, Martinus Nijhoff 1982. (1982) | Zbl 0501.35001

[13J G. N. Watson A treatise on the theory of Bessel functions, Cambridge Univ. Press 1958 (2nd ed.). (1958) | MR 1349110

J. Weidmann Linear operators in Hilbert spaces, Graduate texts in Mathematics 68, Springer-Verlag 1980. (1980) | MR 0566954 | Zbl 0434.47001