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positive definite; compact resolvent; Fourier method; existence theorems; static; transverse static deflection; transverse vibration; thin homogeneous elastic plate; transverse load; dynamic problems; circular plates theory
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.
[1] R. A. Adams: Sobolev spaces. Academic Press 1975. MR 0450957 | Zbl 0314.46030
[2] 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.). MR 0592954
[3] A. Erdélyi, at.: Higher transcendental functions. Vol 2. McGraw Hill 1953.
[4] W. G. Faris: Self-adjoint operators. Lecture Notes in Mathematics 433, Springer-Verlag 1975. MR 0467348 | Zbl 0317.47016
[5] P. Hartman: Ordinary differential equations. J. Wiley and Sons 1964. MR 0171038 | Zbl 0125.32102
[6] E. Jahnke F. Emde: Tables of functions. Dover Publ. 1945 (4th ed.). MR 0015900
[7] W. Magnus F. Oberhettinger: Formeln und Sätze für die speziellen Funktionen der mathematischen Physik. Springer-Verlag 1948 (2nd ed.). MR 0025629
[8] D. E. Mc Farland B. L. Smith W. D. Bernhart: Analysis of plates. Spartan Books 1972.
[9] S. G. Mikhlin: Linear partial differential equations. (Russian.) Vysš. škola, Moscow 1977. MR 0510535
[10] S. Timoshenko D. H. Young W. Weaver, Jr.: Vibrations problems in engineering. John Wiley and Sons 1974 (4th ed.). (Russian: Mashinostrojenije, Moscow 1985.)
[11] S. Timoshenko S. Woinowski-Krieger: Theory of plates and shells. McGraw Hill 1959.
[12] O. Vejvoda, al.: Partial differential equations: time-periodic solutions. Martinus Nijhoff 1982. Zbl 0501.35001
[13J G. N. Watson: A treatise on the theory of Bessel functions. Cambridge Univ. Press 1958 (2nd ed.). MR 1349110
[14] J. Weidmann: Linear operators in Hilbert spaces. Graduate texts in Mathematics 68, Springer-Verlag 1980. MR 0566954 | Zbl 0434.47001
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