Geodesic
A study of an operator arising in the theory of circular plates
Applications of Mathematics, Tome 33 (1988) no. 5, pp. 337-353
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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.
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.
DOI : 10.21136/AM.1988.104315
Classification : 34B20, 35C10, 47B25, 47E05, 73K12, 74H45, 74K20
Keywords: 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 
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Herrmann, Leopold. A study of an operator arising in the theory of circular plates. Applications of Mathematics, Tome 33 (1988) no. 5, pp. 337-353. doi: 10.21136/AM.1988.104315

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