A study of an operator arising in the theory of circular plates

Abstract

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.