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

Leopold Herrmann

Aplikace matematiky (1988)

  • Volume: 33, Issue: 5, page 337-353
  • ISSN: 0862-7940

Abstract

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The operator L 0 : D L 0 ⊂ H → H , L 0 u = 1 r d d r r d d r 1 r d d r r d u d r , D L 0 = { u ∈ 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 0 u = g and u t t + L 0 u = g , respectively.

How to cite

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Herrmann, Leopold. "A study of an operator arising in the theory of circular plates." Aplikace matematiky 33.5 (1988): 337-353. <http://eudml.org/doc/15548>.

@article{Herrmann1988,
abstract = {The operator $L_0:D_\{L_0\}\subset H \rightarrow H$, $L_0u = \frac\{1\}\{r\} \frac\{d\}\{dr\} \left\lbrace r \frac\{d\}\{dr\}\left[\frac\{1\}\{r\} \frac\{d\}\{dr\}\left(r \frac\{du\}\{dr\}\right)\right] \right\rbrace $, $D_\{L_0\}= \lbrace u \in C^4 ([0,R]), u^\{\prime \}(0)=u^\{\prime \prime \prime \prime \}(0)=0, u(R)=u^\{\prime \}(R)=0\rbrace $, $H=L_\{2,r\}(0,R)$ 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.},
author = {Herrmann, Leopold},
journal = {Aplikace matematiky},
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; positive definite; compact resolvent; Fourier method; existence theorems; static; dynamic problems; transverse static deflection; transverse vibration; thin homogeneous elastic plate; transverse load},
language = {eng},
number = {5},
pages = {337-353},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A study of an operator arising in the theory of circular plates},
url = {http://eudml.org/doc/15548},
volume = {33},
year = {1988},
}

TY - JOUR
AU - Herrmann, Leopold
TI - A study of an operator arising in the theory of circular plates
JO - Aplikace matematiky
PY - 1988
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 33
IS - 5
SP - 337
EP - 353
AB - The operator $L_0:D_{L_0}\subset H \rightarrow H$, $L_0u = \frac{1}{r} \frac{d}{dr} \left\lbrace r \frac{d}{dr}\left[\frac{1}{r} \frac{d}{dr}\left(r \frac{du}{dr}\right)\right] \right\rbrace $, $D_{L_0}= \lbrace u \in C^4 ([0,R]), u^{\prime }(0)=u^{\prime \prime \prime \prime }(0)=0, u(R)=u^{\prime }(R)=0\rbrace $, $H=L_{2,r}(0,R)$ 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.
LA - eng
KW - 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; positive definite; compact resolvent; Fourier method; existence theorems; static; dynamic problems; transverse static deflection; transverse vibration; thin homogeneous elastic plate; transverse load
UR - http://eudml.org/doc/15548
ER -

References

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