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Index

averaging of a function, 40

ballclosed,5open, 5

baseof neighborhoods, 9

base of neighborhoods, 4bending moment, 211, 212, 268bifurcation, 235

point, 235boundary of a set, 4

chain rule, 52closure of a set, 4coefficients of the first quadratic

form, 258completion of the set of real

numbers, 3component

of the flexural strain, 258of the tangential strain, 258of the torsional strain, 258

computation of eigenvalues, 127condition

complementing boundary, 169limit density, 103, 112natural, 220of ellipticity, 168of optimality necessary, 90, 95,

135of stability, 220of the free edge, 220of transversality, 220optimality, 192supplementary, 168

condition of filtration, 432cone, 6constitutive equation of a nonlinear

viscous fluid, 431continuity modulus, 93continuity of the spectrum, 118control

discontinuous, 97nonregular, 86nonsmooth, 97regular, 88

convergence, 4strong, 9weak,8

curvilinear coordinates, 260cylindrical stiffness of a plate, 211

distance, 5, 7distribution, 38domain in IRn , 36

eigenfunction, 20approximate, 127

eigenspace, 20eigenvalue, 20

approximate, 127its multiplicity, 20

elasticity characteristics, 243element of the best approximation,

112energy

stored,223strain, 211

finite-dimensional approximation,122

520

formbilinear, 11coercive, 12, 81symmetric, 12

Frechet derivative, 51free oscillations of a shell, 268function

continuous, 11generalized, 37lower semicontinuous, 11test, 37upper semicontinuous, 11

G-closedness of linear operators, 72G-convergence of linear operators,

72Gateaux derivative, 53

homeomorphism, 10

image, 10implicit function, 52index of elliptic operator, 173inequality

Korn,48Schwartz, 17variational, 201

infimum, 3interior of a set, 4isomorphism, 10

Kirchhoff hypotheses, 209, 258Kronecker delta, 19

Lagrange principle, 70lemma

inner point, 183limit

lower, 3upper, 3

Index

majorant,2mapping, 2

k-linear, 11bijective, 2bilinear, 11bounded, 8continuous, 10convex, 7Frechet differentiable, 51

continuously, 51injective, 2sequentially continuous, 10surjective, 2uniformly continuous, 14

materialorthotropic, 259

maximum, 2maximum function, 14, 16

discrete, 14midsurface

of a shell, 257minimizing element

its characterization, 181minimum, 2

local, 70, 303minorant,2model

finite shear, of a shell, 274multi-index, 36

neighborhood of a point, 4norm, 7

equivalent, 7Euclidean, 8

observation, 191operator

coercive, 81compact, 20nonlinear compact, 234selfadjoint, 20

optimizationdomain shape, 173

orthogonalcomplement, 19elements, 18subspaces, 18

plateisotropic, 212orthotropic, 211, 214three-layered, 242

problemcombined, 142, 157control

basic, 154general, 149

eigenvalue, 117eigevalue optimizaton, 162finite-dimensional, 103, 105finite-dimensional regular, 113minimax control, 198of bending of a plate, 215of free oscillations of a plate, 221spectral, 117

productof sets, 2of topologies, 5

prototype, 10

regularization of a function, 40Riesz

method,112operator, 112

scalar product, 17scale of Hilbert spaces, 35segment, 6sequence

Cauchy, 6fundamental, 6

sequence weakly fundamental, 9set

bounded, 2bounded above, 2bounded below, 2closed, 4

Index 521

compact, 13convex, 6dense, 4open, 5relatively compact, 13sequentially *-weakly closed, 102sequentially weakly closed, 9, 110weakly closed, 9

shelllaminated, 282of revolution, 260shallow, 276

spaceBanach,7dual, 8Hausdorff, 4Hilbert, 17metric, 5

complete, 6sequentially weakly complete, 9Sobolev,39topological, 4

compact, 13metrizable, 6separable, 4sequentially compact, 13

vector, 6normed,6

splinefundamental, 299

strain component, 210, 211stress component, 210subspace of functions with

zero-point strain energy, 264supremum, 3system

ellpitic in the sense ofDougHs-Nirenberg, 168

system of operatorsW-coercive, 45coercive, 44

theoremCalderon, 39

522

embedding, 39Lax-Milgram, 18Lebesgue, 37on a composite function, 52on equivalent norms, 41on the invariance of Sobolev

spaces, 76on trace space, 43Riesz, 17

Index

three-layered platefree oscillations, 245natural oscillations, 255

topology, 4induced, 5weak, 8

torque, 211, 212, 268trace of a function, 43