Attouch H., Buttazzo G., Michaille G. — Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization :: Электронная библиотека попечительского совета мехмата МГУ

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Attouch H., Buttazzo G., Michaille G. — Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization

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Название: Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization

Авторы: Attouch H., Buttazzo G., Michaille G.

Аннотация:

For graduate students and researchers, Attouch (mathematics, U. Montpellier II, France) et al. present a guide to variational analysis, optimization, and partial differential equations (PDEs). After discussing the basics, the authors chart new areas of research on BV spaces. Topics include weak solution methods, abstract variational principles, complements on measure theory, Sobolev spaces, examples of classical variational problems, the finite element method, spectral analysis of the Laplacian, convex duality and optimization, relaxation, integral functionals, application in mechanics and computer vision, and shape optimization problems.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2006

Количество страниц: 634

Добавлена в каталог: 02.02.2010

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Предметный указатель

$Aff_{0}(D,R^{m})$       425
$ap\liminf_{x\rightarrow x_{0}}f(x)$       390
$ap\limsup_{x\rightarrow x_{0}}f(x)$       390
$BV(\Omega)$       371
$BV(\Omega,R^{m})$       383
$Cap_{p}(\cdot)$       212
$C^{1}$ -diffeomorphism      174
$C^{m}(\Omega)$       25
$C_{0}(\Omega)$       61
$C_{0}(\Omega,R^{m})$       129
$C_{b}(\Omega,R^{m})$       133
$C_{b}(\Omega;E)$       138
$C_{c}(\Omega)$       16
$C_{c}(\Omega,R^{m})$       129
$C_{c}^{1}(\Omega)$       16
$C_{\#}(Y)$       432
$dim_{H}$       121
$f\#_{e}g$ , f#g      313
$H^{-1}(\Omega)$       167
$H^{1}(\Omega)$       152
$H^{1}_{0}(\Omega)$       154
$H^{s}(R^{N})$       185
$K^{\infty}$       563
$L^{1}_{\lambda}(\Omega, R^{m})$       125
$L_{w}(\Omega,M(E))$       139
$M(\Omega)$       124
$M(\Omega,R^{m})$       124
$M^{m\timesN}$       421
$M_{b}(\Omega)$       124
$SBV(\Omega)$       409
$SBV(\Omega,R^{m})$       409
$S_{f}$       394
$u^{-}$ , $u^{+}$       404
$W^{-m,p^{'}}(\Omega)$       167
$W^{1,p}(\Omega)$       153
$W^{1,p}(\Omega, R^{m})$       421
$W^{1,p}_{0}(\Omega)$       154
$y\succeq_{s}x$       101
$\alpha=ap\lim_{x\rightarrow x_{0}}f(x)$       390
$\Delta$       7
$\Gamma$ -convergence      464
$\Gamma-\liminf_{n\rightarrow +\infty}F_{n}$       465
$\Gamma-\limsup_{n\rightarrow +\infty}F_{n}$       465
$\mathcal{B}(\Omega)$       124
$\mathcal{D}(\Omega)$       16
$\mathcal{D}^{'}(\Omega)$       17
$\mathcal{F}$       181
$\mathcal{H}^{s}$       112
$\mathcal{L}^{N}$       114
$\mathcal{Y}(\Omega;E)$       138
$\mu=(\mu_{x})_{x\in\Omega}\otimes\sigma$       135
$\mu\lfloor A$       124
$\mu^{+}$ , $\mu^{-}$       125
$\overset{nar}{\rightharpoonup}$       138
$\partial f$       331
$\partial_{M}E$       389
$\partial_{r}E$       397
$\rho_{\varepsilon}\ast\mu$       132
$\sigma_{c}$       309
$\widehat{v}$       180
Absolutely continuous      125
Approximate, derivative      408
Approximate, limit      389
Approximate, limit inf      390
Approximate, limit sup      390
Cantor part      407
Cantor set      122
Cantor — Vitali function      408 518
Caratheodory criterion      112
Cauchy — Riemann      9
Coercive      76
Coercive, $\sigma$ -coercive      589
Coercivity      86
Complementary problem      340
Complementary slackness condition      345 356
Concentration      50
Convolution      18
Courant — Fisher      290 291 294
Covering, fine      115
De La Vallee-Poussin criterion      518 519
Density point      388
diam(E)      109
Dirac mass      15 27
Dirichlet      598
Dirichlet, problem      31
distribution      17
Distribution, derivation      24
div      9
dom f      77
Domain      77
Dual function      354 358 361
Dual problem      351 354
Dual value      354
Duality, gap      354 358
Dunford — Pettis theorem      145
Eberlein — Smulian theorem      56
Eigenvalue      279—281
Eigenvector      280 281 286 288 290
Ekeland's $\varepsilon$ -variational principle      98
epi f      79
Epi-sum      312 328
Epiconvergence      466
Ergodic theorem      49
Ergodic, dynamical system      534
Ergodic, subadditive ergodic theorem      534
Exact minorant      331 333
Extension, operator      174
Extension, theorem      179
F*      321
Fenchel, extremality relation      332 364
Fenchel, extremality relations      335
Fenchel, Fenchel — Moreau theorem      321
Fenchel, Legendre — Fenchel, conjugate      320
Fenchel, Legendre — Fenchel, transform      597
Finite perimeter (set of)      396
Galerkin approximating method      257
Galerkin method      73
Gateaux differentiability      98
Gauss      8
Gauss — Green formula      396 401
Generalized solution      420

Hahn — Banach separation theorem      92
Hahn — Banach theorem      307 454
Harmonic function      7
Hat function      261 262
Hausdorff, dimension      120
Hausdorff, measure      109 112
Hausdorff, outer measure      109
hilbert      13
Infcompact function      86
Inner measure-theoretic normal      395
Interpolant      265
Ju, Cu      407
Jump, part      407
Jump, point      394
Jump, set      394 404
Karush — Kuhn — Tucker optimality conditions      341 345
Kernel      10
Lagrange, multiplier vector      346
Lagrange, multiplier vector, characterization of      346
Lagrangian      353
Laplace      8
Lax — Milgram Theorem      67
Lebesgue — Nikodym      592
Legendre — Fenchel, conjugate      361
Limit, analysis      597
Lower semicontinuous regularization      85
Marginal function      348 363
Markov inequality      143
Measure theoretical, boundary      389
Measure theoretical, exterior      388
Measure theoretical, interior      388
Measure, Borel      124
Measure, bounded      125
Measure, counting      593
Measure, Radon      115 125
Measure, regular      61 125
Measure, signed      124
Measure, support      124
Measure, total variation      125
Mollifier      18
Mountain Pass Theorem      100
Narrow      374
Narrow convergence, of Young measures      138
Neumann      598
Neumann, boundary condition      34
Neumann, problem      33
Newtonian potential      7 28
nodes      261 262
Normal cone      338
Optimal value      355
oscillations      49
Palais — Smale compactness condition      99
Perturbation function      360 361 363 365
Picard iterative method      71
Poincare inequality      168
Poincare — Wirtinger inequality      180 400
Poisson equation      8
Primal, problem      353
Primal, value      353
Proper      77
Qf      423
Quasi-continuous representatives      340
Quasi-convex envelope      423 458
Rademacher      379
Radon measure      24
Radon — Nikodym theorem      126
Rarefaction point      388
Rayleigh, Courant — Rayleigh formula      298
Rayleigh, quotient      290
Recession, cone      563
Recession, function      440 478 510 524 592
Recession, functional      555
Reduce boundary      397
Reflexive      55
Regular point      393
Regular, triangulation      264
Relaxation      437
Relaxation scheme      457
Relaxed problem      85 420
Rellich — Kondrakov compact embedding theorem      179
Rellich — Kondrakov Theorem      172
Riemann      9
Riesz, representation theorem      20 48 61 67 129
Riesz, theorem      41 86
Rockafellar theorem      329
Saddle point      354 355
Saddle value problems      354
Self-similar set      123
Separation of variable method      279 305
Set convergence      464
Set of class $C^{1}$       174
singular      125
Slater qualification assumption      341 343 344 348 350 351 355 363
Slater, generalized Slater      363
Slicing decomposition      135
Sobolev spaces      24
spt      16 124
Stokes problem      35
Subadditive      593
Subadditivity      110 592
Subdifferential      331
Support function      309
Tangent cone      338
Test function      31
Test functions      15
Tightness, for nonnegative Borel measures      133
Tightness, for Young measure      139
Uniformly convex      52 55
Uniformly integrable      58 144 145 148 450 451 460
Value function      363
Vitali's covering theorem      115
von Neumann's Minimax Theorem      360
Weak* topology      60
Weak, convergence      41
Weak, solution      31
Weak, topology      41 44
Weierstrass, example      12
Weierstrass, theorem      87
Young measure      138
Young measures, $W^{1,p}$ -Young measures      450
Young measures, associated with functions      142
Young measures, generated by functions      143

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