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Ziemer W.P. — Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation
Ziemer W.P. — Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation

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Название: Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation

Автор: Ziemer W.P.


The major thrust of this book is the analysis of pointwise behavior of Sobolev functions of integer order and BV functions (functions whose partial derivatives are measures with finite total variation). The development of Sobolev functions includes an analysis of their continuity properties in terms of Lebesgue points, approximate continuity, and fine continuity as well as a discussion of their higher order regularity properties in terms of Lp-derivatives. This provides the foundation for further results such as a strong approximation theorem and the comparison of Lp and distributional derivatives. Also included is a treatment of Sobolev-Poincaré type inequalities which unifies virtually all inequalities of this type. Although the techniques required for the discussion of BV functions are completely different from those required for Sobolev functions, there are similarities between their developments such as a unifying treatment of Poincaré-type inequalities for BV functions. This book is intended for graduate students and researchers whose interests may include aspects of approximation theory, the calculus of variations, partial differential equations, potential theory and related areas. The only prerequisite is a standard graduate course in real analysis since almost all of the material is accessible through real variable techniques.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1989

Количество страниц: 332

Добавлена в каталог: 05.07.2008

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Предметный указатель
$(W^{k,p})^*$, measures contained within      4.7(205)
$BV^*$, and measures      5.12(266)
$\mu$-measurable set      1.2(6)
$\varepsilon - \delta$ domains      2.5(64)
(K, p)-extension domain      2 5(63)
Absolute continuity on lines      2.1(44)
Absolutely continuous measures      1.3(15)
Adams, David      2.9(92) Historical
Admissible domain      5.10(256)
Approximate limit, lower      5.9(250)
Approximate limit, upper      5.9(250)
Approximate tangent plane      Historical Notes 5(218)
Approximately continuous      5.9(250)
Aronszajn — Smith      Historical Notes 2(108)
Banach Indicatrix formula      2.7(76)
Banach space      1.5(21)
Beppo Levi      Historical Notes 2(108)
Besicovitch covering theorem      1.3.6(12)
Besicovitch differentiation theorem      5.8(248)
Besicovitch, A, S.      Historical Notes 1(40)
Bessel capacitability, Suslin sets      2.6(71)
Bessel capacity      2.6(66) 4.5(194)
Bessel capacity and Hausdorff measure      2.6(75)
Bessel capacity, and maximal functions      3.10(154)
Bessel capacity, as outer measure      2.6(67)
Bessel capacity, inner regularity      2.6(72)
Bessel capacity, metric properties      2.6(73)
Bessel kernel      2.6(65)
Bessel potentials, and Sobolev functions      2.6(66)
Blow-up technique      5.6(237)
Borel measure      1.2(6)
Boundary, of a set      1.1(1)
Boundary, of a set, measure-theoretic      5.8(249)
Boundary, of a set, reduced      5.5(233)
Bounded variation, on lines      5.3(227)
Boxing inequality      4.9(218)
Brelot      2.6(68)
Brezis — Lieb      Historical Notes 2(110)
Brezis — Wainger      2.9(91) Historical
Burago — Zalgaller      Historical Notes 1(41)
BV function, extension of      5.10(257)
BV function, trace of      5.10(258)
BV functions      2.1(43) 5.1(220)
BV functions, and capacity      5.12(262)
BV functions, and co-area formula      5.4(231)
BV functions, and Poincare inequality      5 11(261)
BV functions, approximation by smooth functions      5.3(225)
BV functions, compactness in $L^1$ of unit ball      5.3(227)
BV norm      5.1(221)
Caecioppoli, R.      Historical Notes 5(281)
Calderdn — Stein      2.5(64)
Calderon Zygmund      Historical Notes 3(175)
Calderon, A.P.      Historical Notes 3(175}
Calderon, A.P., lemma      1.8(36}
Calkin, H.      Historical Notes 2(108)
Capacity and BV functions      5.12(262)
Capacity and Hausdorff measure      5.12(265)
Capacity, inner regularity      5.12(263)
Capacity, regularity of      5.12(267)
Caratheodory, C      Historical Notes 1(41)
Cartan, H.      Historical Notes 2(109)
Cauchy’s inequality      1.5(19)
Cesari, L.      Historical Notes 5(280)
Chain rule, Sobolev functions      2.1(48)
Change of variable, Sobolev functions      2.1(52)
Characteristic function of a set      1.1(2)
Choquet, G.      2.6(70)
Closed ball      1.1(2)
Co-area formula      2.7(76) 4.4(192) 4.9(210)
Co-area formula, for BV functions      5.4(231)
Co-area formula, general formulation      2.7(81)
Continuity of Bessel capacity      2.6(69)
Continuous functions, space of      1—1(2)
Convolution of distributions      1.7(25)
Convolution of functions      1.6(21)
Countably rectifiable sets      5.7(243)
Countably rectifiable sets, and $C^1$ sub-manifolds      5.7(243)
Countably rectifiable sets, and Lipschitz maps      5 7(243)
DeGiorgi, E.      Historical Notes 5(281)
DeGuzman, M.      Historical Notes 1(41)
Deny, J.      Historical Notes 2(109)
Difference quotients, bounded in $L^p$      3.6(137)
Distance function      1.1(1)
Distance function, smooth approximant      3.6(136)
Distance to a set      1.1(1)
Distribution function      1.8(26)
Distribution, derivative of      1.7(25)
Distribution, integrable function      1.7(24)
Distribution, multiplied by a function      1.7(25)
Distribution, Radon measure      1.7(24)
Distribution, Schwartz distribution      1.7(24)
Dual of $W^{m,p}$, representation      4.3(185)
Dual of $W^{m,p}_0$ representation      4.3(187)
Dual of Sobolev spaces      4.3(185)
Essential variation      5.3(227)
Exterior normal      5.5(233)
Exterior normal, generalized      5.5(233)
Exterior normal, measure-theoretic      5.6(240)
Federer, H.      Historical Notes 5(281)
Finely continuous function      3.3 (123)
Finely covered set      1.3(8)
Finite perimeter, and countably rectifiable sets      5.7(243)
Finite perimeter, and relative isoperimetrie inequality      5.4(230)
Finite perimeter, containing smoothly bounded sets      5.4(229)
Finite perimeter, sets of      5 4(229)
Finite perimeter, sets of and Gauss — Green theorem      5.8(248)
Finite perimeter, sets of and Lipschitz domains      5.8(248)
Fleming Rishel      Historical Notes 2(110) Historical
Fleming — Federer      Historical Notes 2(110)
Frehse, J.      Historical Notes 2(109)
Gagliardo, E.      Historical Notes 2(109)
Gauss — Green Theorem      5.5(234)
Gauss — Green theorem, and sets of finite perimeter      5.8(248)
Gelfand, I.M.      Historical Notes 1(41)
Generalized derivative      2.1(42)
Giusti, E.      Historical Notes 5(281)
Goffman, C      Historical Notes 5(281)
Gradient measure, continuity of      5.2(223)
Gradient measure, lower semicontinuity of      5.2(223)
Gradient of a function      1—1(2)
Guslin, W.      Historical Notes 5(281)
Hahn — Banach theorem      4.3(186)
Hardt, Robert      Historical Notes 1(40)
Hardy Littlewood — Wiener maximal theorem      2 8(84)
Hardy’s inequality      1.8(35)
Hausdorff dimension      1.4(18) Exercise
Hausdorff maximal principle      1.3(7)
Hausdorff measure      1.4(15)
Hausdorff measure, as an element of $(W^{m,p})^*$      4.4(191)
Hedberg Wolff      Historical Notes 4(218)
Hedberg, Lars      Historical Notes 2(111)
Holder continuous functions, space of      1.1(3)
Holder’s Inequality      1.5(20)
Hunt, R.      Historical Notes 1(41)
Inner product      1.1(1)
Integral currents      Historical Notes 5(281)
Isodiametric inequality      1.4(17)
Isoperimetrie inequality      2.7(81)
Isoperimetrie inequality, relative      5.4(230)
Jensen’s Inequality      1.5(20)
Jessen — Marrinkiewicz Zygmund      Historical Notes 1(40)
Jones, Peter      2.5(64)
K-quasiconformal mapping      Historical Notes 2(108)
Kondrachov, V.      Historical Notes 2(109)
Krickerberg, K.      Historical Notes 5(280)
Kronrod, A.S.      Historical Notes 2(110)
Lebesgue measurable set      1.2(3)
Lebesgue measure      1.2(3)
Lebesgue measure, as an element of $(W^{m,p})^*$      4.4(188)
Lebesgue point and approximate continuity      4.4(190)
Lebesgue point for higher order Sobolev functions      3.3(122)
Lebesgue point for Sobolev functions      3.3(118) 3.10(157}
Lebesgue point of a Sobolev function      4.4(190)
Lebesgue point, arbitrary measures      1.3(14)
Lebesgue, H.      Historical Notes 1(39)
Leibniz formula      Exercise 1.5 (37)
Length of multi-index      1.1(2)
Lipschitz domain      2.5(64) 4.4(191)
Lorentz space, norm      1.8(28)
Lorentz spaces      1.8(28) 2.10(96)
Lorentz, G.      Historical Notes 1(41)
Lorentz, G., at all points      3.9(152)
Lorentz, G., derivative      3.4(129)
Lorentz, G., difference quotients      2.1(46)
Lorentz, G., norm      1.5(18)
Lorentz, G., related to distributional derivatives      3.9(147)
Lorentz, G., spaces      1.5(18)
Lusin’s theorem for Sobolev functions      3.10(159) 3.11(166)
Marcinkiewicz interpolation theorem      4.7(199)
Marcinkiewicz space      1.8(28)
Maximal function      2.8(84)
Maximal function, fractional      4.7(204)
Maz’ya, V.      Historical Notes 2(109) Historical
Measure densities      3.2(116)
Measure, with finite K, p-energy      4.7(204)
Median, of a function      5.12(269)
Meyers — Serrin      Historical Notes 2(109)
Meyers — Ziemer      Historical Notes 4(218) Historical
Meyers, Norman      Historical Notes 4(217)
Michael — Ziemer      Historical Notes 3(176)
Minimax theorem      2.6(72) 5.12(268)
Minkowski content      2.7(83)
Minkowski’s inequality      1.5(20)
Mollification      1.6(21)
Morrey, C.B.      Historical Notes 2(109)
Morse — Sard theorem      2.7(81) 4.4(192)
Morse, A.P.      Historical Notes 1(40)
Moser, J.      2.9(91)
Multi-index      1.1(2)
Nirenberg, L.      Historical Notes 2(109)
Non-linear potential      4.7(205)
Non-tangential limits      3.9(147)
Norm of vector      1.1(1)
Open ball      1.1(2)
Outer measure      1.2(6)
Outer regularity of Bessel capacity      2.6(69)
O’Neil, R.      Historical Notes 1(41)
Partial derivative operators      1.1(2)
Partial derivative, as a measure      5.1(220)
Partition of unity      2.3(53)
Poincare inequality, abstract version      4.1(179)
Poincare inequality, and BV functions      5.11(261)
Poincare inequality, and measures in (BV)*      5.12(266)
Poincare inequality, and median of functions      5.13(269)
Poincare inequality, extended version with Bessel capacity      4.6(198)
Poincare inequality, general version for Sobolev functions      4.2(183)
Poincare inequality, indirect proof      4.1(177)
Poincare inequality, with Bessel capacity      4.5(195)
Poincare inequality, with Hausdorff measure      4.4(191)
Poincare inequality, with Lebesgue measure      4.4(189)
Polya — Szego      Historical Notes 1(41)
Projection mapping      4.1(178)
Rademacher, H.      Historical Notes 2(108)
Rademacher’s Theorem      2.2(50)
Rademacher’s theorem in $L^p$      3.8(145)
Radon      Historical Notes 1(39)
Radon measure      1—2(6)
Radon — Nikodym derivative      1.3(15)
Radon — Nikodym derivative, and differentiation of measures      5.5(234)
Rearrangement, non-increasing      1.8(26)
Reduced boundary      5 5(233)
Regularization      1.6(21)
Regularization of BV functions      5 3(224)
Regularization of Sobolev functions      2.1(43)
Rellich — Kondrachov Compactness Theorem      2.5(62)
Rellich — Kondrachov imbedding theorem      4.2(182)
Resctnjak, S.      Historical Notes 2(110)
Riesz capacity      2.6(66)
Riesz composition formula      2.8(88)
Riesz potential      2.6(64)
Riesz representation theorem      4.3(186)
Schwartz, L.      Historical Notes 1(41)
Sigma algebra      1.2(3)
Sobolev      Historical Notes 2(109)
Sobolev function      2.1(43)
Sobolev function, $L^p$ approximation by Taylor polynomial      3.4(127)
Sobolev function, approximate continuity of      3.3(122)
Sobolev function, fine continuity of      3.3(124)
Sobolev function, integral averages of      3.1(115)
Sobolev function, represented as a Bessel potential      4.5(194)
Sobolev inequality      2.4(56) 2.7(81)
Sobolev inequality, critical indices      2.9(89)
Sobolev inequality, critical indices in Lorentz spaces      2.10(98)
Sobolev inequality, for Riesz potentials      2.8(86)
Sobolev inequality, involving a general measure      4.7(199)
Sobolev inequality, relative to sub-manifolds      4.7(203)
Sobolev inequality, with general measure and p =1      4.9(209)
Steiner, J.      Historical Notes 1(41)
Strichartz, R.      Historical Notes 2(111)
Strong-type operator      4.7(199)
Suslinset      1.2(6)
Talenti, G.      2.7(82)
Taylor polynomial      3.4(126)
Thin sets      3.3(123)
Tonelli, L.      Historical Notes 5(280) Historical
Total differentiability, Lipschitz functions      2.2(50)
Total variation measure      5.1(221)
Total variation, for functions      2.7(76)
Totally bounded set      2.5(62)
Trace of a Sobolev function      4.4(190)
Trace of a Sobolev function, of BV function in terms of integral averages      5.14(276)
Trudinger, N.      Historical Notes 2(111)
Truncation of Sobolev functions      2.1(47)
Vitali, G.      Historical Notes 1(40)
Vol’pert, A.      Historical Notes 5(281)
Weak $L^P$      1.8(28)
Weak derivative      2.1(42)
Weak-type operator      4.7(199)
Whitney extension theorem      3.5(131)
Whitney extension theorem, $L^p$ version      3 6(141)
Whitney, H.      Historical Notes 1(40)
Young’s inequality      1.5(20)
Young’s inequalityin Lorentz spaces      2.10(96)
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