Aspects of Sobolev-type inequalities /

Focusing on Poincaré, Nash and other Sobolev-type inequalities and their applications to the Laplace and heat diffusion equations on Riemannian manifolds, this text is an advanced graduate book that will also suit researchers.

Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Saloff-Coste, L.
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Cambridge ; New York : Cambridge University Press, 2002.
Colección:London Mathematical Society lecture note series ; 289.
Temas:
Sobolev spaces.
Inequalities (Mathematics)
Espaces de Sobolev.
Inégalités (Mathématiques)
MATHEMATICS > Functional Analysis.
Sobolev spaces
Sobolevsche Ungleichung
Sobolev ruimten.
Vergelijkingen (wiskunde)
Desigualdades (análise matemática)
Acceso en línea:Texto completo
Tabla de Contenidos:
  • 1 Sobolev inequalities in R[superscript n] 7
  • 1.1 Sobolev inequalities 7
  • 1.1.2 The proof due to Gagliardo and to Nirenberg 9
  • 1.1.3 p = 1 implies p [greater than or equal] 1 10
  • 1.2 Riesz potentials 11
  • 1.2.1 Another approach to Sobolev inequalities 11
  • 1.2.2 Marcinkiewicz interpolation theorem 13
  • 1.2.3 Proof of Sobolev Theorem 1.2.1 16
  • 1.3 Best constants 16
  • 1.3.1 The case p = 1: isoperimetry 16
  • 1.3.2 A complete proof with best constant for p = 1 18
  • 1.3.3 The case p> 1 20
  • 1.4 Some other Sobolev inequalities 21
  • 1.4.1 The case p> n 21
  • 1.4.2 The case p = n 24
  • 1.4.3 Higher derivatives 26
  • 1.5 Sobolev
  • Poincare inequalities on balls 29
  • 1.5.1 The Neumann and Dirichlet eigenvalues 29
  • 1.5.2 Poincare inequalities on Euclidean balls 30
  • 1.5.3 Sobolev
  • Poincare inequalities 31
  • 2 Moser's elliptic Harnack inequality 33
  • 2.1 Elliptic operators in divergence form 33
  • 2.1.1 Divergence form 33
  • 2.1.2 Uniform ellipticity 34
  • 2.1.3 A Sobolev-type inequality for Moser's iteration 37
  • 2.2 Subsolutions and supersolutions 38
  • 2.2.1 Subsolutions 38
  • 2.2.2 Supersolutions 43
  • 2.2.3 An abstract lemma 47
  • 2.3 Harnack inequalities and continuity 49
  • 2.3.1 Harnack inequalities 49
  • 2.3.2 Holder continuity 50
  • 3 Sobolev inequalities on manifolds 53
  • 3.1.1 Notation concerning Riemannian manifolds 53
  • 3.1.2 Isoperimetry 55
  • 3.1.3 Sobolev inequalities and volume growth 57
  • 3.2 Weak and strong Sobolev inequalities 60
  • 3.2.1 Examples of weak Sobolev inequalities 60
  • 3.2.2 (S[superscript [theta] subscript r, s])-inequalities: the parameters q and v 61
  • 3.2.3 The case 0 <q <[infinity] 63
  • 3.2.4 The case 1 = [infinity] 66
  • 3.2.5 The case -[infinity] <q <0 68
  • 3.2.6 Increasing p 70
  • 3.2.7 Local versions 72
  • 3.3.1 Pseudo-Poincare inequalities 73
  • 3.3.2 Pseudo-Poincare technique: local version 75
  • 3.3.3 Lie groups 77
  • 3.3.4 Pseudo-Poincare inequalities on Lie groups 79
  • 3.3.5 Ricci [greater than or equal] 0 and maximal volume growth 82
  • 3.3.6 Sobolev inequality in precompact regions 85
  • 4 Two applications 87
  • 4.1 Ultracontractivity 87
  • 4.1.1 Nash inequality implies ultracontractivity 87
  • 4.1.2 The converse 91
  • 4.2 Gaussian heat kernel estimates 93
  • 4.2.1 The Gaffney-Davies L[superscript 2] estimate 93
  • 4.2.2 Complex interpolation 95
  • 4.2.3 Pointwise Gaussian upper bounds 98
  • 4.2.4 On-diagonal lower bounds 99
  • 4.3 The Rozenblum-Lieb-Cwikel inequality 103
  • 4.3.1 The Schrodinger operator [Delta]
  • V 103
  • 4.3.2 The operator T[subscript V] = [Delta superscript -1]V 105
  • 4.3.3 The Birman-Schwinger principle 109
  • 5 Parabolic Harnack inequalities 111
  • 5.1 Scale-invariant Harnack principle 111
  • 5.2 Local Sobolev inequalities 113
  • 5.2.1 Local Sobolev inequalities and volume growth 113
  • 5.2.2 Mean value inequalities for subsolutions 119
  • 5.2.3 Localized heat kernel upper bounds 122
  • 5.2.4 Time-derivative upper bounds 127
  • 5.2.5 Mean value inequalities for supersolutions 128
  • 5.3 Poincare inequalities 130
  • 5.3.1 Poincare inequality and Sobolev inequality 131
  • 5.3.2 Some weighted Poincare inequalities 133
  • 5.3.3 Whitney-type coverings 135
  • 5.3.4 A maximal inequality and an application 139
  • 5.3.5 End of the proof of Theorem 5.3.4 141
  • 5.4 Harnack inequalities and applications 143
  • 5.4.1 An inequality for log u 143
  • 5.4.2 Harnack inequality for positive supersolutions 145
  • 5.4.3 Harnack inequalities for positive solutions 146
  • 5.4.4 Holder continuity 149
  • 5.4.5 Liouville theorems 151
  • 5.4.6 Heat kernel lower bounds 152
  • 5.4.7 Two-sided heat kernel bounds 154
  • 5.5 The parabolic Harnack principle 155
  • 5.5.1 Poincare, doubling, and Harnack 157
  • 5.5.2 Stochastic completeness 161
  • 5.5.3 Local Sobolev inequalities and the heat equation 164
  • 5.5.4 Selected applications of Theorem 5.5.1 168
  • 5.6.1 Unimodular Lie groups 172
  • 5.6.2 Homogeneous spaces 175
  • 5.6.3 Manifolds with Ricci curvature bounded below 176.

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