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Aspects of Sobolev-type inequalitiesLaurent Saloff-Coste.

By: Material type: TextTextSeries: Publication details: Cambridge ; New York : Cambridge University Press, (c)2002.Description: 1 online resource (x, 190 pages) : illustrationsContent type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9781107360747
Subject(s): Genre/Form: LOC classification:
  • QA323 .A874 2002
Online resources: Available additional physical forms:
Contents:
Sobolev inequalities in R[superscript n] -- Sobolev inequalities -- The proof due to Gagliardo and to Nirenberg -- p = 1 implies p [greater than or equal] 1 -- Riesz potentials -- Another approach to Sobolev inequalities -- Marcinkiewicz interpolation theorem -- Proof of Sobolev Theorem 1.2.1 -- Best constants -- The case p = 1: isoperimetry -- A complete proof with best constant for p = 1 -- The case p > 1 -- Some other Sobolev inequalities -- The case p > n -- The case p = n -- Higher derivatives -- Sobolev -- Poincare inequalities on balls -- The Neumann and Dirichlet eigenvalues -- Poincare inequalities on Euclidean balls -- Sobolev -- Poincare inequalities -- Moser's elliptic Harnack inequality -- Elliptic operators in divergence form -- Divergence form -- Uniform ellipticity -- A Sobolev-type inequality for Moser's iteration -- Subsolutions and supersolutions -- Subsolutions -- Supersolutions -- An abstract lemma -- Harnack inequalities and continuity -- Harnack inequalities -- Holder continuity -- Sobolev inequalities on manifolds -- Notation concerning Riemannian manifolds -- Isoperimetry -- Sobolev inequalities and volume growth -- Weak and strong Sobolev inequalities -- Examples of weak Sobolev inequalities -- (S[superscript [theta] subscript r,s])-inequalities: the parameters q and v -- The case 0 < q < [infinity] -- The case 1 = [infinity] -- The case -[infinity] < q < 0 -- Increasing p -- Local versions -- Pseudo-Poincare inequalities -- Pseudo-Poincare technique: local version -- Lie groups -- Pseudo-Poincare inequalities on Lie groups -- Ricci [greater than or equal] 0 and maximal volume growth -- Sobolev inequality in precompact regions -- Two applications -- Ultracontractivity -- Nash inequality implies ultracontractivity -- The converse -- Gaussian heat kernel estimates -- The Gaffney-Davies L[superscript 2. estimate -- Complex interpolation -- Pointwise Gaussian upper bounds -- On-diagonal lower bounds -- The Rozenblum-Lieb-Cwikel inequality -- The Schrodinger operator [Delta] -- V -- The operator T[subscript V] = [Delta superscript -1.V -- The Birman-Schwinger principle -- Parabolic Harnack inequalities -- Scale-invariant Harnack principle -- Local Sobolev inequalities -- Local Sobolev inequalities and volume growth -- Mean value inequalities for subsolutions -- Localized heat kernel upper bounds -- Time-derivative upper bounds -- Mean value inequalities for supersolutions -- Poincare inequalities -- Poincare inequality and Sobolev inequality -- Some weighted Poincare inequalities -- Whitney-type coverings -- A maximal inequality and an application -- End of the proof of Theorem 5.3.4 -- Harnack inequalities and applications -- An inequality for log u -- Harnack inequality for positive supersolutions -- Harnack inequalities for positive solutions -- Holder continuity -- Liouville theorems -- Heat kernel lower bounds -- Two-sided heat kernel bounds -- The parabolic Harnack principle -- Poincare, doubling, and Harnack -- Stochastic completeness -- Local Sobolev inequalities and the heat equation -- Selected applications of Theorem 5.5.1 -- Unimodular Lie groups -- Homogeneous spaces -- Manifolds with Ricci curvature bounded below
Subject: 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.
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Item type Current library Collection Call number URL Status Date due Barcode
Online Book (LOGIN USING YOUR MY CIU LOGIN AND PASSWORD) Online Book (LOGIN USING YOUR MY CIU LOGIN AND PASSWORD) G. Allen Fleece Library ONLINE Non-fiction QA323 (Browse shelf(Opens below)) Link to resource Available ocn839304256

Includes bibliographies and index.

Sobolev inequalities in R[superscript n] -- Sobolev inequalities -- The proof due to Gagliardo and to Nirenberg -- p = 1 implies p [greater than or equal] 1 -- Riesz potentials -- Another approach to Sobolev inequalities -- Marcinkiewicz interpolation theorem -- Proof of Sobolev Theorem 1.2.1 -- Best constants -- The case p = 1: isoperimetry -- A complete proof with best constant for p = 1 -- The case p > 1 -- Some other Sobolev inequalities -- The case p > n -- The case p = n -- Higher derivatives -- Sobolev -- Poincare inequalities on balls -- The Neumann and Dirichlet eigenvalues -- Poincare inequalities on Euclidean balls -- Sobolev -- Poincare inequalities -- Moser's elliptic Harnack inequality -- Elliptic operators in divergence form -- Divergence form -- Uniform ellipticity -- A Sobolev-type inequality for Moser's iteration -- Subsolutions and supersolutions -- Subsolutions -- Supersolutions -- An abstract lemma -- Harnack inequalities and continuity -- Harnack inequalities -- Holder continuity -- Sobolev inequalities on manifolds -- Notation concerning Riemannian manifolds -- Isoperimetry -- Sobolev inequalities and volume growth -- Weak and strong Sobolev inequalities -- Examples of weak Sobolev inequalities -- (S[superscript [theta] subscript r,s])-inequalities: the parameters q and v -- The case 0 < q < [infinity] -- The case 1 = [infinity] -- The case -[infinity] < q < 0 -- Increasing p -- Local versions -- Pseudo-Poincare inequalities -- Pseudo-Poincare technique: local version -- Lie groups -- Pseudo-Poincare inequalities on Lie groups -- Ricci [greater than or equal] 0 and maximal volume growth -- Sobolev inequality in precompact regions -- Two applications -- Ultracontractivity -- Nash inequality implies ultracontractivity -- The converse -- Gaussian heat kernel estimates -- The Gaffney-Davies L[superscript 2. estimate -- Complex interpolation -- Pointwise Gaussian upper bounds -- On-diagonal lower bounds -- The Rozenblum-Lieb-Cwikel inequality -- The Schrodinger operator [Delta] -- V -- The operator T[subscript V] = [Delta superscript -1.V -- The Birman-Schwinger principle -- Parabolic Harnack inequalities -- Scale-invariant Harnack principle -- Local Sobolev inequalities -- Local Sobolev inequalities and volume growth -- Mean value inequalities for subsolutions -- Localized heat kernel upper bounds -- Time-derivative upper bounds -- Mean value inequalities for supersolutions -- Poincare inequalities -- Poincare inequality and Sobolev inequality -- Some weighted Poincare inequalities -- Whitney-type coverings -- A maximal inequality and an application -- End of the proof of Theorem 5.3.4 -- Harnack inequalities and applications -- An inequality for log u -- Harnack inequality for positive supersolutions -- Harnack inequalities for positive solutions -- Holder continuity -- Liouville theorems -- Heat kernel lower bounds -- Two-sided heat kernel bounds -- The parabolic Harnack principle -- Poincare, doubling, and Harnack -- Stochastic completeness -- Local Sobolev inequalities and the heat equation -- Selected applications of Theorem 5.5.1 -- Unimodular Lie groups -- Homogeneous spaces -- Manifolds with Ricci curvature bounded below

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.

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