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A first course in Sobolev spaces:

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Bibliographische Detailangaben
Beteilige Person:	Leoni, Giovanni 1967- (VerfasserIn)
Format:	Buch
Sprache:	Englisch
Veröffentlicht:	Providence, RI American Math. Soc. 2009
Schriftenreihe:	Graduate studies in mathematics 105
Schlagwörter:	Sobolev-Raum
Links:	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018600771&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018600771&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA
Umfang:	XVI, 607 S. graph. Darst.
ISBN:	9780821847688

Internformat

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Datensatz im Suchindex

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adam_text	Titel: A first course in Sobolev spaces Autor: Leoni, Giovanni Jahr: 2009 Contents Preface ix Acknowledgments xv Part 1. Functions of One Variable Chapter 1. Monotone Functions 3 §1.1. Continuity 3 §1.2. Differentiability 8 Chapter 2. Functions of Bounded Pointwise Variation 39 §2.1. Pointwise Variation 39 §2.2. Composition in BPV (I) 55 §2.3. The Space BPV (I) 59 §2.4. Banach Indicatrix 66 Chapter 3. Absolutely Continuous Functions 73 §3.1. AC (I) Versus BPV (I) 73 §3.2. Chain Rule and Change of Variables 94 §3.3. Singular Functions 107 Chapter 4. Curves 115 §4.1. Rectifiable Curves and Arclength 115 §4.2. Frechet Curves 130 §4.3. Curves and Hausdorff Measure 134 §4.4. Jordan s Curve Theorem 146 v Chapter 5. Lebesgue-Stieltjes Measures 155 §5.1. Radon Measures Versus Increasing Functions 155 §5.2. Signed Borel Measures Versus BPV (I) 161 §5.3. Decomposition of Measures 166 §5.4. Integration by Parts and Change of Variables 181 Chapter 6. Decreasing Rearrangement 187 §6.1. Definition and First Properties 187 §6.2. Absolute Continuity of u* 202 §6.3. Derivative of u* 209 Chapter 7. Functions of Bounded Variation and Sobolev Functions 215 §7.1. BV (ft) Versus BPV (ft) 215 §7.2. Sobolev Functions Versus Absolutely Continuous Functions 222 Part 2. Functions of Several Variables Chapter 8. Absolutely Continuous Functions and Change of Variables 231 §8.1. The Euclidean Space RN 231 §8.2. Absolutely Continuous Functions of Several Variables 234 §8.3. Change of Variables for Multiple Integrals 242 Chapter 9. Distributions 255 §9.1. The Spaces VK (ft), V (ft), and V (ft) 255 §9.2. Order of a Distribution 264 §9.3. Derivatives of Distributions and Distributions as Derivatives 266 §9.4. Convolutions 275 Chapter 10. Sobolev Spaces 279 §10.1. Definition and Main Properties 279 §10.2. Density of Smooth Functions 283 §10.3. Absolute Continuity on Lines 293 §10.4. Duals and Weak Convergence 298 §10.5. A Characterization of Wl p (ft) 305 Chapter 11. Sobolev Spaces: Embeddings 311 §11.1. Embeddings: l p N 312 §11.2. Embeddings: p = N 328 §11.3. Embeddings: p N 335 §11.4. Lipschitz Functions 341 Chapter 12. Sobolev Spaces: Further Properties 349 §12.1. Extension Domains 349 §12.2. Poincare Inequalities 359 Chapter 13. Functions of Bounded Variation 377 §13.1. Definition and Main Properties 377 §13.2. Approximation by Smooth Functions 380 §13.3. Bounded Pointwise Variation on Lines 386 §13.4. Coarea Formula for BV Functions 397 §13.5. Embeddings and Isoperimetric Inequalities 401 §13.6. Density of Smooth Sets 408 §13.7. A Characterization of BV (ST) 413 Chapter 14. Besov Spaces 415 §14.1. Besov Spaces Bs p 9, 0 s 1 415 §14.2. Dependence of BS P 6 on s 419 §14.3. The Limit of Bs P e as s -* 0+ and s - 1 421 §14.4. Dependence of B8* 0 on 6 425 §14.5. Dependence of Bs p e on s and p 429 §14.6. Embedding of B3* 9 into L9 437 §14.7. Embedding of W1* into B^ 442 §14.8. Besov Spaces and Fractional Sobolev Spaces 448 Chapter 15. Sobolev Spaces: Traces 451 §15.1. Traces of Functions in W1 1 (ft) 451 §15.2. Traces of Functions in BV (Ct) 464 §15.3. Traces of Functions in W1* (ft), p 465 §15.4. A Characterization of WollP (fi) in Terms of Traces 475 Chapter 16. Sobolev Spaces: Symmetrization 477 §16.1. Symmetrization in Lv Spaces 477 §16.2. Symmetrization of Lipschitz Functions 482 §16.3. Symmetrization of Piecewise Affine Functions 484 §16.4. Symmetrization in Wl* and BV 487 Appendix A. Functional Analysis 493 §A.l. Metric Spaces 493 §A.2. Topological Spaces 494 §A.3. Topological Vector Spaces 497 §A.4. Normed Spaces 501 §A.5. Weak Topologies 503 §A.6. Hilbert Spaces 506 Appendix B. Measures 507 §B.l. Outer Measures and Measures 507 §B.2. Measurable and Integrable Functions 511 §B.3. Integrals Depending on a Parameter 519 §B.4. Product Spaces 520 §B.5. Radon-Nikodym s and Lebesgue s Decomposition Theorems 522 §B.6. Signed Measures 523 §B.7. IP Spaces 526 §B.8. Modes of Convergence 534 §B.9. Radon Measures 536 §B.10. Covering Theorems in RN 538 Appendix C. The Lebesgue and Hausdorff Measures 543 §C.l. The Lebesgue Measure 543 §C2. The Brunn-Minkowski Inequality and Its Applications 545 §C3. Convolutions 550 §C.4. Mollifiers 552 §C5. Differentiable Functions on Arbitrary Sets 560 §C6. Maximal Functions 564 §C.7. Anisotropic IP Spaces 568 §C8. Hausdorff Measures 572 Appendix D. Notes 581 Appendix E. Notation and List of Symbols 587 Bibliography 593 Index 603 Sobolev spaces are a fundamental tool in the modern study of partial differential equations. In this book, Leoni takes a novel approach to the theory by looking at Sobolev spaces as the natural development of monotone, absolutely continuous, and BV functions of one variable. In this way, the majority of the text can be read without the prerequisite of a course in functional analysis. The first part of this text is devoted to studying functions of one variable. Several of the topics treated occur in courses on real analysis or measure theory. Here, the perspective emphasizes their applications to Sobolev functions, giving a very different flavor to the treatment This elementary start to the book makes it suitable for advanced undergraduates or beginning graduate students. Moreover, the one-variable part of the book helps to develop a solid background that facilitates the reading and understanding of Sobolev functions of several vari¬ ables. The second part of the book is more classical, although it also contains some recent results. Besides the standard results on Sobolev functions, this part of the book includes chapters on BV functions, symmetric rearrangement, and Besov spaces. The book contains over 200 exercises.
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spellingShingle	Leoni, Giovanni 1967- A first course in Sobolev spaces Graduate studies in mathematics Sobolev-Raum (DE-588)4055345-0 gnd
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title	A first course in Sobolev spaces
title_auth	A first course in Sobolev spaces
title_exact_search	A first course in Sobolev spaces
title_full	A first course in Sobolev spaces Giovanni Leoni
title_fullStr	A first course in Sobolev spaces Giovanni Leoni
title_full_unstemmed	A first course in Sobolev spaces Giovanni Leoni
title_short	A first course in Sobolev spaces
title_sort	a first course in sobolev spaces
topic	Sobolev-Raum (DE-588)4055345-0 gnd
topic_facet	Sobolev-Raum
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