.

0 (hodnocen0 x )

BK

2nd., rev. a augm. ed.

Heidelberg : Springer, c2011

xxviii, 866 s. ; 24 cm

ISBN 978-3-642-15563-5 (váz.)

Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen : : a series of comprehensive studies in mathematics ; 342

Obsahuje bibliografické odkazy a rejstřík

000196434

Short contents: 1 Basic Properties of Sobolev Spaces 1 // 1.1 The Spaces Llp(Ω), Vlp(Ω) and Wlp(Ω) 1 // 1.2 Facts from Set Theory and Function Theory 32 // 1.3 Some Inequalities for Functions of One Variable 40 // 1.4 Embedding Theorems of Sobolev Type 63 // 1.5 More on Extension of Functions in Sobolev Spaces 87 // 1.6 Inequalities for Functions with Zero Incomplete Cauchy Data 99 // 1.7 Density of Bounded Functions in Sobolev Spaces 107 // 1.8 Maximal Algebra in Wlp(Ω) 117 // 2 Inequalities for Functions Vanishing at the Boundary 123 // 2.1 Conditions for Validity of Integral Inequalities (the Case p = 1) 124 // 2.2 (p,Φ)-Capacity 141 // 2.3 Conditions for Validity of Integral Inequalities (the Case p>1) 152 // / 2.4 Continuity and Compactness of Embedding Operators of L1p(Ω) and W1p(Ω) into Birnbaum-Orlicz Spaces 179 // 2.5 Structure of the Negative Spectrum of the Multidimensional Schrödinger Operator 188 // 2.6 Properties of Sobolev Spaces Generated by Quadratic Forms with Variable Coefficients 205 // 2.7 Dilation Invariant Sharp Hardy’s Inequalities 213 // 2.8 Sharp Hardy-Leray Inequality for Axisymmetric Divergence-Free Fields 220 // 3 Conductor and Capacitary Inequalities with Applications to Sobolev-Type Embeddings 231 // 3.1 Introduction 231 // 3.2 Comparison of Inequalities (3.1.4) and (3.1.5) 233 // 3.3 Conductor Inequality (3.1.1) 234 // 3.4 Applications of the Conductor Inequality (3.1.1) 236 // 3.5 p-Capacity Depending on v and Its Applications to a Conductor Inequality and Inequality (3.4.1) 241 // 3.6 Compactness and Essential Norm 243 // 3.7 Inequality (3.1.10) with Integer I > 2 245 // 3.8 Two-Weight Inequalities Involving Fractional Sobolev Norms 249 // 3.9 Comments to Chap. 3 252 // 4 Generalizations for Functions on Manifolds and Topological Spaces 255 // 4.1 Introduction 255 //

4.2 Integral Inequalities for Functions on Riemannian Manifolds 257 // 4.3 The First Dirichlet-Laplace Eigenvalue and Isoperimetric Constant 261 // 4.4 Conductor Inequalities for a Dirichlet-Type Integral with a Locality Property 265 // 4.5 Conductor Inequality for a Dirichlet-Type Integral without Locality Conditions 270 // 4.6 Sharp Capacitary Inequalities and Their Applications 273 // 4.7 Capacitary Improvement of the Faber-Krahn Inequality 278 // 4.8 Two-Weight Sobolev Inequality with Sharp Constant 282 // 4.9 Comments to Chap. 4 286 // 5 Integrability of Functions in the Space 287 // 5.1 Preliminaries 288 // 5.2 Classes of Sets ?α, ?α and the Embedding L11(Ω) ⊂ Lq(Ω) 290 // 5.3 Subareal Mappings and the Classes ?α and ?α 300 // 5.4 Two-Sided Estimates for the Function Л for the Domain in Nikodým’s Example 308 // 5.5 Compactness of the Embedding L11(Ω) ⊂ Lq(Ω) (q > 1) 311 // 5.6 Embedding W11,R(Ω,δΩ) ⊂ Lq(Ω) 314 // 5.7 Comments to Chap. 5 319 // 6 Integrability of Functions in the Space L1p (Ω) 323 // 6.1 Conductivity 324 // 6.2 Multiplicative Inequality for Functions Which Vanish on a Subset off? 329 // 6.3 Classes of Sets ?p,α 331 // 6.4 Embedding W1 p,s(Ω) ⊂ Lq.(Ω) for q* < p 341 // 6.5 More on the Nikodým Example 352 // 6.6 Some Generalizations 360 // 6.7 Inclusion W1 p,r(Ω) ⊂ Lq{Ω) (r > q) for Domains with Infinite Volume 364 // 6.8 Compactness of the Embedding L1р(Ω) ⊂ Lq(Ω) 374 // 6.9 Embedding L1p(Ω) ⊂ Lq(Ω) 379 // 6.10 Applications to the Neumann Problem for Strongly Elliptic Operators 380 // 6.11 Inequalities Containing Integrals over the Boundary 390 // 6.12 Comments to Chap. 6 401 // 7 Continuity and Boundedness of Functions in Sobolev Spaces 405 // 7.1 The Embedding W1p(Ω) ⊂ C(Ω) ∩ L∞(Ω) 406 // 7.2 Multiplicative Estimate for Modulus of a Function in W1p(Ω) 412 //

7.3 Continuity Modulus of Functions in L1p(Ω) 416 // 7.4 Boundedness of Functions with Derivatives in Birnbaum-Orlicz Spaces 419 // 7.5 Compactness of the Embedding Wp1(Ω) ⊂ C(Ω) ∩ L∞(Ω) 422 // 7.6 Generalizations to Sobolev Spaces of an Arbitrary Integer Order 426 // 7.7 Comments to Chap. 7 434 // 8 Localization Moduli of Sobolev Embeddings for General Domains 435 // 8.1 Localization Moduli and Their Properties 437 // 8.2 Counterexample for the Case p = q 442 // 8.3 Critical Sobolev Exponent 444 // 8.4 Generalization 446 // 8.5 Measures of Noncompactness for Power Cusp-Shaped Domains 447 // 8.6 Finiteness of the Negative Spectrum of a Schrödinger Operator on β-Cusp Domains 452 // 8.7 Relations of Measures of Noncompactness with Local Isoconductivity and Isoperimetric Constants 456 // 8.8 Comments to Chap. 8 457 // 9 Space of Functions of Bounded Variation 459 // 9.1 Properties of the Set Perimeter and Functions in BV(Ω) 459 // 9.2 Gauss-Green Formula for Lipschitz Functions 467 // 9.3 Extension of Functions in BV(Ω) onto Rn 477 // 9.4 Exact Constants for Certain Convex Domains 484 // 9.5 Rough Trace of Functions in BV(Ω) and Certain Integral Inequalities 489 // 9.6 Traces of Functions in BV(Ω) on the Boundary and Gauss-Green Formula 500 // 9.7 Comments to Chap. 9 507 // 10 Certain Function Spaces, Capacities, and Potentials 511 // 10.1 Spaces of Functions Differentiable of Arbitrary Positive Order 512 // 10.2 Bourgain, Brezis, and Mironescu Theorem Concerning Limiting Embeddings of Fractional Sobolev Spaces 521 // 10.3 On the Brezis and Mironescu Conjecture Concerning a Gagliardo-Nirenberg Inequality for Fractional Sobolev Norms 530 // 10.4 Some Facts from Nonlinear Potential Theory 536 // 10.5 Comments to Chap. 10 545 // 11 Capacitary and Trace Inequalities for Functions in Rn with Derivatives of an Arbitrary Order 549 //

11.1 Description of Results 549 // 11.2 Capacitary Inequality of an Arbitrary Order 552 // 11.3 Conditions for the Validity of Embedding Theorems in Terms of Isocapacitary Inequalities 556 // 11.4 Counterexample to the Capacitary Inequality for the Norm in L22(Ω) 558 // 11.5 Ball and Pointwise Criteria 564 // 11.6 Conditions for Embedding into Lq(μ) for p > q > 0 570 // 11.7 Cartan-Type Theorem and Estimates for Capacities 575 // 11.8 Embedding Theorems for the Space Sp (Conditions in Terms of Balls, p > 1) 579 // 11.9 Applications 582 // 11.10 Embedding Theorems for p = 1 588 // 11.11 Criteria for an Upper Estimate of a Difference Seminorm (the Case p > 1) 597 // 12 Pointwise Interpolation Inequalities for Derivatives and Potentials 611 // 12.1 Pointwise Interpolation Inequalities for Riesz and Bessel Potentials 612 // 12.2 Sharp Pointwise Inequalities for ?u 624 // 12.3 Pointwise Interpolation Inequalities Involving “Fractional Derivatives” 638 // 12.4 Application of (12.3.11) to Composition Operator in Fractional Sobolev Spaces 643 // 12.5 Comments to Chap. 12 653 // 13 A Variant of Capacity 657 // 13.1 Capacity Cap 657 // 13.2 On (p, l)-Polar Sets 663 // 13.3 Equivalence of Two Capacities 664 // 13.4 Removable Singularities of l-Harmonic Functions in Lm2 666 // 13.5 Comments to Chap. 13 668 // 14 Integral Inequality for Functions on a Cube 669 // 14.1 Connection Between the Best Constant and Capacity (Case k = 1) 670 // 14.1.1 Definition of a (p, Z)-Negligible Set 670 // 14.2 Connection Between Best Constant and the (p,l)-Inner Diameter (Case k = 1) 675 // 14.3 Estimates for the Best Constant С in the General Case 679 // 14.4 Comments to Chap. 14 691 // 15 Embedding of the Space Llp(Ω) into Other Function Spaces 693 // 15.1 Preliminaries 693 // 15.2 Embedding Llp(Ω) ⊂ D’(Ω) 694 // 15.3 Embedding Llp(Ω) ⊂ Lq(Ω,loс) 701 //

15.4 Embedding Llp(Ω) ⊂ Lq(Ω) (the Case p < q) 703 // 15.5 Embedding Llp(Ω) ⊂ Lq(Ω) (the Case p > q > 1) 707 // 15.6 Compactness of the Embedding Llp(Ω) ⊂ Lq(Ω) 714 // 15.7 Application to the Dirichlet Problem for a Strongly Elliptic Operator 716 // 15.8 Applications to the Theory of Quasilinear Elliptic Equations 721 // 15.9 Comments to Chap. 15 733 // 16 Embedding Llp(Ω,νu) ⊂ Wmr(Ω) 737 // 16.1 Auxiliary Assertions 737 // // 16.2 Continuity of the Embedding Operator Llp(Ω) -> Wrm(Ω) 739 // 16.3 Compactness of the Embedding Operator Llp(Ω,v) -> Wmr(Ω) 742 // 16.4 Closability of Embedding Operators 746 // 16.5 Application: Positive Definiteness and Discreteness of the Spectrum of a Strongly Elliptic Operator 749 // 16.6 Comments to Chap. 16 751 // 17 Approximation in Weighted Sobolev Spaces 755 // 17.1 Main Results and Applications 755 // 17.2 Capacities 757 // 17.3 Applications of Lemma 17.2/3 761 // 17.4 Proof of Theorem 17.1 765 // 17.5 Comments to Chap. 17 768 // 18 Spectrum of the Schrödinger Operator and the Dirichlet Laplacian 769 // 18.1 Main Results on the Schrödinger Operator 770 // 18.2 Discreteness of Spectrum: Necessity 773 // 18.3 Discreteness of Spectrum: Sufficiency 781 // 18.4 A Sufficiency Example 783 // 18.5 Positivity of Ну 787 // 18.6 Structure of the Essential Spectrum of Ну 787 // 18.7 Two-Sided Estimates of the First Eigenvalue of the Dirichlet Laplacian 789 // References 803 // List of Symbols 849 // Subject Index 853 // Author Index 859