Quasiconformal Mappings and Sobolev Spaces (Mathematics and its Applications)

Gol'dshtein, V.M. M.; Reshetnyak, Yu.G.

ISBN 10: 9401073589 / ISBN 13: 9789401073585

Editore: Springer, 2011

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• Titolo: Quasiconformal Mappings and Sobolev Spaces (Mathematics and its Applications)
• Editore: Springer
• Data di pubblicazione: 2011
• Lingua: Inglese
• ISBN 10: 9401073589
• ISBN 13: 9789401073585
• Rilegatura: Brossura
• Condizione: New

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Riassunto

'Ht moi, ... , si j'avait su comment en revenir, One lemce mathematics has rendered the je n'y serai. point aile.' human race. It has put common sense back Jule. Verne ..... "'" it belong., on the topmost shelf next to the dusty caniller labelled 'discarded non- The series is divergent; therefore we may be sense'. able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non­ linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com­ puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'~re of this series.

Contenuti

1. Preliminary Information about Integration Theory.- §1. Notation and Terminology.- 1.1. Sets in Rn.- 1.2. Classes of Functions in Rn.- §2. Some Auxiliary Information about Sets and Functions in Rn.- 2.1. Averaging of Functions.- 2.2. The Whitney Partition Theorem.- 2.3. Partition of Unitiy.- §3. General Information about Measures and Integrals.- 3.1. Notion of a Measure.- 3.2. Decompositions in the Sense of Hahn and Jordan.- 3.3. The Radon-Nikodym Theorem and the Lebesgue Decomposition of Measure.- §4. Differentiation Theorems for Measures in Rn.- 4.1. Definitions.- 4.2. The Vitali Covering Lemma.- 4.3. The Lp-Continuity Theorem for Functions of the Class Lp,loc.- 4.4. The Differentiability Theorem for the Measure in Rn.- §5. Generalized Functions.- 5.1. Definition and Examples of Generalized Functions.- 5.2. Operations with Generalized Functions.- 5.3. Support of a Generalized Function. The Order of Singularity of a Generalized Function.- 5.4. The Generalized Function as a Derivative of the Usual Function. Averaging Operation.- 2. Functions with Generalized Derivatives.- §1. Sobolev-Type Integral Representations.- 1.1. Preliminary Remarks.- 1.2. Integral Representations in a Curvilinear Cone.- 1.3. Domains of the Class J.- 1.4. Integral Representations of Smooth Functions in Domains of the Class J.- §2. Other Integral Representations.- 2.1. Sobolev-Type Integral Representations for Simple Domains.- 2.2. Differential Operators with the Complete Integrability Condition.- 2.3. Integral Representations of a Function in Terms of a System of Differential Operators with the Complete Integrability Condition.- 2.4. Integral Representations for the Deformation Tensor and for the Tensor of Conformal Deformation.- §3. Estimates for Potential-Type Integrals.- 3.1. Preliminary Information.- 3.2. Lemma on the Compactness of Integral Operators.- 3.3. Basic Inequalities.- §4. Classes of Functions with Generalized Derivatives.- 4.1. Definition and the Simplest Properties.- 4.2. Integral Representations for Elements of the Space W?1,locl.- 4.3. The Imbedding Theorem.- 4.4. Corollaries of Theorem 4.2. Normalization of the Spaces Wpl(U).- 4.5. Approximation of Functions from Wpl by Smooth Functions.- 4.6. Change of Variables for Functions with Generalized Derivatives.- 4.7. Compactness of the Imbedding Operators.- 4.8. Estimates with a Small Coefficient for the Norm in Lpl.- 4.9. Functions of One Variable.- 4.10. Differential Description of Convex Functions.- 4.11. Functions Satisfying the Lipschitz Condition.- §5. Theorem on the Differentiability Almost Everywhere.- 5.1. Definitions.- 5.2. Auxiliary Propositions.- 5.3 The Main Result.- 5.4. Corollaries of the General Theorem on the Differentiability Almost Everywhere.- 5.5. The Behaviour of Functions of the Class Wpl on Almost All Planes of Smaller Dimensionality.- 5.6. The ACL-Classes.- 3. Nonlinear Capacity.- §1. Capacity Induced by a Linear Positive Operator.- 1.1. Definition and the Simplest Properties.- 1.2. Capacity as the Outer Measure.- 1.3. Sets of Zero Capacity.- 1.4. Extension of the Set of Admissible Functions.- 1.5. Extremal Function for Capacity.- 1.6. Comparison of Various Capacities.- §2. The Classes W(T, p, V).- 2.1. Definition of Classes.- 2.2. Theorems of Egorov and Luzin for Capacity.- 2.3. Dual (T, p)-Capacity, p > 1. Definition and Basic Properties.- 2.4. Calculation of Dual (T, p)-Capacity.- §3. Sets Measurable with Respect to Capacity.- 3.1. Definition and the Simplest Properties of Generalized Capacity.- 3.2. (T, p)-Capacity as Generalized Capacity.- §4. Variational Capacity.- 4.1. Definition of Variational Capacity.- 4.2. Comparison of Variational Capacity and (T, p)-Capacity.- 4.3. Sets of Zero Variational Capacity.- 4.4. Examples of Variational Capacity.- 4.5. Refined Functions.- 4.6. Theorems of Imbedding into the Space of Continuous Functions.- §5. Capacity in Sobolev Spaces.- 5.1. Three Types of Capacity.- 5.2. Extremal Functions for Capacity.- 5.3. Capacity and the Hausdorff h-Measure.- 5.4. Sufficient Conditions for the Vanishing of (l, p)-Capacity.- §6. Estimates of [l, p]-Capacity for Some Pairs of Sets.- 6.1. Estimates of Capacity for Spherical Domains.- 6.2. Estimates of Capacity for Pairs of Continuums Connecting Concentric Spheres.- §7. Capacity in Besov-Nickolsky Spaces.- 7.1. Preliminary Information.- 7.2. Capacities in bl,p,?,G,hl. Simplest Properties.- 7.3. Comparison of Capacity of a Pair of Points to Capacity of a Point Relative to a Complement of a Ball.- 7.4. Capacity of the Spherical Layer.- 4. Density of Extremal Functions in Sobolev Spaces with First Generalized Derivatives.- §1. Extremal Functions for (l, p)-Capacity.- 1.1. Simplest Properties of Extremal Functions.- 1.2. The Dirichlet Problem and Extremal Functions.- 1.3. Extremal Functions for Pairs of Smooth Compacts.- §2. Theorem on the Approximation of Functions from Lpl by Extremal Functions.- 2.1. Auxiliary Statements.- 2.2. The Class Extp(G).- 2.3. Proof of the Theorem on Approximation.- 2.4. Representation in Form of a Series.- §3. Removable Singularities for the Spaces Lpl (G).- 3.1. Two Ways of Describing Removable Singularities.- 3.2. Properties of NCp-Sets. Localization Principle.- 5. Change of Variables.- §1. Multiplicity of Mapping, Degree of Mapping, and Their Analogies.- 1.1. The Multiplicity Function of Mapping.- 1.2. The Approximate Differential.- 1.3. The K-Differential.- 1.4. The Change of Variable Theorem for the Multiplicity Function.- 1.5. The Degree of Mapping.- §2. The Change of Variable in the Integral for Mappings of Sobolev Spaces.- 2.1. The Change of Variable Theorem for Continuous Mappings of the Class Lnl.- 2.2. The Linking Index.- 2.3. The Change of Variable Theorem for Discontinuous Mappings of the Class Lnl.- §3. Sufficient Conditions of Monotonicity and Continuity for the Approximation Functions of the Class Lnl.- §4. Invariance of the Spaces Lpl(G)(Lnl(G)) for Quasiisometric (Quasiconformal) Homeomorphisms.- 4.1. Preliminary Information on the Mappings.- 4.2. Differentiation of Composition.- 4.3. Representation of Operators Preserving the Order.- 6. Extension of Differentiate Functions.- §1. Arc Diameter Condition.- 1.1. Analysis of the Ahlfors Condition.- 1.2. The Arc Diameter Condition.- 1.3. Properties of Domains Satisfying the Arc Diameter Condition.- §2. Necessary Extension Conditions for Seminormed Spaces.- 2.1. The Extension Operator. Capacitary Extension Condition.- 2.2. Additional Properties of Capacity.- 2.3. The Invisibility Condition.- 2.4. The Extension Theorem.- 2.5. Verification of the Conditions of the Theorem for the Spaces Lpl (G), Wpl(G).- §3. Necessary Extension Conditions for Sobolev Spaces.- 3.1. Necessary Extension Conditions for Lpl, Wpl at lp=n.- 3.2. Necessary Conditions for Lpl, Wpl at 1 ? lp ? 2 in Plane Domains.- 3.3 Necessary Conditions Different from the Arc Diameter Condition.- 3.4. Refinement for the Space Wpl.- §4. Necessary Extension Conditions for Besov and Nickolsky Spaces.- 4.1. Extension Theorem for lp > n.- 4.2. Extension Conditions for lp = n.- §5. Sufficient Extension Conditions.- 5.1. Quasiconformal Extension.- 5.2. Extension Conditions for Sobolev Classes.- 5.3. Example of Estimating the Norm of an Extension Operator.- 5.4. The Extension Condition for Nickolsky-Besov Spaces.- Comments.- References.