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9789810227678

Differentiable Functions on "Bad" Domains

by ; ;
  • ISBN13:

    9789810227678

  • ISBN10:

    9810227671

  • Format: Hardcover
  • Copyright: 1998-04-01
  • Publisher: World Scientific Pub Co Inc
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Table of Contents

I. INTRODUCTION TO SOBOLEV SPACES FOR DOMAINS 1(142)
1. Basic Properties of Sobolev Spaces
1(88)
1.1. Preliminaries
1(6)
1.1.1. Notation
1(2)
1.1.2. The Space L(p) and Integral Inequalities
3(4)
1.2. Functions with Generalized Derivatives
7(10)
1.2.1. Mollification
7(2)
1.2.2. Generalized Derivatives
9(3)
1.2.3. The Spaces (XXX)
12(2)
1.2.4. Absolute Continuity of Functions in L(p)^1(XXX)
14(2)
1.2.5. On Removable Singularities for Functions in V(p)^l(XXX)
16(1)
1.3. Classes of Domains
17(7)
1.3.1. Domains of Class C and Domains Having the Segment Property
17(3)
1.3.2. Domains Starshaped with Respect to a Ball and Domains of Class C^(0,1)
20(2)
1.3.3. Domains Having the Cone Property
22(1)
1.3.4. Domains of Class C^(0,1) and Lipschitz Domains
23(1)
1.4. Density of Smooth Functions in Sobolev Spaces
24(7)
1.4.1. Approximation of Functions in Sobolev Spaces by Functions in C(Infinity)(XXX)
24(3)
1.4.2. Approximation by Functions in C^(Infinity)(XXX)
27(2)
1.4.3. Density of Bounded Smooth Functions in L(p)^1(XXX) and W(p)^1(XXX)
29(2)
1.5. Poincare's Inequality and Equivalent Norms in Sobolev Spaces
31(11)
1.5.1. Sobolev's Integral Representation
31(4)
1.5.2. Generalized Poincare Inequality
35(4)
1.5.3. The Space L(p)^l(XXX) and Normings in L(p)^l(XXX)
39(1)
1.5.4. Equivalent Norms in W(p)^l(XXX)
40(2)
1.6. Extendability of Functions in Sobolev Spaces
42(4)
1.6.1. Extension Across the "Plane" Part of a Boundary
42(3)
1.6.2. Domains of Class EV(p)^l
45(1)
1.7. Change of Coordinates for Sobolev Functions
46(1)
1.8. Summability and Continuity of Functions in Sobolev Spaces
47(10)
1.8.1. On Continuity of the Imbedding Operator: (XXX)
47(7)
1.8.2. Sobolev's Theorem
54(3)
1.9. Equivalence of Integral and Isoperimetric Inequalities
57(3)
1.10. Compactness Theorems
60(4)
1.11. The Maximal Algebra in W(p)^l(XXX)
64(4)
1.12. Application to the Neumann Problem for Elliptic Operators of Arbitrary Order
68(5)
1.12.1. Necessary and Sufficient Condition for the Continuity of the Imbedding (XXX)
68(1)
1.12.2. Solvability of the Neumann Problem
69(4)
Exercises for Chapter 1
73(5)
Comments to Chapter 1
78(11)
2. Examples of "Bad" Domains in the Theory of Sobolev Spaces
89(54)
2.1. The Property (XXX) does not Ensure the Density of C(Infinity)(XXX) in Sobolev Spaces
89(3)
2.2. Functions with Bounded Gradients are not Always Dense in L(p)^1(XXX)
92(2)
2.3. A Planar Bounded Domain for Which (XXX) is not Dense in L(1)^2(XXX)
94(4)
2.4. On Density of Bounded Functions in L(p)^2(XXX) for Paraboloids in R^n
98(5)
2.5. Imbedding and Compactness Properties May Fail for the Intersection of "Good" Domains
103(2)
2.6. A Domain for Which the Imbedding (XXX) is Continuous but Noncompact
105(2)
2.7. Nikodym's Domain
107(7)
2.7.1. A Domain with the Property (XXX) for l = 1, 2, ..., q (Greater than) 0 and p (XXX) [1, (Infinity)
107(1)
2.7.2. A Domain for Which V(p)^l(XXX) is Noncompactly Imbedded into L(p)(XXX) for p (XXX) [1, Infinity] and l = 1, 2, ...
108(1)
2.7.3. Equivalence of the Imbeddings (XXX) and (XXX)
109(5)
2.7.4. The Neumann Problem for Nikodym's Domain
114(1)
2.8. The Space (XXX) is not Always a Banach Algebra
114(1)
2.9. The Second Gradient of a Function May Be Better Than the First One
115(1)
2.10. Counterexample to the Generalized Poincare Inequality
116(5)
2.11. Counterexample to the Sharpened Friedrichs Inequality
121(3)
2.12. Planar Domains in EV(p)^1 Which are not Quasidisks
124(5)
2.13. Counterexample to the Strong Capacitary Inequality for the Norm in L(2)^2(XXX)
129(7)
Exercises for Chapter 2
136(1)
Comments to Chapter 2
137(6)
II. SOBOLEV SPACES FOR DOMAINS DEPENDING ON PARAMETERS 143(120)
3. Extension of Functions Defined on Parameter Dependent Domains
143(64)
Introduction 143(2)
3.1. Estimates for the Norm of an Extension Operator to the Exterior and Interior of a Small Domain
145(11)
3.1.1. Generalized Poincare Inequality for Domains in EV(p)^l
145(1)
3.1.2. An Extension from a Small Domain to Another One
146(2)
3.1.3. The Interior of a Small Domain
148(1)
3.1.4. Inequalities for Functions Defined on a Ball
149(1)
3.1.5. The Exterior of a Small Domain
150(6)
3.2. Extension with Zero Boundary Conditions
156(6)
3.3. On the "Best" Extension Operator from a Small Domain
162(6)
3.4. The Interior of a Thin Cylinder
168(7)
3.4.1. An Extension Operator with Uniformly Bounded Norm
169(3)
3.4.2. The Case n = 1
172(3)
3.5. A Mollification Operator
175(8)
3.6. Extension to the Exterior of a Thin Cylinder
183(11)
3.6.1. Three Lemmas on Functions Defined in a Thin Cylinder
184(6)
3.6.2. An Extension Operator from a Thin Cylinder
190(4)
3.7. Extension Operators for Particular Domains
194(11)
3.7.1. Examples of Extension Operators for Domains Depending on a Small Parameter
194(6)
3.7.2. Extension from a Domain Depending on Two Small Parameters
200(5)
Comments to Chapter 3
205(2)
4. Boundary Values of Functions with First Derivatives in L(p) on Parameter Dependent Domains
207(56)
Introduction 207(2)
4.1. Traces on Small and Large Components of a Boundary
209(10)
4.1.1. Gagliardo's Theorem and its Consequences
209(2)
4.1.2. The Interior of a Small and Large Domain
211(2)
4.1.3. The Exterior of a Small Domain
213(6)
4.2. On the Trace Space for a Narrow Cylinder
219(14)
4.2.1. An Explicit Norm in the Trace Space for a Narrow Cylinder
219(6)
4.2.2. Equivalent Seminorms
225(3)
4.2.3. Traces on the Boundary of an Infinite Funnel
228(5)
4.3. Inequalities for Functions Defined on a Cylindrical Surface
233(5)
4.4. A Norm in the Space TW(p)^1 for the Exterior of an n-Dimensional Cylinder, p Less than n - 1
238(6)
4.5. The Exterior of a Cylinder, p Greater than n - 1
244(7)
4.6. An (XXX)-Dependent Norm in the Space TW(p)^1 for the Exterior of a Cylinder of Width (XXX), p = n - 1
251(8)
Comments to Chapter 4
259(4)
III. SOBOLEV SPACES FOR DOMAINS WITH CUSPS 263(198)
5. Extension of Functions to the Exterior of a Domain with the Vertex of a Peak on the Boundary
263(64)
Introduction 263(2)
5.1. Integral Inequalities for Functions on Domains with Peaks
265(6)
5.1.1. Friedrichs' Inequality for Functions on Domain with Outer Peak
266(1)
5.1.2. Hardy's Inequalities in Domains with Outer Peaks
267(4)
5.2. Outer Peak. Extension Operator: (XXX), lp Less than n - 1
271(4)
5.3. The Case lp = n - 1
275(10)
5.3.1. Positive Homogeneous Functions of Degree Zero as Multipliers in the Space (XXX)
275(2)
5.3.2. Lemma on Differentiation of a Cut-off Function
277(2)
5.3.3. Extension Operator: (XXX), lp = n - 1
279(6)
5.4. Outer Peak. Extension for lp Greater than n - 1
285(12)
5.4.1. Extension from a Peak to a Circular Peak and to a Cone
285(7)
5.4.2. Extension Operator: (XXX) for lp Greater than n - 1
292(5)
5.5. Inner Peaks
297(7)
5.5.1. The Case n Greater than 2
297(1)
5.5.2. Planar Domains with Inner Peaks
298(6)
5.6. Extension Operator: (XXX), q Less than p
304(14)
5.6.1. Outer Peak, the Case lq Less than n - 1
304(4)
5.6.2. Extension Operator: (XXX), lq = n - 1
308(3)
5.6.3. The Case lq Greater than n - 1
311(4)
5.6.4. Inner Peak, the Case n = 2
315(3)
5.7. Small Perturbations of Peaks in the Vicinity of the Vertex
318(7)
5.7.1. Truncated Outer Peak. Extension Operators: (XXX)
318(4)
5.7.2. Inner Truncated Peaks, n = 2
322(3)
Comments to Chapter 5
325(2)
6. Boundary Values of Sobolev Functions on Non-Lipschitz Domains Bounded by Lipschitz Surfaces
327(36)
Introduction 327(2)
6.1. Ball Coverings of an Open Set Associated with a Lipschitz Function
329(5)
6.2. Domains Between Two Lipschitz Graphs
334(12)
6.2.1. Description of Domains and Approximation Lemma
334(3)
6.2.2. Trace Theorems for Domains Between Two Lipschitz Graphs
337(9)
6.3. The Space TW(p)^l(XXX) for a Planar Domain with Zero Angle
346(6)
6.4. Traces of Functions in W(p)^1(XXX) for Domains Complementary to Those Between Lipschitz Graphs
352(7)
6.5. A Planar Domain with the Vertex of an Inner Peak on the Boundary
359(3)
Comments to Chapter 6
362(1)
7. Boundary Values of Functions in Sobolev Spaces for Domains with Peaks
363(46)
Introduction 363(2)
7.1. Traces of Functions with Gradient in L(1)
365(9)
7.1.1. Outer Peaks
365(7)
7.1.2. Inner Peaks
372(2)
7.2. The Space TW(p)^1(XXX), p Greater than 1, for a Domain with Outer Peak
374(7)
7.3. Boundary Values of Functions in W(p)^1(XXX) for a Domain (XXX) with Inner Peak, p (XXX) (1, n - 1)
381(4)
7.4. Inner Peak, the Case p = n - 1
385(7)
7.4.1. Equivalent Norms for Functions Defined on (XXX) in the Vicinity of the Vertex of a Peak
386(3)
7.4.2. Trace Theorem
389(3)
7.5. Application to the Dirichlet Problem for Second Order Elliptic Equations
392(2)
7.6. Inequalities for Functions Defined on a Surface with Cusp
394(6)
7.7. The Space TW(p)^1(XXX) for a Domain with Inner Peak, p Greater than n - 1
400(8)
Comments to Chapter 7
408(1)
8. Imbedding and Trace Theorems for Domains with Outer Peaks and for General Domains
409(52)
Introduction 409(2)
8.1. Lemma on Averaged Functions
411(5)
8.2. Continuity of the Imbedding Operator: (XXX) for Domains with Outer Peaks
416(9)
8.2.1. Smoothing of the Function Describing a Cusp
416(1)
8.2.2. Summability and Continuity of Functions in Sobolev Spaces on Domains with Outer Peaks
417(8)
8.2.3. Imbedding into a Weighted L(q)
425(1)
8.3. Compactness Theorem
425(6)
8.3.1. Criteria for Compactness of the Imbedding Operators: (XXX) and (XXX) for a Domain with Outer Peak
425(5)
8.3.2. On the Neumann Problem for a Domain with Outer Peak
430(1)
8.4. Imbedding Theorems for Perturbed Peaks
431(10)
8.4.1. Truncated Peaks
432(2)
8.4.2. Union of a Peak and a Small Ball
434(7)
8.5. Capacitary Criteria for the Continuity of the Trace Operator: (XXX)
441(8)
8.5.1. Three Lemmas
441(2)
8.5.2. Capacitary Isoperimetric Inequality as a Criterion for the Continuity of the Trace Operator: (XXX), q Greater than or equal to p
443(1)
8.5.3. The Case q Less than p
444(5)
8.6. Compactness of the Trace Operator: (XXX)
449(9)
8.6.1. Compactness of Continuous Convolution Operators (XXX), q Less than p
450(2)
8.6.2. The Equivalence of the Continuity and Compactness of the Trace Operator: (XXX), q Less than p, p Greater than 1
452(3)
8.6.3. Compactness Theorem in the Case q Greater than or equal to p
455(3)
Comments to Chapter 8
458(3)
References 461(16)
Index 477(4)
List of Symbols
481

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