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El. knyga: Regularity Estimates for Nonlinear Elliptic and Parabolic Problems: Cetraro, Italy 2009

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  • Formatas: PDF+DRM
  • Serija: C.I.M.E. Foundation Subseries 2045
  • Išleidimo metai: 02-Mar-2012
  • Leidėjas: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Kalba: eng
  • ISBN-13: 9783642271458
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Regularity Estimates for Nonlinear Elliptic and Parabolic Problems: Cetraro, Italy 2009 Didesnis paveikslėlis
  • Formatas: PDF+DRM
  • Serija: C.I.M.E. Foundation Subseries 2045
  • Išleidimo metai: 02-Mar-2012
  • Leidėjas: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Kalba: eng
  • ISBN-13: 9783642271458
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The issue of regularity has played a central role in the theory of Partial Differential Equations almost since its inception, and despite the tremendous advances made it still remains a very fruitful research field. In particular considerable strides have been made in regularity estimates for degenerate and singular elliptic and parabolic equations over the last several years, and in many unexpected and challenging directions. Because of all these recent results, it seemed high time to create an overview that would highlight emerging trends and issues in this fascinating research topic in a proper and effective way. The course aimed to show the deep connections between these topics and to open new research directions through the contributions of leading experts in all of these fields.
Applications of Boundary Harnack Inequalities for p Harmonic Functions and Related Topics
1(72)
J. Lewis
1 Outline of the Course
1(1)
1.1 Ode to the p Laplacian
1(1)
1.2 My Introduction to p Harmonic Functions
2(1)
2 Basic Estimates for the p Laplacian
2(7)
2.1 p Harmonic Functions in NTA Domains
4(2)
2.2 The p Laplacian and Elliptic PDE
6(1)
2.3 Degenerate Elliptic Equations
7(2)
3 p Harmonic Measure
9(14)
3.1 p Harmonic Measure in Simply Connected Domains
15(1)
3.2 Preliminary Reductions for Theorem 2.6
15(1)
3.3 Proof of Theorem 2.8
16(3)
3.4 The Final Proof
19(2)
3.5 p Harmonic Measure in Space
21(1)
3.6 Open Problems for p Harmonic Measure
22(1)
4 Boundary Harnack Inequalities and the Martin Boundary Problem for p Harmonic Functions
23(23)
4.1 History of Theorem 3.1
24(2)
4.2 Proof of Step 1
26(1)
4.3 Proof of Step 2
27(3)
4.4 Proof of Step 3
30(3)
4.5 Proof of Step 4 and Theorem 3.1
33(4)
4.6 More on Boundary Harnack Inequalities
37(1)
4.7 The Martin Boundary Problem
38(4)
4.8 Proof of Theorem 3.9
42(4)
4.9 Further Remarks
46(1)
5 Uniqueness and Regularity in Free Boundary: Inverse Type Problems
46(27)
5.1 History of Theorem 4.1
46(3)
5.2 Proof of Theorem 4.1
49(1)
5.3 Further Uniqueness Results
50(1)
5.4 Boundary Regularity of p Harmonic Functions
51(1)
5.5 Proof of Theorem 4.3
52(3)
5.6 Proof of Theorem 4.4
55(2)
5.7 Proof of Theorem 4.5
57(2)
5.8 Regularity in a Lipschitz Free Boundary Problem
59(1)
5.9 History of Theorem 4.11
60(1)
5.10 Proof of Theorem 4.11
60(1)
5.11 Enlargement of the Cone of Monotonicity in the Interior
61(1)
5.12 Enlargement of the Cone of Monotonicity at the Free Boundary
61(2)
5.13 An Application of Theorem 4.11
63(2)
5.14 Proof of (161)
65(3)
5.15 Closing Remarks
68(1)
References
69(4)
Regularity of Supersolutions
73(60)
Peter Lindqvist
1 Introduction
73(5)
2 The Stationary Equation
78(13)
3 The Evolutionary Equation
91(20)
3.1 Definitions
92(2)
3.2 Bounded Supersolutions
94(8)
3.3 Unbounded Supersolutions
102(7)
3.4 Reduction to Zero Boundary Values
109(2)
4 Weak Supersolutions are Semicontinuous
111(11)
5 The Equation with Measure Data
122(1)
6 Pointwise Behaviour
123(10)
6.1 The Stationary Equation
123(2)
6.2 The Evolutionary Equation
125(5)
References
130(3)
Introduction to Random Tug-of-War Games and PDEs
133(20)
Juan J. Manfredi
1 Introduction
133(1)
2 Probability Background
133(8)
3 The p-Laplacian Gambling House
141(3)
4 p-harmonious Functions
144(3)
5 Directed Trees
147(3)
6 Epilogue
150(3)
References
151(2)
The Problems of the Obstacle in Lower Dimension and for the Fractional Laplacian
153(78)
Sandro Salsa
1 Introduction
153(7)
2 The Zero Obstacle Problem
160(41)
2.1 Setting of the Problem
160(2)
2.2 Lipschitz Continuity and Semiconvexity
162(4)
2.3 Local C1,α Estimate
166(4)
2.4 Optimal Regularity for Tangentially Convex Global Solutions
170(3)
2.5 Almgren's Frequency Formula
173(4)
2.6 Asymptotic Profiles and Optimal Regularity
177(3)
2.7 Lipschitz Continuity of the Free Boundary at Stable Points
180(3)
2.8 Boundary Harnack Principles and C1,α Regularity of the Free Boundary at Stable Points
183(5)
2.9 Structure of the Singular Set
188(13)
3 Obstacle Problem for the Fractional Laplacian
201(30)
3.1 Construction of the Solution and Basic Properties
202(1)
3.2 Lipschitz Continuity, Semiconvexity and C1,α Estimates
203(1)
3.3 Thin Obstacle for the Operator La: Local C1,α Estimates
204(2)
3.4 Minimizers of the Weighted Rayleigh Quotient and a Monotonicity Formula
206(1)
3.5 Optimal Regularity for Tangentially Convex Global Solutions
207(4)
3.6 Frequency Formula
211(6)
3.7 Blow-up Sequences and Optimal Regularity
217(8)
3.8 Nondegenerate Case: Lipschitz Continuity of the Free Boundary
225(2)
3.9 Boundary Harnack Principles and C1,α Regularity of the Free Boundary
227(4)
Appendix A The Fractional Laplacian
231(3)
Definition and Basic Properties
231(1)
Supersolutions and comparison
232(2)
Appendix B The Operator La
234(6)
Definition and Preliminary Facts
234(2)
Harnack inequality, Liouville theorem and mean value property
236(4)
Poincare inequalities
240(1)
Appendix C Relation between (---Δ)S and La
240(5)
References
243(2)
List of Participants 245
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