**Memoirs of the American Mathematical Society**

2010;
269 pp;
Softcover

MSC: Primary 31;
Secondary 76

Print ISBN: 978-0-8218-4810-4

Product Code: MEMO/206/969

List Price: $98.00

Individual Member Price: $58.80

**Electronic ISBN: 978-1-4704-0583-0
Product Code: MEMO/206/969.E**

List Price: $98.00

Individual Member Price: $58.80

# Small Modifications of Quadrature Domains

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*Makoto Sakai*

For a given plane domain, the author adds a constant multiple of the Dirac measure at a point in the domain and makes a new domain called a quadrature domain. The quadrature domain is characterized as a domain such that the integral of a harmonic and integrable function over the domain equals the integral of the function over the given domain plus the integral of the function with respect to the added measure. The family of quadrature domains can be modeled as the Hele-Shaw flow with a free-boundary problem. The given domain is regarded as the initial domain and the support point of the Dirac measure as the injection point of the flow.

#### Table of Contents

# Table of Contents

## Small Modifications of Quadrature Domains

- Chapter 1. Introduction and Main Results 18 free
- Chapter 2. Quadrature Domains 916 free
- Chapter 3. Construction of Measures for Localization 2128
- Chapter 4. Generalizations of the Reflection Theorem 2532
- Chapter 5. Continuous Reflection Property and Smooth Boundary Points 3138
- Chapter 6. Proofs of (1) and (3) in Theorem 1.1 3744
- Chapter 7. Corners with Right Angles 4956
- Chapter 8. Properly Open Cusps 8592
- Chapter 9. Microlocalization and the Local-Reflection Theorem 95102
- Chapter 10. Modifications of Measures in R+ 129136
- Chapter 11. Modifications of Measures in R- 181188
- Chapter 12. Sufficient Conditions for a Cusp to be a Laminar-Flow Point 213220
- Chapter 13. Turbulent-Flow Points 251258
- Chapter 14. The Set of Stationary Points 261268
- Chapter 15. Open Questions 263270
- Bibliography 265272
- Symbol Index 267274
- Index 269276 free