Electronic ISBN:  9781470405830 
Product Code:  MEMO/206/969.E 
List Price:  $98.00 
MAA Member Price:  $88.20 
AMS Member Price:  $58.80 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 206; 2010; 269 ppMSC: Primary 31; Secondary 76;
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 HeleShaw flow with a freeboundary 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

Chapters

1. Introduction and Main Results

2. Quadrature Domains

3. Construction of Measures for Localization

4. Generalizations of the Reflection Theorem

5. Continuous Reflection Property and Smooth Boundary Points

6. Proofs of (1) and (3) in Theorem 1.1

7. Corners with Right Angles

8. Properly Open Cusps

9. Microlocalization and the LocalReflection Theorem

10. Modifications of Measures in $R^+$

11. Modifications of Measures in $R^$

12. Sufficient Conditions for a Cusp to be a LaminarFlow Point

13. TurbulentFlow Points

14. The Set of Stationary Points

15. Open Questions


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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 HeleShaw flow with a freeboundary 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.

Chapters

1. Introduction and Main Results

2. Quadrature Domains

3. Construction of Measures for Localization

4. Generalizations of the Reflection Theorem

5. Continuous Reflection Property and Smooth Boundary Points

6. Proofs of (1) and (3) in Theorem 1.1

7. Corners with Right Angles

8. Properly Open Cusps

9. Microlocalization and the LocalReflection Theorem

10. Modifications of Measures in $R^+$

11. Modifications of Measures in $R^$

12. Sufficient Conditions for a Cusp to be a LaminarFlow Point

13. TurbulentFlow Points

14. The Set of Stationary Points

15. Open Questions