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# memo_has_moved_text();Small modifications of quadrature domains

Makoto Sakai, Mukaihara 1-13-15, Asao-ku, Kawasaki 215-0007, Japan

Publication: Memoirs of the American Mathematical Society
Publication Year: 2010; Volume 206, Number 969
ISBNs: 978-0-8218-4810-4 (print); 978-1-4704-0583-0 (online)
DOI: http://dx.doi.org/10.1090/S0065-9266-10-00596-X
Published electronically: February 19, 2010
MathSciNet review: 2667421
Keywords:Quadrature domains, Hele-Shaw flow, harmonic measure
MSC: Primary 30C20; Secondary 31-02, 31A05, 35Q35, 35R35, 76D27, 76M40

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Chapters

• Chapter 1. Introduction and main results
• Chapter 3. Construction of measures for localization
• Chapter 4. Generalizations of the reflection theorem
• Chapter 5. Continuous reflection property and smooth boundary points
• Chapter 6. Proofs of (1) and (3) in Theorem 1.1
• Chapter 7. Corners with right angles
• Chapter 8. Properly open cusps
• Chapter 9. Microlocalization and the local-reflection theorem
• Chapter 10. Modifications of measures in $R^+$
• Chapter 11. Modifications of measures in $R^-$
• Chapter 12. Sufficient conditions for a cusp to be a laminar-flow point
• Chapter 13. Turbulent-flow points
• Chapter 14. The set of stationary points
• Chapter 15. Open questions

### Abstract

For a given plane domain, we add a constant multiple of the Dirac measure at a point in the domain and make 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. We regard the given domain as the initial domain and the support point of the Dirac measure as the injection point of the flow. We treat the case in which the initial domain has a corner on the boundary and discuss the shape of the time-dependent domain around the corner immediately after the initial time. If the interior angle of the corner is less than $\pi/2$, then it is a laminar-flow stationary corner, and if the angle is greater than $\pi/2$ but less than $2\pi$, then it is a laminar-flow point. The critical values of the interior angles are $\pi/2$ and $2\pi$. We give a criterion for whether a corner with interior angle $\pi/2$ is stationary. If the interior angle of the corner is $2\pi$, we call the corner a cusp. We give a sufficient condition for a cusp to be a laminar-flow point and prove that cusps appearing in the Hele-Shaw flow with a free-boundary problem are laminar-flow points. We give a sufficient condition for a cusp to be a turbulent-flow point as well.