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Small Modifications of Quadrature Domains
Makoto Sakai, Kawasaki, Japan

Memoirs of the American Mathematical Society
2010; 269 pp; softcover
Volume: 206
ISBN-10: 0-8218-4810-0
ISBN-13: 978-0-8218-4810-4
List Price: US$98
Individual Members: US$58.80
Institutional Members: US$78.40
Order Code: MEMO/206/969
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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 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

  • Introduction and main results
  • Quadrature domains
  • Construction of measures for localization
  • Generalizations of the reflection theorem
  • Continuous reflection property and smooth boundary points
  • Proofs of (1) and (3) in Theorem 1.1
  • Corners with right angles
  • Properly open cusps
  • Microlocalization and the local-reflection theorem
  • Modifications of measures in \(R^+\)
  • Modifications of measures in \(R^-\)
  • Sufficient conditions for a cusp to be a laminar-flow point
  • Turbulent-flow points
  • The set of stationary points
  • Open questions
  • Bibliography
  • Symbol index
  • Index
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