The Schwarz Function and Its Generalization to Higher Dimensions

The Schwarz function originates in classical complex analysis and potential theory. Here the author presents the advantages favoring a mode of treatment which unites the subject with modern theory of distributions and partial differential equations thus bridging the gap between two–dimensional geometric and multi–dimensional analysts. Examines the Schwarz function and its relationship to recent investigations regarding inverse problems of Newtonian gravitation, free boundaries, Hele–Shaw flows and the propagation of singularities for holomorphic p.d.e.

Spis treści:
The Schwarz Principle of Reflection.
The Logarithmic Potential, Balayage, and Quadrature Domains.
Examples of ``Quadrature Identities′′.
Quadrature Domains: Basic Properties, 1.
Quadrature Domains: Basic Properties, 2.
Schwarzian Reflection, Revisited.
Projectors from L? (dOmega) to H? (dOmega).
The Friedrichs Operator.
Concluding Remarks.
Bibliography.
Index.