Linear Holomorphic Partial Differential Equations and Classical Potential Theory

Select Guide Rating

• Introduction: Some motivating questions; The Cauchy-Kovalevskaya theorem with estimates; Remarks on the Cauchy-Kovalevskaya theorem; Zerner's theorem; The method of globalizing families; Holmgren's uniqueness theorem; The continuity method of F. John; The Bony-Schapira theorem; Applications of the Bony-Schapira theorem: Part I - Vekua hulls; Applications of the Bony-Schapira theorem: Part II - Szego's theorem revisited; The reflection principle; The reflection principle (continued); Cauchy problems and the Schwarz potential conjecture; The Schwarz potential conjecture for spheres; Potential theory on ellipsoids: Part I - The mean value property; Potential theory on ellipsoids: Part II - There is no gravity in the cavity; Potential theory on ellipsoids: Part III - The Dirichlet problem; Singularities encountered by the analytic continuation of solutions to the Dirichlet problem; An introduction to J. Leray's principle on propagation of singularities through $\mathbb{C}^n$; Global propagation of singularities in $\mathbb{C}^n$; Quadrature domains and Laplacian growth; Other varieties of quadrature domains; Bibliography; Index.