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On First and Second Order Planar Elliptic Equations with Degeneracies
Abdelhamid Meziani, Florida International University, Miami, FL

Memoirs of the American Mathematical Society
2012; 77 pp; softcover
Volume: 217
ISBN-10: 0-8218-5312-0
ISBN-13: 978-0-8218-5312-2
List Price: US$60
Individual Members: US$36
Institutional Members: US$48
Order Code: MEMO/217/1019
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This paper deals with elliptic equations in the plane with degeneracies. The equations are generated by a complex vector field that is elliptic everywhere except along a simple closed curve. Kernels for these equations are constructed. Properties of solutions, in a neighborhood of the degeneracy curve, are obtained through integral and series representations. An application to a second order elliptic equation with a punctual singularity is given.

Table of Contents

  • Introduction
  • Preliminaries
  • Basic Solutions
  • Example
  • Asymptotic behavior of the basic solutions of \(\mathcal{L}\)
  • The kernels
  • The homogeneous equation \(\mathcal{L}u = 0\)
  • The nonhomogeneous equation \(\mathcal{L}u = F\)
  • The semilinear equation
  • The second order equation: Reduction
  • The homogeneous equation \(Pu = 0\)
  • The nonhomogeneous equation \(Pu = F\)
  • Normalization of a class of second order equations with a singularity
  • Bibliography
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