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On first and second order planar elliptic equations with degeneracies
About this Title
Abdelhamid Meziani, Department of Mathematics Florida International University Miami, Florida 33199
Publication: Memoirs of the American Mathematical Society
Publication Year:
2012; Volume 217, Number 1019
ISBNs: 978-0-8218-5312-2 (print); 978-0-8218-8750-9 (online)
DOI: https://doi.org/10.1090/S0065-9266-2011-00634-9
Published electronically: May 18, 2011
Keywords: CR equations,
degenerate elliptic,
spectral values,
fundamental matrix,
asymptotic behavior,
kernels,
semilinear,
normalization,
vector fields.
MSC: Primary 35J70, Secondaries, 35F05, 30G20
Table of Contents
Chapters
- Introduction
- 1. Preliminaries
- 2. Basic Solutions
- 3. Example
- 4. Asymptotic behavior of the basic solutions of $\mathcal {L}$
- 5. The kernels
- 6. The homogeneous equation $\mathcal {L} u=0$
- 7. The nonhomogeneous equation $\mathcal {L} u=F$
- 8. The semilinear equation
- 9. The second order equation: Reduction
- 10. The homogeneous equation $Pu=0$
- 11. The nonhomogeneous equation $Pu=F$
- 12. Normalization of a Class of Second Order Equations with a Singularity
Abstract
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.- Heinrich Begehr and Dao-Qing Dai, On continuous solutions of a generalized Cauchy-Riemann system with more than one singularity, J. Differential Equations 196 (2004), no. 1, 67–90. MR 2025186, DOI 10.1016/j.jde.2003.07.013
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