**Memoirs of the American Mathematical Society**

2012;
77 pp;
Softcover

MSC: Primary 35;
Secondary 30

Print ISBN: 978-0-8218-5312-2

Product Code: MEMO/217/1019

List Price: $60.00

AMS Member Price: $36.00

MAA Member Price: $54.00

**Electronic ISBN: 978-0-8218-8750-9
Product Code: MEMO/217/1019.E**

List Price: $60.00

AMS Member Price: $36.00

MAA Member Price: $54.00

# On First and Second Order Planar Elliptic Equations with Degeneracies

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*Abdelhamid Meziani*

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

# Table of Contents

## On First and Second Order Planar Elliptic Equations with Degeneracies

- Introduction 18 free
- Chapter 1. Preliminaries 512 free
- Chapter 2. Basic Solutions 916
- 2.1. Properties of basic solutions 916
- 2.2. The spectral equation and Spec(L0) 1118
- 2.3. Existence of basic solutions 1320
- 2.4. Properties of the fundamental matrix of (E,) 1421
- 2.5. The system of equations for the adjoint operator L* 1623
- 2.6. Continuation of a simple spectral value 1724
- 2.7. Continuation of a double spectral value 1926
- 2.8. Purely imaginary spectral value 2229
- 2.9. Main result about basic solutions 2431

- Chapter 3. Example 2734
- Chapter 4. Asymptotic behavior of the basic solutions of L 2936
- Chapter 5. The kernels 3744
- Chapter 6. The homogeneous equation L u=0 4350
- Chapter 7. The nonhomogeneous equation L u=F 5158
- Chapter 8. The semilinear equation 5764
- Chapter 9. The second order equation: Reduction 6168
- Chapter 10. The homogeneous equation Pu=0 6370
- Chapter 11. The nonhomogeneous equation Pu=F 6976
- Chapter 12. Normalization of a Class of Second Order Equations with a Singularity 7380
- Bibliography 7784