## Properties of solutions of a class of planar elliptic operators with degeneracies

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- by P. L. Dattori da Silva and A. Meziani PDF
- Proc. Amer. Math. Soc.
**139**(2011), 3937-3949 Request permission

## Abstract:

In this paper we investigate properties of solutions of first and second order elliptic equations that degenerate along a simple closed curve in $\mathbb {R}^2$. These equations are generated by a $\mathbb {C}$-valued vector field $L$. To the vector field $L$, we associate the second order operator $\mathbb {P}=\mathrm {Re}\left [L\overline {L}+p L \right ]$, where $p$ is a $\mathbb {C}$-valued function. We establish a one-to-one correspondence between the solutions of the equation $\mathbb {P}u=0$ and those of an associated first order equation of type $Lw=Aw+B\overline {w}$.## References

- Heinrich G. W. Begehr,
*Complex analytic methods for partial differential equations*, World Scientific Publishing Co., Inc., River Edge, NJ, 1994. An introductory text. MR**1314196**, DOI 10.1142/2162 - A. Bergamasco, P. Cordaro, and J. Hounie,
*Global properties of a class of vector fields in the plane*, J. Differential Equations**74**(1988), no. 2, 179–199. MR**952894**, DOI 10.1016/0022-0396(88)90001-0 - Adalberto P. Bergamasco, Paulo D. Cordaro, and Gerson Petronilho,
*Global solvability for a class of complex vector fields on the two-torus*, Comm. Partial Differential Equations**29**(2004), no. 5-6, 785–819. MR**2059148**, DOI 10.1081/PDE-120037332 - Shiferaw Berhanu, Paulo D. Cordaro, and Jorge Hounie,
*An introduction to involutive structures*, New Mathematical Monographs, vol. 6, Cambridge University Press, Cambridge, 2008. MR**2397326**, DOI 10.1017/CBO9780511543067 - Paulo L. Dattori da Silva,
*Nonexistence of global solutions for a class of complex vector fields on two-torus*, J. Math. Anal. Appl.**351**(2009), no. 2, 543–555. MR**2473960**, DOI 10.1016/j.jmaa.2008.10.039 - Paulo Leandro Dattori da Silva,
*$C^k$-solvability near the characteristic set for a class of planar complex vector fields of infinite type*, Ann. Mat. Pura Appl. (4)**189**(2010), no. 3, 403–413. MR**2657417**, DOI 10.1007/s10231-009-0115-8 - Abdelhamid Meziani,
*Representation of solutions of planar elliptic vector fields with degeneracies*, Geometric analysis of PDE and several complex variables, Contemp. Math., vol. 368, Amer. Math. Soc., Providence, RI, 2005, pp. 357–370. MR**2127042**, DOI 10.1090/conm/368/06791 - Abdelhamid Meziani,
*Representation of solutions of a singular Cauchy-Riemann equation in the plane*, Complex Var. Elliptic Equ.**53**(2008), no. 12, 1111–1130. MR**2467386**, DOI 10.1080/17476930802509239 - Abdelhamid Meziani,
*Properties of solutions of a planar second-order elliptic equation with a singularity*, Complex Var. Elliptic Equ.**54**(2009), no. 7, 677–688. MR**2538058**, DOI 10.1080/17476930902998928 - A. Meziani,
*On first and second order planar elliptic equations with degeneracies*, to appear in Memoirs of the AMS (see also arXiv:0910.0539v1). - François Trèves,
*Remarks about certain first-order linear PDE in two variables*, Comm. Partial Differential Equations**5**(1980), no. 4, 381–425. MR**567779**, DOI 10.1080/0360530800882143 - François Trèves,
*Hypo-analytic structures*, Princeton Mathematical Series, vol. 40, Princeton University Press, Princeton, NJ, 1992. Local theory. MR**1200459** - I. N. Vekua,
*Generalized analytic functions*, Pergamon Press, London-Paris-Frankfurt; Addison-Wesley Publishing Company, Inc., Reading, Mass., 1962. MR**0150320**

## Additional Information

**P. L. Dattori da Silva**- Affiliation: Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Caixa Postal 668, São Carlos, SP, 13560-970 Brazil
- MR Author ID: 785140
- Email: dattori@icmc.usp.br
**A. Meziani**- Affiliation: Department of Mathematics, Florida International University, Miami, Florida 33199
- MR Author ID: 239413
- Email: meziani@fiu.edu
- Received by editor(s): April 22, 2010
- Received by editor(s) in revised form: September 9, 2010
- Published electronically: March 11, 2011
- Additional Notes: The first author was supported in part by CNPq and FAPESP
- Communicated by: Mei-Chi Shaw
- © Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**139**(2011), 3937-3949 - MSC (2010): Primary 35C10; Secondary 35F05
- DOI: https://doi.org/10.1090/S0002-9939-2011-10826-8
- MathSciNet review: 2823040