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