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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Authors: P. L. Dattori da Silva and A. Meziani
Journal: Proc. Amer. Math. Soc. 139 (2011), 3937-3949
MSC (2010): Primary 35C10; Secondary 35F05
Published electronically: March 11, 2011
MathSciNet review: 2823040
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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}$.


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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
Email: dattori@icmc.usp.br

A. Meziani
Affiliation: Department of Mathematics, Florida International University, Miami, Florida 33199
Email: meziani@fiu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10826-8
PII: S 0002-9939(2011)10826-8
Keywords: Elliptic equations, series representation, normalization
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
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.