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Abstract and Applied Analysis
Volume 2016, Article ID 7210540, 8 pages
http://dx.doi.org/10.1155/2016/7210540
Research Article

Local Hypoellipticity by Lyapunov Function

Instituto de Ciências Matemáticas e de Computaçao, Universidade de São Paulo, Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos, SP, Brazil

Received 7 July 2015; Accepted 20 December 2015

Academic Editor: Maria Grazia Naso

Copyright © 2016 E. R. Aragão-Costa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We treat the local hypoellipticity, in the first degree, for a class of abstract differential operators complexes; the ones are given by the following differential operators: , , where is a self-adjoint linear operator, positive with , in a Hilbert space , and is a series of nonnegative powers of with coefficients in , being an open set of , for any , different from what happens in the work of Hounie (1979) who studies the problem only in the case . We provide sufficient condition to get the local hypoellipticity for that complex in the elliptic region, using a Lyapunov function and the dynamics properties of solutions of the Cauchy problem , , being the first coefficient of . Besides, to get over the problem out of the elliptic region, that is, in the points such that = 0, we will use the techniques developed by Bergamasco et al. (1993) for the particular operator .