Abstract and Applied Analysis
- Abstr. Appl. Anal.
- Volume 2016 (2016), Article ID 7210540, 8 pages.
Local Hypoellipticity by Lyapunov Function
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 .
Abstr. Appl. Anal., Volume 2016 (2016), Article ID 7210540, 8 pages.
Received: 7 July 2015
Accepted: 20 December 2015
First available in Project Euclid: 10 February 2016
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Aragão-Costa, E. R. Local Hypoellipticity by Lyapunov Function. Abstr. Appl. Anal. 2016 (2016), Article ID 7210540, 8 pages. doi:10.1155/2016/7210540. https://projecteuclid.org/euclid.aaa/1455115145