Abstract and Applied Analysis

Local Hypoellipticity by Lyapunov Function

E. R. Aragão-Costa

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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: Lj=/tj+(ϕ/tj)(t,A)A, j=1,2,,n, where A:D(A)HH is a self-adjoint linear operator, positive with 0ρ(A), in a Hilbert space H, and ϕ=ϕ(t,A) is a series of nonnegative powers of A-1 with coefficients in C(Ω), Ω being an open set of Rn, for any nN, different from what happens in the work of Hounie (1979) who studies the problem only in the case n=1. 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 t(s)=-Reϕ0(t(s)), s0, t(0)=t0Ω,ϕ0:ΩC being the first coefficient of ϕ(t,A). Besides, to get over the problem out of the elliptic region, that is, in the points tΩ such that Reϕ0(t) = 0, we will use the techniques developed by Bergamasco et al. (1993) for the particular operator A=1-Δ:H2(RN)L2(RN)L2(RN).

Article information

Source
Abstr. Appl. Anal. Volume 2016 (2016), Article ID 7210540, 8 pages.

Dates
Received: 7 July 2015
Accepted: 20 December 2015
First available in Project Euclid: 10 February 2016

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1455115145

Digital Object Identifier
doi:10.1155/2016/7210540

Mathematical Reviews number (MathSciNet)
MR3454567

Citation

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


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