Open Access
2016 Local Hypoellipticity by Lyapunov Function
E. R. Aragão-Costa
Abstr. Appl. Anal. 2016: 1-8 (2016). DOI: 10.1155/2016/7210540


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).


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E. R. Aragão-Costa. "Local Hypoellipticity by Lyapunov Function." Abstr. Appl. Anal. 2016 1 - 8, 2016.


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

zbMATH: 06929387
MathSciNet: MR3454567
Digital Object Identifier: 10.1155/2016/7210540

Rights: Copyright © 2016 Hindawi

Vol.2016 • 2016
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