Abstract and Applied Analysis

Local Hypoellipticity by Lyapunov Function

E. R. Aragão-Costa

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


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

Export citation


  • F. Trèves, “Concatenations of second-order evolution equations applied to local solvability and hypoellipticity,” Communications on Pure and Applied Mathematics, vol. 26, pp. 201–250, 1973.
  • F. Treves, “Study of a model in the theory of complexes of pseudodifferential operators,” Annals of Mathematics, vol. 104, no. 2, pp. 269–324, 1976.
  • L. C. Yamaoka, Resolubilidade Local de uma Classe de Sistemas Subdeterminados Abstratos, IME-USP Tese de Doutorado, 2011.
  • Z. Han, “Local solvability of analytic pseudodifferential complexes in top degree,” Duke Mathematical Journal, vol. 87, no. 1, pp. 1–28, 1997.
  • A. P. Bergamasco, P. D. Cordaro, and P. A. Malagutti, “Globally hypoelliptic systems of vector fields,” Journal of Functional Analysis, vol. 114, no. 2, pp. 267–285, 1993.
  • D. Henry, Geometric Theory of Semilinear Parabolic Equations, vol. 840 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1981.
  • L. Hörmander, Linear Partial Differential Operators, Springer, New York, NY, USA, 1963.
  • F. Treves, Topological Vector Spaces, Distributions and Kernels, Academic Press, New York, NY, USA, 1967.
  • J. Hounie, “Globally hypoelliptic and globally solvable first order evolution equations,” Transactions of the American Mathematical Society, vol. 252, pp. 233–248, 1979.
  • A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, NY, USA, 1963.
  • E. R. Aragao-Costa, A. N. Carvalho, P. Marín-Rubio, and G. Planas, “Gradient-like nonlinear semigroups with infinitely many equilibria and applications to cascade systems,” Topological Methods in Nonlinear Analysis, vol. 42, no. 2, pp. 345–376, 2013. \endinput