Abstract and Applied Analysis

Local Hypoellipticity by Lyapunov Function

E. R. Aragão-Costa

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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

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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

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  • 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