2020/02/09 by Fernando de Ávila Silva, Silva, Fernando de Ávila
Mathematics · Physics and Astronomy · #35B10 #35B65 #35H10 #35S05 #Advanced Algebra and Geometry #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Waves and Solitons #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2002.03373
openalex publication_date 2020/02/09 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
This article presents an investigation on the global hypoellipticity problem for systems belonging to the class P = Dt + Q(t,Dx), where Q(t,Dx) is a m× m matrix with entries cj,k(t)Qj,k(Dx). The coefficients cj,k(t) are smooth, complex-valued functions on the torus \mathbbT ≃ ℝ/2πℤ and Qj,k(Dx) are pseudo-differential operators on \mathbbTn. The approach consists in establishing conditions on the matrix symbol Q(t,ξ) such that it can be transformed into a suitable triangular form Λ(t,ξ) + N(t,ξ), where Λ(t,ξ) is the diagonal matrix diag(λ1(t,ξ) … λm(t,ξ)) and N(t,ξ) is a nilpotent upper triangular matrix. Hence, the global hypoellipticity of P is studied by analyzing the behavior of the eigenvalues λj(t,ξ) and its averages λ0,j(ξ), as |ξ| → ∞.