Local Hypoellipticity by Lyapunov Function
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)⊂H→H is a self-adjoint linear operator, positive with 0∈ρ(A), in a Hilbert sp...
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Wiley
2016-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2016/7210540 |
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| author | E. R. Aragão-Costa |
| author_facet | E. R. Aragão-Costa |
| author_sort | E. R. Aragão-Costa |
| collection | DOAJ |
| description | 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)⊂H→H is a self-adjoint linear operator, positive with 0∈ρ(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 n∈N, 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)), s≥0, t(0)=t0∈Ω,ϕ0:Ω→C being the first coefficient of ϕ(t,A). Besides, to get over the problem out of the elliptic region, that is, in the points t∗ ∈Ω 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). |
| format | Article |
| id | doaj-art-cde7cd46493e47a6a3ffee3c5e3dae24 |
| institution | OA Journals |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2016-01-01 |
| publisher | Wiley |
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| series | Abstract and Applied Analysis |
| spelling | doaj-art-cde7cd46493e47a6a3ffee3c5e3dae242025-08-20T02:20:02ZengWileyAbstract and Applied Analysis1085-33751687-04092016-01-01201610.1155/2016/72105407210540Local Hypoellipticity by Lyapunov FunctionE. R. Aragão-Costa0Instituto de Ciências Matemáticas e de Computaçao, Universidade de São Paulo, Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos, SP, BrazilWe 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)⊂H→H is a self-adjoint linear operator, positive with 0∈ρ(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 n∈N, 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)), s≥0, t(0)=t0∈Ω,ϕ0:Ω→C being the first coefficient of ϕ(t,A). Besides, to get over the problem out of the elliptic region, that is, in the points t∗ ∈Ω 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).http://dx.doi.org/10.1155/2016/7210540 |
| spellingShingle | E. R. Aragão-Costa Local Hypoellipticity by Lyapunov Function Abstract and Applied Analysis |
| title | Local Hypoellipticity by Lyapunov Function |
| title_full | Local Hypoellipticity by Lyapunov Function |
| title_fullStr | Local Hypoellipticity by Lyapunov Function |
| title_full_unstemmed | Local Hypoellipticity by Lyapunov Function |
| title_short | Local Hypoellipticity by Lyapunov Function |
| title_sort | local hypoellipticity by lyapunov function |
| url | http://dx.doi.org/10.1155/2016/7210540 |
| work_keys_str_mv | AT eraragaocosta localhypoellipticitybylyapunovfunction |