Gebremeskel, Abraham Hailu (2016) On Applications of FBI Transforms to Wave Front Sets. PhD thesis, Addis Ababa University.
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Abstract
In this thesis, we study the application of FBI transforms to the C1; analytic and Gevrey wave front sets of functions. We characterize the C1 wave front set of a function by providing a simpler proof of a result by Berhanu and Hounie. To characterize the analytic wave front set, we generalize the work of Berhanu and Hounie [10] to two polynomials in the generating function of the FBI transform they define. The Gevrey wave front set is characterized first as in the paper of Berhanu and Hounie and then generalized to two polynomials. Finally, we apply the standard FBI transform to study the microlocal smoothness of C2 solutions u of the first-order nonlinear partial differential equation ut = f(x; t; u; ux) where f(x; t; ζ0; ζ) is a complex-valued function which is C1 in all the variables (x; t; ζ0; ζ) and holomorphic in the variables (ζ0; ζ): If the solution u is C2; σ 2 Char(Lu) and 1i σ([Lu; L¯u]) < 0; then we show that σ = 2 WF(u): Here WF(u) denotes the C1 wave front set of u and Char(Lu) denotes the characteristic set of the linearized operator Lu = @ @t − mX j =1 @f @ζj (x; t; u; ux)@x @j :
| Item Type: | Thesis (PhD) |
|---|---|
| Subjects: | Q Science > Q Science (General) Q Science > QA Mathematics |
| Divisions: | Africana |
| Depositing User: | Selom Ghislain |
| Date Deposited: | 18 Jun 2018 09:47 |
| Last Modified: | 18 Jun 2018 09:47 |
| URI: | http://thesisbank.jhia.ac.ke/id/eprint/4351 |
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