Propagation of Singularities for a Class of Operators with Double Characteristics

Hanges, Nicholas

doi:10.1515/9781400881581-006

Cited by 9 publications

(16 citation statements)

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“…This equality is a complex version of the branching of singularities of solutions in real domains (cf. S. Alinhac [1], N. Ranges [5] and T. Oaku [11]). Consider the following Cauchy problem as well.…”

Section: \P A} U--={j%-zipt+'zl-ib J Dj+c}u = Q (1)mentioning

confidence: 99%

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Propagation of Singularities in the Ramified Cauchy Problem for a Class of Operators with Non-involutive Multiple Characteristics

Igari

1999

Publ. Res. Inst. Math. Sci.

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In the ramified Cauchy problem, analytic continuation of holomorphic solutions has been mainly studied. In this paper, we prove the propagation of singularities for a class of linear partial differential equations with non-involutive multiple characteristics. §1. IntroductionIn this paper, we consider the Cauchy problem in a complex domain with singular initial data and study the propagation of singularities of the solution.It is one of the most fundamental problems in the theory of partial differential equations, and its study goes back to J. Leray

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Section: \P A} U--={j%-zipt+'zl-ib J Dj+c}u = Q (1)mentioning

confidence: 99%

“…It is a complex version of the branching of singularities in real domains (Cf. [1], [5] and [11]) and gives an answer to the first problem mentioned in Introduction. …”

Section: Propagation Of Singularitiesmentioning

confidence: 99%

Propagation of Singularities in the Ramified Cauchy Problem for a Class of Operators with Non-involutive Multiple Characteristics

Igari

1999

Publ. Res. Inst. Math. Sci.

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“…As for microhyperbolic operators with non-involutive characteristics, there are many papers for the case of order two (c.f. [1,2,4,5,6,10,11,12,14,15]), and some papers for the case of higher orders (c.f. [16,17,18]).…”

Section: Introductionmentioning

confidence: 99%

Stokes operators for microhyperbolic equations

Uchikoshi

1997

New Trends in Microlocal Analysis

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Abstract.We consider microhyperbolic equations degenerated precisely on a hyperplane, and study the propagation of the singularities. IntroductionIn this paper we study the propagation of the singularity for some class of microhyperbolic operators, containing the case of non-involutive characteristics. As for microhyperbolic operators with non-involutive characteristics, there are many papers for the case of order two (c.f. [1,2,4,5,6,10,11,12,14,15]), and some papers for the case of higher orders (c.f. [16,17,18]). In these cases it is well-known that the propagation of the singularity is closely related to the classical theory for ordinary differential equations. In this paper, we generalize such a result, and give a general representation of the elementary solution: It is the composite of holomorphic microlocal operators and quantized contact transformations. As a natural consequence, we obtain the notion of Stokes operators. Such operators were previously known only for a very special case. Using these operators, we can study the branching of the singularity. Note that we do not assume any restrictive conditions for the lower order terms.Let P (x, D) be a microdifferential operator defined at x * = (0; 0, · · · , 0, √ −1) ∈ T * C n of order m, written in the formHere we have written D = (D 2 , · · · , D n ) as usual. Sometimes we also write

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“…Namely, the characteristic variety {(£, x, r, f) ; (r-t£ ) (r + *f)=0} satisfies {r-*£, r + *f} =2=£0 and d(r -t^ /\d(r + t£) ^0 for t = 0, f^O, where the bracket stands for Poisson bracket. Ivrii [9] and Hanges [7] studied a more general operator with doubly noninvolutive characteristics. The subprincipal symbol plays an essential role in their condition of branching of singularities.…”

mentioning

confidence: 99%

“…Also Nakane [14] considered the same operator and discussed the propagation of zeroes in the analytic category as a counter example of Treve's conjecture [23]. Recently, in the analytic category, Oaku [16] generalized the results cf Ivrii [9] and Hanges [7] to a system with doubly non-involutive characteristic. Moreover, in the C°° category, Shinkai [20] considered a first order pseudo -differential system with higher order degeneracy and obtained results similar to ours.…”

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confidence: 99%