Characteristic Cauchy problems and solutions of formal power series

Ouchi, Sunao

doi:10.5802/aif.907

Cited by 12 publications

(12 citation statements)

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“…But the existence of solutions with an asymptotic expansion is not studied in those papers. It is studied in [4], where characteristic Cauchy problems are considered. Characteristic Cauchy problems have formal power series solutions.…”

Section: §0 Introductionmentioning

confidence: 99%

Existence of Solutions with Asymptotic Expansion of Linear Partial Differential Equations in the Complex Domain

Ouchi¹

2004

Publ. Res. Inst. Math. Sci.

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Consider the linear partial differential equation P (z, ∂ z )u(z) = f (z) in C d+1 , where f (z) is not holomorphic on K = {z 0 = 0}, but it has an asymptotic expansion with respect to z 0 as z 0 → 0 in some sectorial region. We show under some conditions on P (z, ∂ z ) that there exists a solution u(z) which has an asymptotic expansion of the same type as that of f (z). §0. Introduction Let P (z, ∂ z ) be a linear partial differential operator with holomorphic coefficients in a neighborhood Ω of z = 0 in C d+1 and K = {z 0 = 0}. Consider the equationwhere f (z) is holomorphic except on K, but f (z) has an asymptotic expansion f (z) ∼ n f n (z )z n 0 as z 0 → 0 in some sectorial region with respect to z 0 . In the present paper we study the existence of solutions. Firstly we remark that if we require nothing about the behavior of u(z) near K, there exists a solution u(z) with singularities on K under some conditions on the principal symbol of P (z, ∂ z ). But the singularities of u(z) may be much stronger than those of f (z) (see [1], [2], [5] and [9]).

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Section: §0 Introductionmentioning

confidence: 99%

Existence of Solutions with Asymptotic Expansion of Linear Partial Differential Equations in the Complex Domain

Ouchi¹

2004

Publ. Res. Inst. Math. Sci.

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“…From a different point of view, Ouchi developed the theory concerning the analytic meaning of formal solutions [9]. It is certain that his theory implies one part of our theorem.…”

Section: To Consider Cauchy Problemsmentioning

confidence: 99%

“…Definition, The Newton polygon of P, denoted by N(P) 9 Remark. If P is a differential operator with holomorphic coefficients, this definition is a special case of more general one [4,5,6]: If we choose…”

Section: To Consider Cauchy Problemsmentioning

confidence: 99%

Newton Polygons and Formal Gevrey Classes

Yonemura¹

1990

Publ. Res. Inst. Math. Sci.

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“…Using techniques due to Boutet de Monvel and Kree, Yonemura simplified the proof (Boutet de Monvel and Krée 1967;Ouchi 1983;Yonemura 1990). Our proof will be self-contained.…”

Section: Order S Is Of Gevrey Class αS In the Time Variable That Ismentioning

confidence: 99%

“…In the 1980s, Ouchi made an important step, when he discovered that the divergent series associated to time evolution of a single linear partial differential equation are always asymptotic expansions of sectorial solutions (Ouchi 1983) (see also Tahara 2011a, b;Yonemura 1990 and references therein). We will extend the results of Kovalevskaïa thesis and Ouchi's theorem to arbitrary systems of linear partial differential equations with Gevrey coefficients.…”

Section: Introductionmentioning

confidence: 99%