Fuchsian Mild Microfunctions with Fractional Order and their Applications to Hyperbolic Equations
Kataoka introduced a concept of mildness in boundary value problems. He defined mild microfunctions with boundary values. This theory has effective results in propagation of singularities of diffraction. Furthermore, Oaku introduced F-mild microfunctions and applied them to Fuchsian partial differential equations. Based on these theories, we introduce Fuchsian mild microfunctions with fractional order. We show the properties of such microfunctions and their applications to partial differential equations of hyperbolic type. By using a fractional coordinate transform and a quantised Legendre transform, degenerate hyperbolic equations are transformed into equations with derivatives of fractional order. We present a correspondence between solutions for the hyperbolic equations and those for the transformed equations.
KeywordsBoundary value problems microlocal analysis hyperbolic equations hyperfunctions F-mildness
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- 4.M. Kashiwara and P. Schapira, Sheaves on Manifolds, Grundlehren der Math. Wiss. 292, Springer, 1990.Google Scholar
- 5.K. Kataoka, Micro-local theory of boundary value problems. I. Theory of mild hyperfunctions and Green’s formula, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), 355–399.Google Scholar
- 6.K. Kataoka, Microlocal theory of boundary value problems. II. Theorems on regularity up to the boundary for reflective and diffractive operators, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), 31–56.Google Scholar
- 7.K. Kataoka, Microlocal analysis of boundary value problems with regular or fractional power singularities, in Structure of Solutions of Differential Equations, Katata/Kyoto, 1995, World Scientific, 1996, 215–225.Google Scholar
- 8.T. Oaku, A canonical form of a system of microdifferential equations with noninvolutory characteristics and branching of singularities, Invent. Math. 65 (1981/82), 491–525.Google Scholar
- 9.T. Oaku, Microlocal boundary value problem for Fuchsian operators. I. F-mild microfunctions and uniqueness theorem, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32 (1985), 287–317.Google Scholar
- 10.T. Oaku, Higher-codimensional boundary value problem and F-mild hyperfunctions, in Algebraic Analysis, Academic Press, 1988, 571–586.Google Scholar