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On the analyticity of the locus of singularity of real analytic solutions with minimal dimension

Published online by Cambridge University Press:  22 January 2016

Akira Kaneko
Akira Kaneko*
Affiliation:
Department of Mathematics, College of General Education, University of Tokyo, 3-8-1, Komaba, Meguro-ku Tokyo 153, Japan
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Let P(x, D) be a linear partial differential operator with real analytic coefficients and let C ⊂ Rn be a germ of closed subset, say at the origin. We say that C is (the locus of) an irremovable singularity of a real analytic solution u of P(x, D)u = 0 if u is defined outside C on a neighborhood Ω of 0 but cannot be extended to the whole neighborhood Ω even as a hyperfunction solution of P(x, D)u = 0. This usage of the word “singularity” is the same as the one for the analytic functions in complex analysis, and is different of the usual usage of “singularities of solutions” in the theory of partial differential equations.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 104 , December 1986 , pp. 63 - 84
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1986

References

[1] Ahlfors, L. & Sario, L. [1] Riemann Surfaces, Princeton Univ. Press, Princeton, 1960.Google Scholar
[1] Bochner, S. & Martin, W. T. [1] Several Complex Variables, Princeton Univ. Press, Princeton, 1948.Google Scholar
[1] Hartogs, F. [1] Über die aus den singulären Stellen einer analytischen Funktion mehrerer Veränderlichen bestehenden Gebilde, Acta Math., 32 (1908), 57–79.Google Scholar
[1] Hitotumatu, S. [1] Theory of Analytic Functions of Several Variables (in Japanese), Baifukan, Tokyo, 1960.Google Scholar
[1] John, F. [1] Plane Waves and Spherical Means, Interscience, New York, 1955.Google Scholar
[1] Kajiwara, J. [1] Theory of Complex Functions (in Japanese), Morikita, Tokyo, 1968.Google Scholar
[1] Kaneko, A. [1] On the singular spectrum of boundary values of real analytic solutions, J. Math. Soc. Japan, 29 (1977), 385–398.Google Scholar
[2] Kaneko, A. Singular spectrum of boundary values of solutions of partial differential equations with real analytic coefficients, Sci. Papers College Gen. Ed. Univ. Tokyo, 25 (1975), 59–68; Correction in Appendix B of Kaneko [5].Google Scholar
[3] Kaneko, A. Analyticity of minimal dimensional singularity of real analytic solutions, ibid., 26 (1976), 1–5.Google Scholar
[4] Kaneko, A. On continuation of regular solutions of linear partial differential equations, Publ. RIMS Kyoto Univ., 12, Suppl. (1977), 113–121.CrossRefGoogle Scholar
[5] Kaneko, A. Estimation of singular spectrum of boundary values for some semihyperbolic operators, J. Fac. Sci. Univ. Tokyo Sec. 1A, 27 (1980), 401–461.Google Scholar
[6] Kaneko, A. On continuation of real analytic solutions of linear partial differential equations, Astérisque, 89–90, Soc. Math. France, 1981, pp. 11–44.Google Scholar
[7] Kaneko, A. On the propagation of micro-analyticity along the boundary, J. Fac. Sci. Univ. Tokyo Sec. 1A, 29 (1982), 319–352.Google Scholar
[8] Kaneko, A. On the analyticity of the locus of singularity of real analytic solutions with minimal dimension (in Japanese), Sûrikaiseki-kenkyûsho Kôkyûroku, 533 (1984), 256–271.Google Scholar
[1] Kataoka, K. [1] On the theory of Radon transformations of hyperfunctions, J. Fac. Sci. Univ. Tokyo Sec. 1A, 28 (1981), 331–413.Google Scholar
[2] Kataoka, K. Micro-local theory of boundary value problems I, J. Fac. Sci. Univ. Tokyo Sec. 1A, 27 (1980), 355–399.Google Scholar
[3] Kataoka, K. Micro-local theory of boundary value problems II, J. Fac. Sci. Univ. Tokyo Sec. 1A, 28 (1981), 31–56.Google Scholar
[1] Kawai, T. [1] Construction of elementary solutions for I-hyperbolic operators and solutions with small singularities, Proc. Japan Acad., 46 (1970), 912–916.Google Scholar
[1] Komatsu, H. & Kawai, T. [1] Boundary values of hyperfunction solutions of linear partial differential equations, Publ. RIMS Kyoto Univ., 7 (1971), 95–104.CrossRefGoogle Scholar
[1] Kurosh, A. G. [1] Theory of Groups I, II, Chelsea, New York, 1955–56.Google Scholar
[1] Nevanlinna, R. [1] Uniformisierung, Springer, Berlin-Göttingen-Heidelberg, 1953.Google Scholar
[1] Nishino, T. [1] Nouvelles recherches sur les fonctions entières de plusieurs variables complexes (II). Fonctions entières qui se reduisent à celles d’une variable, J. Math. Kyoto Univ., 9 (1969), 221–274.Google Scholar
[2] Nishino, T. Ditto (III). Sur quelques propriétés topologiques des surfaces premières, ibid., 10 (1970), 245–271.Google Scholar
[3] Nishino, T. Ditto (IV). Types de surfaces premières, ibid., 13 (1973), 217–272.Google Scholar
[1] Ohsawa, T. [1] Personal correspondences, September 1984 to January 1985.Google Scholar
[1] Oka, K. [1] Sur les fonctions analytiques de plusieurs variables IX. Domaines finis sans point critique intérieur, Japan. J. Math., 27 (1953), 97–155; Reprinted in Collected Papers from Iwanami, Tokyo 1961.Google Scholar
[1] Schapira, P. [1] Propagation au bord et réflexion des singularités analytiques des solutions des équations aux dérivées partielles, Séminaire Goulaouic-Schwartz 1975–76, exposé 6.Google Scholar
[2] Schapira, P. Propagation au bord et réflexion des singularités analytiques des solutions des équations aux dérivées partielles II, Séminaire Goulaouic-Schwartz 1976–77, exposé 9.Google Scholar
[1] Siciak, J. [1] Analyticity and separate analyticity of functions defined on lower dimensional subsets of Cn , Zeszyty Nauk. Uniw. Jagielloń, Prace Mat., 13 (1969), 53–70.Google Scholar
[1] Stein, K. [1] Überlagerungen holomorph-voliständiger komplexer Räurne, Arch. Math., 7 (1956), 354–361.Google Scholar
[1] Tsuji, M. [1] Potential Theory, Maruzen, Tokyo, 1959.Google Scholar
[1] Yamaguchi, H. [1] Parabolicité d’une fonction entière, J. Math. Kyoto Univ., 16 (1976), 71–92.Google Scholar