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2000 | 74 | 1 | 143-159
Tytuł artykułu

Liouville type theorem for solutions of linear partial differential equations with constant coefficients

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Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We discuss existence of global solutions of moderate growth to a linear partial differential equation with constant coefficients whose total symbol P(ξ) has the origin as its only real zero. It is well known that for such equations, global solutions tempered in the sense of Schwartz reduce to polynomials. This is a generalization of the classical Liouville theorem in the theory of functions. In our former work we showed that for infra-exponential growth the corresponding assertion is true if and only if the complex zeros of P(ξ) are absent in a strip at infinity. In this article we discuss the growth in between and present a characterization employing the space of ultradistributions corresponding to the growth.
Rocznik
Tom
74
Numer
1
Strony
143-159
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-05-31
poprawiono
1999-11-20
Twórcy
autor
  • Department of Information Sciences, Ochanomizu University, 2-1-1, Otsuka, Bunkyo-ku, Tokyo 112-8610, Japan
Bibliografia
  • [G] A. Grothendieck, Topological Vector Spaces, Gordon and Breach, 1973.
  • [H1] L. Hörmander, Linear Partial Differential Operators, Springer, 1963.
  • [H2] L. Hörmander, Between distributions and hyperfunctions, Astérisque 131 (1985), 89-106.
  • [Kn1] A. Kaneko, On the structure of Fourier hyperfunctions, Proc. Japan Acad. 48 (1972), 651-653.
  • [Kn2] A. Kaneko, Introduction to Hyperfunctions, Univ. of Tokyo Press, 1980-1982 (in Japanese); English translation, Kluwer, 1988.
  • [Kn3] A. Kaneko, Liouville type theorem for solutions of infra-exponential growth of linear partial differential equations with constant coefficients, Nat. Sci. Report Ochanomizu Univ. 49 (1998), 1-5.
  • [Kw] T. Kawai, On the theory of Fourier hyperfunctions and its applications to partial differential equations with constant coefficients, J. Fac. Sci. Univ. Tokyo Sect. 1A 17 (1970), 467-517.
  • [Km] H. Komatsu, Ultradistributions I, ibid. 20 (1973), 25-105.
  • [MY] M. Morimoto and K. Yoshino, Some examples of analytic functionals with carrier at the infinity, Proc. Japan Acad. 56 (1980), 357-361.
  • [P] V. P. Palamodov, From hyperfunctions to analytic functionals, Soviet Math. Dokl. 18 (1977), 975-979.
  • [PM] Y. S. Park and M. Morimoto, Fourier ultra-hyperfunctions in the Euclidean n-space, J. Fac. Sci. Univ. Tokyo Sect. 1A 20 (1973), 121-127.
  • [dR] J. W. de Roever, Hyperfunctional singular support of ultradistributions, ibid. 31 (1985), 585-631.
  • [SM] P. Sargos et M. Morimoto, Transformation des fonctionnelles analytiques à porteurs non compacts, Tokyo J. Math. 4 (1981), 457-492.
  • [S] L. Schwartz, Théorie des Distributions, new ed., Hermann, Paris, 1966.
  • [Z] B. Ziemian, The Mellin transformation and multidimensional generalized Taylor expansions of singular functions, J. Fac. Sci. Univ. Tokyo Sect. 1A 36 (1989), 263-295.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-apmv74z1p143bwm
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