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2000 | 140 | 1 | 15-40
Tytuł artykułu

Localizations of partial differential operators and surjectivity on real analytic functions

Treść / Zawartość
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EN
Abstrakty
EN
Let P(D) be a partial differential operator with constant coefficients which is surjective on the space A(Ω) of real analytic functions on an open set $ Ω ⊂ ℝ^n$. Then P(D) admits shifted (generalized) elementary solutions which are real analytic on an arbitrary relatively compact open set ω ⊂ ⊂ Ω. This implies that any localization $P_{m,Θ}$ of the principal part $P_m$ is hyperbolic w.r.t. any normal vector N of ∂Ω which is noncharacteristic for $P_{m,Θ}$. Under additional assumptions $P_m$ must be locally hyperbolic.
Czasopismo
Rocznik
Tom
140
Numer
1
Strony
15-40
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-02-01
poprawiono
1999-12-13
Twórcy
Bibliografia
  • [1] K. G. Andersson, Propagation of analyticity of solutions of partial differential equations with constant coefficients, Ark. Mat. 8 (1971), 277-302.
  • [2] K. G. Andersson, Global solvability of partial differential equations in the space of real analytic functions, in: Analyse Fonctionnelle et Applications (Coll. Analyse, Rio de Janeiro, August 1972), Actualités Sci. Indust. 1367, Hermann, Paris, 1975, 1-4.
  • [3] A. Andreotti and M. Nacinovich, Analytic Convexity and the Principle of Phragmén-Lindelöf, Scuola Norm. Sup., Pisa, 1980.
  • [4] G. Bengel, Das Weylsche Lemma in der Theorie der Hyperfunktionen, Math. Z. 96 (1967), 373-392.
  • [5] J. Bochnak, M. Coste et M. F. Roy, Géométrie Algébrique Réelle, Ergeb. Math. Grenzgeb. (3) 12, Springer, Berlin, 1987.
  • [6] J. M. Bony, Extensions du théorème de Holmgren, Sém. Goulaouic-Schwartz, Exp. 17, Centre Math., École Polytechnique, Palaiseau, 1976, 13 pp.
  • [7] J. M. Bony, Propagation of analytic and differentiable singularities for solutions of partial differential equations, Publ. Res. Inst. Math. Sci. Suppl. 12 (1976/77), 5-17.
  • [8] J. M. Bony and P. Schapira, Propagation des singularités analytiques pour les solutions des équations aux dérivées partielles, Ann. Inst. Fourier (Grenoble) 26 (1976), no. 1, 81-140.
  • [9] R. W. Braun, The surjectivity of a constant coefficient homogeneous differential operator on the real analytic functions and the geometry of its symbol, ibid. 45 (1995), 223-249.
  • [10] R. W. Braun, R. Meise and D. Vogt, Application of the projective limit functor to convolution and partial differential equations, in: T. Terzioğlu (ed.), Advances in the Theory of Fréchet Spaces, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 287, Kluwer, 1989, 29-46.
  • [11] R. W. Braun, R. Meise and D. Vogt, Characterization of the linear partial differential operators with constant coefficients, which are surjective on non-quasianalytic classes of Roumieu type on, Math. Nachr 168 (1994), 19-54.
  • [12] L. Cattabriga ed E. de Giorgi, Una dimostrazione diretta dell'esistenza di soluzioni analitiche nel piano reale di equazioni a derivate parziali a coefficienti costanti, Boll. Un. Mat. Ital. (4) 4 (1971), 1015-1027.
  • [13] M. L. de Cristoforis, Soluzioni con lacune di certi operatori differenziali lineari, Rend. Accad. Naz. Sci. XL Mem. Mat. (5) 8 (1984), 137-142.
  • [14] J. Fehrman, Hybrids between hyperbolic and elliptic differential operators with constant coefficients, Ark. Mat. 13 (1975), 209-235.
  • [15] L. Gårding, Local hyperbolicity, Israel J. Math. 13 (1972), 65-81.
  • [16] A. Grigis, P. Schapira et J. Sjöstrand, Propagation de singularités analytiques pour les solutions des opérateurs à caractéristiques multiples, C. R. Acad. Sci. Paris Sér. I 293 (1981), 397-400.
  • [17] N. Hanges, Propagation of analyticity along real bicharacteristics, Duke Math. J. 48 (1981), 269-277.
  • [18] N. Hanges and J. Sjöstrand, Propagation of analyticity for a class of non-micro-characteristic operators, Ann. of Math. 116 (1982), 559-577.
  • [19] L. Hörmander, Uniqueness theorems and wave front sets for solutions of linear differential equations with analytic coefficients, Comm. Pure Appl. Math. 24 (1971), 671-704.
  • [20] L. Hörmander, On the singularities of solutions of partial differential equations with constant coefficients, Israel J. Math. 13 (1972), 82-105.
  • [21] L. Hörmander, On the existence of real analytic solutions of partial differential equations with constant coefficients, Invent. Math. 21 (1973), 151-183.
  • [22] L. Hörmander, The Analysis of Linear Partial Differential Operators I,II, Grundlehren Math. Wiss. 256, 257, Springer, Berlin, 1983.
  • [23] A. Kaneko, On the global existence of real analytic solutions of linear partial differential equations on unbounded domain, J. Fac. Sci. Tokyo Sect. IA Math. 32 (1985), 319-372.
  • [24] M. Kashiwara and T. Kawai, Microhyperbolic pseudodifferential operators I, J. Math. Soc. Japan 27 (1975), 359-404.
  • [25] T. Kawai, On the global existence of real analytic solutions of linear differential equations I, ibid. 24 (1972), 481-517.
  • [26] M. Langenbruch, Surjective partial differential operators on spaces of ultradifferentiable functions of Roumieu type, Results Math. 29 (1996), 254-275.
  • [27] M. Langenbruch, Continuation of Gevrey regularity for solutions of partial differential operators, in: S. Dierolf, S. Dineen and P. Domański (eds.), Functional Analysis, Proc. First Workshop at Trier University, de Gruyter, 1996, 249-280.
  • [28] M. Langenbruch, Surjective partial differential operators on Gevrey classes and their localizations at infinity, Linear Topol. Spaces Complex Anal. 3 (1997), 95-111,
  • [29] M. Langenbruch, Surjectivity of partial differential operators in Gevrey classes and extension of regularity, Math. Nachr. 196 (1998), 103-140.
  • [30] M. Langenbruch, Extension of analyticity for solutions of partial differential operators, Note Mat. 17 (1997), 29-59.
  • [31] M. Langenbruch, Surjective partial differential operators on spaces of real analytic functions, preprint.
  • [32] P. Laubin, Propagation des singularités analytiques pour des opérateurs à caractéristiques involutives de multiplicité variable, Portugal. Math. 41 (1982), 83-90.
  • [33] P. Laubin, Analyse microlocale des singularités analytiques, Bull. Soc. Roy. Sci. Liège 52 (1983), 103-212.
  • [34] O. Liess, Necessary and sufficient conditions for propagation of singularities for systems of partial differential equations with constant coefficients, Comm. Partial Differential Equations 8 (1983), 89-198.
  • [35] R. Meise und D. Vogt, Einführung in die Funktionalanalysis, Vieweg, Braunschweig, 1992.
  • [36] T. Miwa, On the global existence of real analytic solutions of systems of linear differential equations with constant coefficients, Proc. Japan Acad. 49 (1973), 500-502.
  • [37] L. C. Piccinini, Non surjectivity of the Cauchy-Riemann operator on the space of the analytic functions on $ℝ^n$, Boll. Un. Mat. Ital. (4) 7 (1973), 12-28.
  • [38] J. Sjöstrand, Singularités Analytiques Microlocales, Astérisque 95 (1982), 1-166.
  • [39] G. Zampieri, Operatori differenziali a coefficienti costanti di tipo iperbolico-(ipo) ellittico, Rend. Sem. Mat. Univ. Padova 72 (1984), 27-44.
  • [40] G. Zampieri, Propagation of singularity and existence of real analytic solutions of locally hyperbolic equations, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31 (1984), 373-390.
  • [41] G. Zampieri, An application of the fundamental principle of Ehrenpreis to the existence of global Gevrey solutions of linear differential equations, Boll. Un. Mat. Ital. B (6) 5 (1986), 361-392.
Typ dokumentu
Bibliografia
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