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1996 | 33 | 1 | 149-160
Tytuł artykułu

On the global solvability of linear partial differential equations with constant coefficients in the space of real analytic functions

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
This article surveys results on the global surjectivity of linear partial differential operators with constant coefficients on the space of real analytic functions. Some new results are also included.
Słowa kluczowe
Rocznik
Tom
33
Numer
1
Strony
149-160
Opis fizyczny
Daty
wydano
1996
Twórcy
autor
  • Department of Mathematical Sciences, University of Tokyo, 3-8-1, Komaba, meguro-ku, Tokyo 153, Japan
Bibliografia
  • [A] K. G. Andersson, Global solvability of partial differential equations in the space of real analytic functions, in: Coll. on Analysis, Rio de Janeiro, August 1972, Analyse Fonctionnelle, Hermann, 1974, 1-4.
  • [AN] A. Andreotti and M. Nacinovich, Analytic convexity and the principle of Phragmén-Lindelöf, Lecture Notes, Università di Pisa, 1980.
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  • [BT] G. Bratti and N. Trevisan, Un teorema di riducibilità per la risoluzione di qualche sistema differenziale sovradeterminato, Rend. Sem. Mat. Univ. Politec. Torino, Fasc. Speciale, Settembre (1983), 75-79.
  • [Br] R. W. Braun, A partial differential operator which is surjective on Gevrey classes $Γ^d(R^3)$ with 1 ≤ d < 2 and d ≥ 6 but not for 2 ≤ d < 6, Studia Math. 107 (1993), 157-169.
  • [BMV] 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 $ℝ^N$, preprint.
  • [C1] L. Cattabriga, Sull'esistenza di soluzioni analitiche reali di equazioni a derivate parziali a coefficienti costanti, Boll. Un. Mat. Ital. (4) 12 (1975), 221-234.
  • [C2] L. Cattabriga, Soluzioni di equazioni differenziali a coefficienti costanti appartenenti in un semispazio a certe classi di Gevrey, ibid. 12 (1975), 324-348.
  • [C3] L. Cattabriga, Esistenza di una soluzione fondamentale con supporto singolare contenuto in un semispazio e suriettivita di operatori differenziali a coefficienti costanti, preprint, Univ. di Bologna, 1981.
  • [C4] L. Cattabriga, On the surjectivity of differential polynomials on Gevrey spaces, in: Proceedings 'Linear Partial and Pseudo-Differential Operators', Torino, 1982, Rend. Sem. Mat. Univ. Politec. Torino, Special Issue (1984), 81-89.
  • [C5] L. Cattabriga, Solutions in Gevrey spaces of partial differential equations with constant coefficients, preprint, Univ. di Bologna.
  • [CD1] L. Cattabriga and E. De Giorgi, Sull'esistenza di soluzioni analitiche di equazioni a derivate parziali a coefficienti costanti in un qualunque numero di variabili, Boll. Un. Mat. Ital. (4) 6 (1972), 301-311.
  • [CD2] L. Cattabriga and E. De Giorgi, Soluzioni di equazioni differenziali a coefficienti costanti appartenenti in un semispazio a certe classi di Gevrey, ibid. 12 Suppl. (1975), 324-348.
  • [D] E. De Giorgi, Solutions analytiques des équations aux dérivées partielles à coefficients constants, in: Séminaire Goulaouic-Schwartz, 1971-72, Exposé No. 24.
  • [DC1] E. De Giorgi and L. Cattabriga, 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.
  • [E1] L. Ehrenpreis, Solution of some problems of division I, Amer. J. Math. 76 (1954), 883-903.
  • [E2] L. Ehrenpreis, Mean periodic functions I, ibid. 77 (1955), 293-328.
  • [E3] L. Ehrenpreis, Solution of some problems of division III, ibid. 78 (1956), 685-715.
  • [E4] L. Ehrenpreis, Solution of some problems of division IV, ibid. 82 (1960), 522-588.
  • [E5] L. Ehrenpreis, A fundamental principle for systems of linear partial differential equations with constant coefficients and some of its applications, in: Proc. Intern. Symp. on Linear Spaces, Jerusalem, 1961, 161-174.
  • [E6] L. Ehrenpreis, Fourier Analysis in Several Complex Variables, Wiley-Interscience, 1970.
  • [H1] L. Hörmander, On the range of convolution operators, Ann. of Math. 76 (1962), 148-170.
  • [H2] L. Hörmander, Linear Partial Differential Operators, Springer, 1963.
  • [H3] L. Hörmander, On the existence of real analytic solutions of partial differential equations with constant coefficients, Invent. Math. 21 (1973), 151-182.
  • [Kan1] A. Kaneko, On the global existence of real analytic solutions of linear partial differential equations on unbounded domain, J. Fac. Sci. Univ. Tokyo Sec. 1A 32 (1985), 319-372.
  • [Kan2] A. Kaneko, A sharp sufficient geometric condition for the existence of global real analytic solutions on a bounded domain, J. Math. Soc. Japan 39 (1987), 163-170.
  • [Kan3] A. Kaneko, Remarks on necessary conditions for the existence of global real analytic solutions of linear partial differential equations on a compact set, ibid. 32 (1985), 417-427.
  • [Kan4] A. Kaneko, Introduction to Hyperfunctions, Univ. of Tokyo Press, 1980-1982 (in Japanese); English translation, Kluwer, 1988.
  • [Kan5] A. Kaneko, On the flabbiness of the sheaf of Fourier microfunctions, Sci. Pap. Coll. Gen. Educ. Univ. Tokyo 36 (1986), 1-14.
  • [Kaw1] T. Kawai, On the theory of Fourier hyperfunctions and its applications to partial differential equations with constant coefficients, J. Fac. Sci. Univ. Tokyo Sec. 1A 17 (1970), 467-517.
  • [Kaw2] T. Kawai, On the global existence of real analytic solutions of linear differential equations (I), J. Math. Soc. Japan 24 (1972), 481-517.
  • [Kaw3] T. Kawai, On the global existence of real analytic solutions of linear differential equations (II), ibid. 25 (1973), 644-647.
  • [L1] J.-L. Lieutenant, Application de décompositions des fonctions analytiques à la théorie des microfonctions, Thèse, Univ. de Liège 1981; Résumé in Astérisque 89-90, (1981).
  • [L2] J.-L. Lieutenant, Microlocalization at the boundary of a convex set, J. Fac. Sci. Univ. Tokyo Sec. 1A 33 (1986), 83-130.
  • [Mal1] B. Malgrange, Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, Ann. Inst. Fourier (Grenoble) 6 (1955), 271-355.
  • [Mal2] B. Malgrange, Sur les ouverts convexes par rapport à un opérateur différentiel, C. R. Acad. Sci. Paris 254 (1962), 614-615.
  • [Mi] 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.
  • [Pa1] V. P. Palamodov, Linear Differential Operators with Constant Coefficients, Moscow, 1967 (in Russian); English translation, Springer, 1970; Japanese translation, Yoshioka, 1973.
  • [Pa2] V. P. Palamodov, Functor of projective limit in the category of topological linear spaces, Mat. Sb. 75 (1968), 567-603.
  • [Pi] L. C. Piccinini, Non surjectivity of the Cauchy-Riemann operator on the space of the analytic functions on $R^n$. Generalization to the parabolic operators, Boll. Un. Mat. Ital. (4) 7 (1973), 12-28.
  • [SKK] M. Sato, T. Kawai and M. Kashiwara, Microfunctions and Pseudo-Differential Equations, in: Lecture Notes in Math. 287, Springer, 1973, 263-352.
  • [Sch1] P. Schapira, Front d'onde analytique au bord I, C. R. Acad. Sci. Paris 302 (1986), 383-386.
  • [Sch2] P. Schapira, Front d'onde analytique au bord II, in: Séminaire E.D.P., Ecole Polytech. 1985-86, Exposé No. 1.
  • [Z1] G. Zampieri, A link between $C^∞$ and analytic solvability for P.D.E. with constant coefficients, Rend. Sem. Mat. Univ. Padova 63 (1980), 145-150.
  • [Z2] G. Zampieri, On the stability under localization of the Phragmén-Lindelöf principle, Rend. Accad. Naz. Sci. Mat. 99 (1982), 87-114.
  • [Z3] G. Zampieri, Propagation of singularity and existence of real analytic solutions of locally hyperbolic equations, J. Fac. Sci. Univ. Tokyo Sec. 1A 31 (1984), 373-390.
  • [Z4] 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. (6) 5 (1986), 361-392.
Typ dokumentu
Bibliografia
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