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

Akira Kaneko

Banach Center Publications (1996)

  • Volume: 33, Issue: 1, page 149-160
  • ISSN: 0137-6934

Abstract

top
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.

How to cite

top

Kaneko, Akira. "On the global solvability of linear partial differential equations with constant coefficients in the space of real analytic functions." Banach Center Publications 33.1 (1996): 149-160. <http://eudml.org/doc/262544>.

@article{Kaneko1996,
abstract = {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.},
author = {Kaneko, Akira},
journal = {Banach Center Publications},
keywords = {micro-local analysis},
language = {eng},
number = {1},
pages = {149-160},
title = {On the global solvability of linear partial differential equations with constant coefficients in the space of real analytic functions},
url = {http://eudml.org/doc/262544},
volume = {33},
year = {1996},
}

TY - JOUR
AU - Kaneko, Akira
TI - On the global solvability of linear partial differential equations with constant coefficients in the space of real analytic functions
JO - Banach Center Publications
PY - 1996
VL - 33
IS - 1
SP - 149
EP - 160
AB - 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.
LA - eng
KW - micro-local analysis
UR - http://eudml.org/doc/262544
ER -

References

top
  1. [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. 
  2. [AN] A. Andreotti and M. Nacinovich, Analytic convexity and the principle of Phragmén-Lindelöf, Lecture Notes, Università di Pisa, 1980. Zbl0458.35004
  3. [Bj] G. B. Björk, Linear partial differential operators and generalized distributions, Ark. Mat. 6 (1966), 351-407. 
  4. [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. 
  5. [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. 
  6. [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 N , preprint. Zbl0848.35023
  7. [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. Zbl0328.35009
  8. [C2] L. Cattabriga, Soluzioni di equazioni differenziali a coefficienti costanti appartenenti in un semispazio a certe classi di Gevrey, ibid. 12 (1975), 324-348. Zbl0336.35018
  9. [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. 
  10. [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. 
  11. [C5] L. Cattabriga, Solutions in Gevrey spaces of partial differential equations with constant coefficients, preprint, Univ. di Bologna. 
  12. [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. Zbl0258.35006
  13. [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. Zbl0336.35018
  14. [D] E. De Giorgi, Solutions analytiques des équations aux dérivées partielles à coefficients constants, in: Séminaire Goulaouic-Schwartz, 1971-72, Exposé No. 24. Zbl0244.35017
  15. [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. 
  16. [E1] L. Ehrenpreis, Solution of some problems of division I, Amer. J. Math. 76 (1954), 883-903. Zbl0056.10601
  17. [E2] L. Ehrenpreis, Mean periodic functions I, ibid. 77 (1955), 293-328. 
  18. [E3] L. Ehrenpreis, Solution of some problems of division III, ibid. 78 (1956), 685-715. 
  19. [E4] L. Ehrenpreis, Solution of some problems of division IV, ibid. 82 (1960), 522-588. Zbl0098.08401
  20. [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. 
  21. [E6] L. Ehrenpreis, Fourier Analysis in Several Complex Variables, Wiley-Interscience, 1970. 
  22. [H1] L. Hörmander, On the range of convolution operators, Ann. of Math. 76 (1962), 148-170. Zbl0109.08501
  23. [H2] L. Hörmander, Linear Partial Differential Operators, Springer, 1963. 
  24. [H3] L. Hörmander, On the existence of real analytic solutions of partial differential equations with constant coefficients, Invent. Math. 21 (1973), 151-182. Zbl0282.35015
  25. [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. 
  26. [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. Zbl0619.35019
  27. [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. 
  28. [Kan4] A. Kaneko, Introduction to Hyperfunctions, Univ. of Tokyo Press, 1980-1982 (in Japanese); English translation, Kluwer, 1988. 
  29. [Kan5] A. Kaneko, On the flabbiness of the sheaf of Fourier microfunctions, Sci. Pap. Coll. Gen. Educ. Univ. Tokyo 36 (1986), 1-14. 
  30. [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. 
  31. [Kaw2] T. Kawai, On the global existence of real analytic solutions of linear differential equations (I), J. Math. Soc. Japan 24 (1972), 481-517. Zbl0234.35012
  32. [Kaw3] T. Kawai, On the global existence of real analytic solutions of linear differential equations (II), ibid. 25 (1973), 644-647. 
  33. [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). 
  34. [L2] J.-L. Lieutenant, Microlocalization at the boundary of a convex set, J. Fac. Sci. Univ. Tokyo Sec. 1A 33 (1986), 83-130. 
  35. [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. Zbl0071.09002
  36. [Mal2] B. Malgrange, Sur les ouverts convexes par rapport à un opérateur différentiel, C. R. Acad. Sci. Paris 254 (1962), 614-615. Zbl0117.06301
  37. [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. Zbl0273.35013
  38. [Pa1] V. P. Palamodov, Linear Differential Operators with Constant Coefficients, Moscow, 1967 (in Russian); English translation, Springer, 1970; Japanese translation, Yoshioka, 1973. 
  39. [Pa2] V. P. Palamodov, Functor of projective limit in the category of topological linear spaces, Mat. Sb. 75 (1968), 567-603. Zbl0175.41801
  40. [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. Zbl0264.35003
  41. [SKK] M. Sato, T. Kawai and M. Kashiwara, Microfunctions and Pseudo-Differential Equations, in: Lecture Notes in Math. 287, Springer, 1973, 263-352. 
  42. [Sch1] P. Schapira, Front d'onde analytique au bord I, C. R. Acad. Sci. Paris 302 (1986), 383-386. 
  43. [Sch2] P. Schapira, Front d'onde analytique au bord II, in: Séminaire E.D.P., Ecole Polytech. 1985-86, Exposé No. 1. 
  44. [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. 
  45. [Z2] G. Zampieri, On the stability under localization of the Phragmén-Lindelöf principle, Rend. Accad. Naz. Sci. Mat. 99 (1982), 87-114. Zbl0514.35013
  46. [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. 
  47. [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. Zbl0624.35011

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.