The overdetermined Cauchy problem

Chiara Boiti; Mauro Nacinovich

Annales de l'institut Fourier (1997)

  • Volume: 47, Issue: 1, page 155-199
  • ISSN: 0373-0956

Abstract

top
We consider the (characteristic and non-characteristic) Cauchy problem for a system of constant coefficients partial differential equations with initial data on an affine subspace of arbitrary codimension. We show that evolution is equivalent to the validity of a principle on the complex characteristic variety and we study the relationship of this condition with the one introduced by Hörmander in the case of scalar operators and initial data on a hypersurface.

How to cite

top

Boiti, Chiara, and Nacinovich, Mauro. "The overdetermined Cauchy problem." Annales de l'institut Fourier 47.1 (1997): 155-199. <http://eudml.org/doc/75225>.

@article{Boiti1997,
abstract = {We consider the (characteristic and non-characteristic) Cauchy problem for a system of constant coefficients partial differential equations with initial data on an affine subspace of arbitrary codimension. We show that evolution is equivalent to the validity of a principle on the complex characteristic variety and we study the relationship of this condition with the one introduced by Hörmander in the case of scalar operators and initial data on a hypersurface.},
author = {Boiti, Chiara, Nacinovich, Mauro},
journal = {Annales de l'institut Fourier},
keywords = {overdetermined systems; Cauchy problem; Phragmén-Lindelöf principle},
language = {eng},
number = {1},
pages = {155-199},
publisher = {Association des Annales de l'Institut Fourier},
title = {The overdetermined Cauchy problem},
url = {http://eudml.org/doc/75225},
volume = {47},
year = {1997},
}

TY - JOUR
AU - Boiti, Chiara
AU - Nacinovich, Mauro
TI - The overdetermined Cauchy problem
JO - Annales de l'institut Fourier
PY - 1997
PB - Association des Annales de l'Institut Fourier
VL - 47
IS - 1
SP - 155
EP - 199
AB - We consider the (characteristic and non-characteristic) Cauchy problem for a system of constant coefficients partial differential equations with initial data on an affine subspace of arbitrary codimension. We show that evolution is equivalent to the validity of a principle on the complex characteristic variety and we study the relationship of this condition with the one introduced by Hörmander in the case of scalar operators and initial data on a hypersurface.
LA - eng
KW - overdetermined systems; Cauchy problem; Phragmén-Lindelöf principle
UR - http://eudml.org/doc/75225
ER -

References

top
  1. [AHLM] A. ANDREOTTI, C. D. HILL, S. LOJASIEWICZ, B. MACKICHAN, Complexes of Differential operators. The Majer Vietoris sequence, Invent. Math., 26 (1976), 43-86. Zbl0332.58016MR54 #11404
  2. [AN1] A. ANDREOTTI, M. NACINOVICH, Analytic convexity, Ann. S.N.S. Pisa, IV, vol. VII, n. 2 (1980). Zbl0435.35039MR81m:32025
  3. [AN2] A. ANDREOTTI, M. NACINOVICH, Analytic Convexity and the Principle of Phragmén-Lindelöf, Quaderni della Scuola Normale Superiore, Pisa (1980). Zbl0458.35004
  4. [AN3] A. ANDREOTTI, M. NACINOVICH, Noncharacteristic hypersurfaces for complexes of differential operators, Ann. Mat. Pura e Appl., (IV), 125 (1980), 13-83. Zbl0456.58024MR83e:58079
  5. [AT] A. ANDREOTTI, G. TOMASSINI, Spazi vettoriali topologici, Quaderni dell'Unione Matematica Italiana, Bologna (1978). 
  6. [Bae] A. BAERNSTEIN II, Representation of holomorphic functions by boundary integrals, Transact. AMS, 160 (1971), 27-37. Zbl0225.30044MR44 #415
  7. [BMS] K.D. BIERSTEDT, R. MEISE, W.H. SUMMERS, A projective description of weighted inductive limits, Transact. AMS, 272 (1982), 107-160. Zbl0599.46026MR84g:46037
  8. [BN] C. BOITI, M. NACINOVICH, Evolution and hyperbolic pairs, Preprint n. 2.185.836, Sezione di Analisi Matematica e Probabilità, Dipartimento di Matematica, Università di Pisa, Dicembre 1994. 
  9. [Eh] L. EHRENPREIS, Fourier analysis in several complex variables, Wiley-Interscience Publisher, New York, 1970. Zbl0195.10401MR44 #3066
  10. [FW] K. FLORET, J. WLOKA, Einführung in die Theorie der lokalkonvexen Räume, Lecture Notes in Mathematics, 56, Springer, 1968. Zbl0155.45101MR37 #1945
  11. [F] U. FRANKEN, On the equivalence of holomorphic and plurisubharmonic Phragmén-Lindelöf principles, Michigan Math. J., 42 (1995), 163-173. Zbl0839.32007MR96d:32014
  12. [GS] I.M. GEL'FAND e G.E. SHILOV, Generalized functions, vol. 1, 2, Academic Press, New York, 1967. 
  13. [GR] H. GRAUERT, R. REMMERT, Coherent Analytic Sheaves, Springer, 1984, Grundlehren. Zbl0537.32001MR86a:32001
  14. [Gr] A. GROTHENDIECK, Espaces vectoriels topologiques, Sociatade de Matemática de S.Paulo, São Paulo, 1964. 
  15. [Hö1] L. HÖRMANDER, The analysis of linear partial differential operators, vol. I, II, Springer-Verlag, Berlin, 1983. Zbl0521.35001
  16. [Hö2] L. HÖRMANDER, On the existence of analytic solutions of partial differential equations with constant coefficients, Invent. Math., 21 (1973), 151-182. Zbl0282.35015MR49 #817
  17. [Hö3] L. HÖRMANDER, Complex analysis in several variables, 3a ediz., North-Holland, Amsterdam, 1991. 
  18. [Hö4] L. HÖRMANDER, Notions of convexity, Birkhäuser, Boston, 1994. Zbl0835.32001
  19. [Ko] H. KOMATSU, Projective and inductive limits of weakly compact sequences of locally convex spaces, J. Math. Soc. Japan, 19 (1967), 366-383. Zbl0168.10603MR36 #646
  20. [Kr] S.G. KRANTZ, Function theory of several complex variables, A Wiley-Interscience publication, Pure & Applied Mathematics, New York, 1982. Zbl0471.32008MR84c:32001
  21. [MTV] R. MEISE, B.A. TAYLOR, D. VOGT, Equivalence of Analytic and Plurisubharmonic Phragmén-Lindelöf Conditions, Proceedings of Symposia in Pure Mathematics, vol. 52 (1991), Part 3. Zbl0745.32004MR93a:32023
  22. [N1] M. NACINOVICH, On boundary Hilbert differential complexes, Annales Polonici Mathematici, XLVI (1985). Zbl0606.58046MR88a:58184
  23. [N2] M. NACINOVICH, Cauchy problem for overdetermined systems, Annali di Matematica pura ed applicata, (IV), vol. CLVI (1990), 265-321. Zbl0734.35054MR92a:35117
  24. [N3] M. NACINOVICH, Overdetermined Hyperbolic Systems on l.e. Convex Sets, Rend. Sem. Mat. Univ. Padova, vol. 83 (1990). Zbl0736.35019MR91f:35189
  25. [N4] M. NACINOVICH, Approximation and extension of Whitney CR forms in “Complex Analysis and Geometry”, pp. 271-283, Plenum Press, N.Y., 1993. Zbl0802.32025MR94a:32024
  26. [Pa] V. P. PALAMODOV, Linear differential operators with constant coefficients, Springer Verlag, Berlin, 1970. Zbl0191.43401MR41 #8793
  27. [Sc] H.H. SCHAEFER, Topological vector spaces, The Macmillan Company, New-York, 1966. Zbl0141.30503MR33 #1689
  28. [Sch] L. SCHWARTZ, Théorie des distributions, Hermann, Paris, 1966. 
  29. [Tou] J.C. TOUGERON, Idéaux de fonctions différentiables, Springer-Verlag, Berlin, 1972. Zbl0251.58001MR55 #13472

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.