The overdetermined Cauchy problem
Annales de l'Institut Fourier, Volume 47 (1997) no. 1, pp. 155-199.

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.

On considère le problème de Cauchy (caractéristique et non-caractéristique) pour les systèmes d’équations aux dérivées partielles à coefficients constants et données initiales sur un sous-espace affine de codimension arbitraire. On montre que l’évolution est équivalente à la validité d’un principe de Phragmén-Lindelöf sur la variété caractéristique complexe et on étudie ensuite la relation avec les conditions formulées par Hörmander dans le cas d’un opérateur scalaire et données sur une hypersurface.

     author = {Boiti, Chiara and Nacinovich, Mauro},
     title = {The overdetermined {Cauchy} problem},
     journal = {Annales de l'Institut Fourier},
     pages = {155--199},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {47},
     number = {1},
     year = {1997},
     doi = {10.5802/aif.1564},
     mrnumber = {98a:35095},
     zbl = {0865.35091},
     language = {en},
     url = {}
AU  - Boiti, Chiara
AU  - Nacinovich, Mauro
TI  - The overdetermined Cauchy problem
JO  - Annales de l'Institut Fourier
PY  - 1997
SP  - 155
EP  - 199
VL  - 47
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  -
DO  - 10.5802/aif.1564
LA  - en
ID  - AIF_1997__47_1_155_0
ER  - 
%0 Journal Article
%A Boiti, Chiara
%A Nacinovich, Mauro
%T The overdetermined Cauchy problem
%J Annales de l'Institut Fourier
%D 1997
%P 155-199
%V 47
%N 1
%I Association des Annales de l’institut Fourier
%R 10.5802/aif.1564
%G en
%F AIF_1997__47_1_155_0
Boiti, Chiara; Nacinovich, Mauro. The overdetermined Cauchy problem. Annales de l'Institut Fourier, Volume 47 (1997) no. 1, pp. 155-199. doi : 10.5802/aif.1564.

[AHLM] A. Andreotti, C. D. Hill, S. Lojasiewicz, B. Mackichan, Complexes of Differential operators. The Majer Vietoris sequence, Invent. Math., 26 (1976), 43-86. | MR | Zbl

[AN1] A. Andreotti, M. Nacinovich, Analytic convexity, Ann. S.N.S. Pisa, IV, vol. VII, n. 2 (1980). | Numdam | MR | Zbl

[AN2] A. Andreotti, M. Nacinovich, Analytic Convexity and the Principle of Phragmén-Lindelöf, Quaderni della Scuola Normale Superiore, Pisa (1980). | Zbl

[AN3] A. Andreotti, M. Nacinovich, Noncharacteristic hypersurfaces for complexes of differential operators, Ann. Mat. Pura e Appl., (IV), 125 (1980), 13-83. | MR | Zbl

[AT] A. Andreotti, G. Tomassini, Spazi vettoriali topologici, Quaderni dell'Unione Matematica Italiana, Bologna (1978).

[Bae] A. Baernstein Ii, Representation of holomorphic functions by boundary integrals, Transact. AMS, 160 (1971), 27-37. | MR | Zbl

[BMS] K.D. Bierstedt, R. Meise, W.H. Summers, A projective description of weighted inductive limits, Transact. AMS, 272 (1982), 107-160. | MR | Zbl

[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.

[Eh] L. Ehrenpreis, Fourier analysis in several complex variables, Wiley-Interscience Publisher, New York, 1970. | MR | Zbl

[FW] K. Floret, J. Wloka, Einführung in die Theorie der lokalkonvexen Räume, Lecture Notes in Mathematics, 56, Springer, 1968. | MR | Zbl

[F] U. Franken, On the equivalence of holomorphic and plurisubharmonic Phragmén-Lindelöf principles, Michigan Math. J., 42 (1995), 163-173. | MR | Zbl

[GS] I.M. Gel'Fand e G.E. Shilov, Generalized functions, vol. 1, 2, Academic Press, New York, 1967.

[GR] H. Grauert, R. Remmert, Coherent Analytic Sheaves, Springer, 1984, Grundlehren. | MR | Zbl

[Gr] A. Grothendieck, Espaces vectoriels topologiques, Sociatade de Matemática de S.Paulo, São Paulo, 1964.

[Hö1] L. Hörmander, The analysis of linear partial differential operators, vol. I, II, Springer-Verlag, Berlin, 1983. | Zbl

[Hö2] L. Hörmander, On the existence of analytic solutions of partial differential equations with constant coefficients, Invent. Math., 21 (1973), 151-182. | MR | Zbl

[Hö3] L. Hörmander, Complex analysis in several variables, 3a ediz., North-Holland, Amsterdam, 1991.

[Hö4] L. Hörmander, Notions of convexity, Birkhäuser, Boston, 1994. | Zbl

[Ko] H. Komatsu, Projective and inductive limits of weakly compact sequences of locally convex spaces, J. Math. Soc. Japan, 19 (1967), 366-383. | MR | Zbl

[Kr] S.G. Krantz, Function theory of several complex variables, A Wiley-Interscience publication, Pure & Applied Mathematics, New York, 1982. | MR | Zbl

[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. | MR | Zbl

[N1] M. Nacinovich, On boundary Hilbert differential complexes, Annales Polonici Mathematici, XLVI (1985). | MR | Zbl

[N2] M. Nacinovich, Cauchy problem for overdetermined systems, Annali di Matematica pura ed applicata, (IV), vol. CLVI (1990), 265-321. | MR | Zbl

[N3] M. Nacinovich, Overdetermined Hyperbolic Systems on l.e. Convex Sets, Rend. Sem. Mat. Univ. Padova, vol. 83 (1990). | Numdam | MR | Zbl

[N4] M. Nacinovich, Approximation and extension of Whitney CR forms in “Complex Analysis and Geometry”, pp. 271-283, Plenum Press, N.Y., 1993. | MR | Zbl

[Pa] V. P. Palamodov, Linear differential operators with constant coefficients, Springer Verlag, Berlin, 1970. | MR | Zbl

[Sc] H.H. Schaefer, Topological vector spaces, The Macmillan Company, New-York, 1966. | MR | Zbl

[Sch] L. Schwartz, Théorie des distributions, Hermann, Paris, 1966.

[Tou] J.C. Tougeron, Idéaux de fonctions différentiables, Springer-Verlag, Berlin, 1972. | MR | Zbl

Cited by Sources: