Cauchy problem in multi-anisotropic Gevrey classes for weakly hyperbolic operators
Calvo, Daniela
Bollettino dell'Unione Matematica Italiana, Tome 9-A (2006), p. 21-50 / Harvested from Biblioteca Digitale Italiana di Matematica

We prove the well-posedness of the Cauchy Problem for first order weakly hyperbolic systems in the multi-anisotropic Gevrey classes, that generalize the standard Gevrey spaces. The result is obtained under the following hypotheses: the principal part is weakly hyperbolic with constant coefficients, the lower order terms satisfy some Levi-type conditions; and lastly the coefficients of the lower order terms belong to a suitable anisotropic Gevrey class. In the proof it is used the quasi-symmetrization of Sylvster-type systems, adapted to the case of the multi-anisotropic Gevrey classes and taking into account the lower order terms.

Si dimostra la buona positura del problema di Cauchy per sistemi debolmente iperbolici nell'ambito delle classi Gevrey multi-anisotrope, generalizzanti le classi Gevrey standard. Il risultato è ottenuto sotto le seguenti ipotesi: la parte principale ha e coefficienti costanti; i termini di ordine inferiore soddisfano delle condizioni di tipo Levi; infine i coefficienti dei termini di ordine inferiore appartengono a un'opportuna classe Gevrey anisotropa. Nella dimostrazione viene utilizzata la tecnica della quasi-simmetrizzazione di sistemi di tipo Sylvester, adattata alle classi Gevrey multi-anisotrope e tenendo conto dei termini di ordine inferiore.

Publié le : 2006-02-01
@article{BUMI_2006_8_9B_1_21_0,
     author = {Daniela Calvo},
     title = {Cauchy problem in multi-anisotropic Gevrey classes for weakly hyperbolic operators},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {9-A},
     year = {2006},
     pages = {21-50},
     zbl = {1178.35236},
     mrnumber = {2204898},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2006_8_9B_1_21_0}
}
Calvo, Daniela. Cauchy problem in multi-anisotropic Gevrey classes for weakly hyperbolic operators. Bollettino dell'Unione Matematica Italiana, Tome 9-A (2006) pp. 21-50. http://gdmltest.u-ga.fr/item/BUMI_2006_8_9B_1_21_0/

[1] Boggiatto, P. - Buzano, E. - Rodino, L., Global hypoellipticity and spectral theory,Akademie Verlag, Berlin, 1996. | MR 1435282 | Zbl 0878.35001

[2] Bronstein, M. D., The Cauchy problem for hyperbolic operators with characteristics of variable multiplicity, Trudy Moscow Math. Soc., 41 (1980), 87-103. | MR 611140

[3] Calvo, D., Generalized Gevrey classes and multi-quasi-hyperbolic operators, Rend. Sem. Mat. Univ. Pol. Torino, 6, N. 2 (2002), 73-100. | MR 1980363 | Zbl 1177.35050

[4] Calvo, D., Cauchy problem in generalized Gevrey classes, Evolution Equations, Propagation Phenomena - Global Existence - Influence of Non-Linearities, Eds. R. Picard, M. Reissig, W. Zajaczkowski, Warszawa2003, Banach Center Publications, Vol. 60, 269-278. | MR 1993054 | Zbl 1024.35063

[5] Calvo, D., Cauchy problem in inhomogeneous Gevrey classes, Progress in Analysis, Proceeding of the 3rd International ISAAC Congress, Vol. II, Eds. G. W. Bgehr, R. B. Gilber, M. W. Wong, World Scientific (Singapore, 2003), 1015-1033. | MR 2032781 | Zbl 1060.35077

[6] Cattabriga, L., Alcuni problemi per equazioni differenziali lineari con coefficienti costanti, Quad. Un. Mat. It., 24, Pitagora 1983, Bologna.

[7] Colombini, F. - Spagnolo, S., An example of weakly hyperbolic Cauchy problem not well-posed in C I , Acta Math., 148 (1982), 243-253. | MR 666112 | Zbl 0517.35053

[8] Corli, A., Un Teorema di rappresentazione per certe classi generalizzate di Gevrey, Boll. Un. Mat. It. Serie VI, Vol. 4-C, N.1 (1985), 245-257. | MR 805217

[9] D'Ancona, P. - Spagnolo, S., Quasi-symmetrization of hyperbolic systems and propagation of the analytic regularity, Boll. Un. Mat. It., 1-B (1998), 165-185. | MR 1618976

[10] Friberg, J., Multi-quasielliptic polynomials, Ann. Sc. Norm. Sup. Pisa, Cl. di Sc., 21 (1967), 239-260. | MR 221090

[11] Garello, G., Generalized Sobolev algebras and regularity for solutions of multiquasi-elliptic semilinear equations, Comm. Appl. Anal., 3, N. 4 (1999), 563-574. | MR 1706710 | Zbl 0933.35204

[12] Garello, G., Pseudodifferential operators with symbols in weighted Sobolev spaces and regularity for non linear partial differential equations, Math. Nachr., 239-240 (2002), 62-79. | MR 1905664 | Zbl 1027.35170

[13] Gindikin, G. - Volevich, L. R., The method of Newton's polyhedron in the theory of partial differential equations, Mathematics and its applications, Soviet Series, 86 (1992). | MR 1256484 | Zbl 0779.35001

[14] Hakobyan, G. H. - Markaryan, V. N., On Gevrey type solutions of hypoelliptic equations, Journal of Contemporary Math. Anal. (Armenian Akad. of Sciences), 31, N. 2 (1996), 33-47. | MR 1683925

[15] Hakobyan, G. H. - Markaryan, V. N., Gevrey class solutions of hypoellipticequations, IZV. Nat. Acad. Nauk Arm., Math., 33, N. 1 (1998), 35-47. | MR 1714534

[16] Hormander, L., The analysis of linear partial differential operators, I, II, III, IV, Springer-Verlag, Berlin, 1983-1985. | MR 705278

[17] Jannelli, E., Sharp quasi-symmetrizer for hyperbolic Sylvester matrices, Communication in the «Workshop on Hyperbolic PDE», Venezia 11-12 April 2002, to appear.

[18] Jannelli, E., On the symmetrization of the principal symbol of hyperbolic equations, Comm. Partial Differential Equations, 14 (1989), 1617-1634. | MR 1039912

[19] Kajitani, K., Local solution of the Cauchy problem for nonlinear hyperbolic systems in Gevrey classes, Hokkaido Math. J., 12 (1983), 434-460. | MR 725589

[20] Larsson, E., Generalized hyperbolicity, Ark. Mat., 7 (1967), 11-32. | MR 221062

[21] Leray, J., Équations hyperboliques non-strictes: contre-exemples, du type De Giorgi, aux théorèmes d'existence et d'unicité, Math. Ann., 162 (1966), pp. 228-236. | MR 217418 | Zbl 0135.15001

[22] Leray, J. - Ohya, Y., Équations et systemes non-lineaires, hyperboliques non-stricts, Math. Ann., 70 (1967), 167-205. | MR 208136 | Zbl 0146.33701

[23] Liess, O. - Rodino, L., Inhomogeneous Gevrey classes and related pseudodifferential operators, Anal. Funz. Appl. Boll. Un. Mat. It., 3-C, N. 1 (1984), 233-323. | MR 749292 | Zbl 0557.35131

[24] Mizohata, S., The theory of partial differential equations, Universiy Press, Cambridge, 1973. | MR 599580 | Zbl 0263.35001

[25] Rodino, L., Linear partial differential operators in Gevrey spaces, World Scientific Publishing Co., Singapore, 1993. | MR 1249275 | Zbl 0869.35005

[26] Steinberg, S., Existence and uniqueness of solutions of hyperbolic equations which are not necessarily strictly hyperbolic, J. Differential Equations, 17 (1975), 119-153. | MR 355359 | Zbl 0287.35065

[27] Zanghirati, L., Iterati di una classe di operatori ipoellittici e classi generalizzate di Gevrey, Suppl. Boll. Un. Mat. It., 1 (1980), 177-195. | MR 629415

[28] Zanghirati, L., Iterati di operatori e regolarita Gevrey microlocale anisotropa, Rend. Sem. Mat. Univ. Padova, 67 (1982), 85-104. | MR 682703