On weighted hyperbolic polynomials. (English. Russian original) Zbl 1314.35021
J. Contemp. Math. Anal., Armen. Acad. Sci. 49, No. 5, 212-222 (2014); translation from Izv. Nats. Akad. Nauk Armen., Mat. 49, No. 5, 23-39 (2014).
Summary: The results by Gårding, Larsson, Cattabriga, Rodino, Calvo on correctness of the Cauchy problem for \(N\)-hyperbolic equations are generalized. We prove that in the general case where the vector \(N = (N_1,\ldots, N_n)\) is different from the vector \((1, 0, \ldots, 0)\), for the correctness of the Cauchy problem a stronger condition is required, which we call weighted hyperbolicity condition. We also discuss the properties of polynomials possessing the weighted hyperbolicity property.
Keywords:
hypoelliptic operator; weighted hyperbolic polynomial; Gevrey multianisotropic space; completely regular Newton polyhedronReferences:
| [1] | V.P. Mikhailov, “On behavior at infinity of a class of polynomials”, Trudy MIAN SSSR, 91, 59-81, 1967. · Zbl 0185.33901 |
| [2] | L.R. Volevich, S.G. Gindikin, “On a class of hypoelliptic operators”, Mat. Sbornik, 75 (117)(3), 400-416, 1968. · Zbl 0155.43401 |
| [3] | E.E. Levi, “Caracterische multiple e probleme di Cauchy”, Ann. Mat. Pure Apple., 16(3), 161-201, 1909. · JFM 40.0415.02 · doi:10.1007/BF02419694 |
| [4] | I.G. Petrovskii, “On Cauchy problem for systems of linear differential equations with partial derivatives in the domain non-analytic functions”, Bull. Mosk. Uni. (A), 1(7), 1-72, 1951. |
| [5] | L. Görding, “Linear hyperbolic partial differential equations with constant coefficients”, Acta Math., 85, 1-62, 1951. · Zbl 0045.20202 · doi:10.1007/BF02395740 |
| [6] | L. Hörmander, The Analysis of Linear Partial Differential Operators. 2 (Springer — Verlag, Berlin, 1983. · Zbl 0521.35002 |
| [7] | S. Mizohata, On the Cauchy Problem. Notes and Repotes in Mathematics, in Science and Enginering 3 (Academic Press, Orlando; Science Press, Beiging, 1985). · Zbl 0616.35002 |
| [8] | W. Matsumoto, H. Yamahara, “On Cauchy — Kowalevskaya theorem for system”, Proc. Japan Acad. ser. A, Math Sci., 67(6), 181-185, 1991. · Zbl 0797.35004 · doi:10.3792/pjaa.67.181 |
| [9] | M. Gevre, “Sur la nature analitique des solutions des equations aux derivatives partielles”, Ann. Ec. Norm. Sup., Paris, 35, 129-190, 1918. · JFM 46.0721.01 |
| [10] | P.D. Lax, “Asimptotic solutions of oscillatory initial value problems”, Duke Math. J., 24, 627-646, 1974. · Zbl 0083.31801 · doi:10.1215/S0012-7094-57-02471-7 |
| [11] | S.L. Svensson, “Necessary and sufficient conditions for the hyperbolisity of polynomials with hyperbolic principal parts”, Ark. Mat., 8, 145-162, 1969. · Zbl 0203.40903 · doi:10.1007/BF02589555 |
| [12] | E. Larsson, “Generalized hyperbolisity”, Ark. Mat., 7, 11-32, 1967. · Zbl 0183.41302 · doi:10.1007/BF02591674 |
| [13] | L. Cattabriga, “Alcuni problemi per equazioni differenziali lineari con coefficienti constanti”, Quad. Un. Mat. It., Bologna, 24, 1983. · Zbl 0557.35131 |
| [14] | O. Liess, L. Rodino, “Inhomogeneous Gevrey klasses and related pseudodifferetial operators”, Anal. Funz. Appl. Suppl. Boll. Un. Mat. It., 3C(1), 233-323, 1984. · Zbl 0557.35131 |
| [15] | L. Görding, “Local hyperbolisity”, Proc. of the International Simposium on Partial Diff. Equations (Jerusalim, 1972) Israel J. Math., 13, 65-81, 1972. · Zbl 0263.35062 |
| [16] | L. Rodino, Linear Partial Diff. Operators in Gevrey Spaces (World Scientific, Singapore, 1993). · Zbl 0869.35005 · doi:10.1142/9789814360036 |
| [17] | D. Calvo, Multianisotropic Gevrey Classes and Caushy Problem, Ph. D. Thesis in Mathematics (Universita degli Studi di Pisa). |
| [18] | K.A. Yagjian, “Necessary conditions for correctness of Cauchy problem for operators with multiple characteristics”, Izv. ANArm. SSR, Mat., 23(5), 439-462, 1988. · Zbl 0689.35020 |
| [19] | K.A. Yagjian, The Cachy Problem for Hyperbolic Operators. Multiple Characteristics. Micro-local Approach (Mathematical Topics, Akademie Verlag, Berlin, 1997). · Zbl 0887.35002 |
| [20] | V.N. Margaryan, G.H. Hakobyan, “On Gevrey type solutions of hypoelliptic equations”, Izv. Nat. Acad. Armenii, Math., 31(2), 33-47, 2002. · Zbl 0900.35094 |
| [21] | A. Corli, “Un teorema di rappresentazione per certe classi generelizzate di Gevrey”, Boll. Un. Mat. It., 6, 4-C(1), 245-257, 1985. · Zbl 0599.35030 |
| [22] | L. Zanghirati, “Iterati di operatori e regolarita Gevrey microlocale anisotropa”, Rend. Sem. Mat. Univ. Padova, 67, 85-104, 1982. · Zbl 0516.35020 |
| [23] | H. Komatsu, “Ultradistributions. Structure theorems and a characterisation” J. Pac. Sci. Univ. Tokyo, 20, 25-105, 1973. · Zbl 0258.46039 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
