Hyperbolicity with weight of polynomials in terms of comparing their power

H. G. Ghazaryan ; V. N. Margaryan

Eurasian mathematical journal, Tome 11 (2020) no. 2, pp. 40-51.

Résumé

For a given completely regular Newton polyhedron $\mathfrak{R}$, and a given vector $N\in\mathbb{R}^n$, we give conditions under which a weakly hyperbolic polynomial (with respect to the vector $N$) $P(\xi)=P(\xi_1,\dots,\xi_n)$ is $\mathfrak{R}$-hyperbolic (with respect to the vector $N$). For polynomials of two variables, the largest number $s >0$ is determined for which an $\mathfrak{R}$-hyperbolic (with respect to the vector $N$) polynomial is $s$-hyperbolic.

Bibliographie

[1] D. Calvo, Multianisotropic Gevrey classes and Cauchy problem, PhD Thesis in Mathematics, Universita degli Studi di Pisa, 2000

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

[3] A. Corli, “Un teorema di rappresentazione per certe classi generelizzate di Gevrey”, Boll. Un. Mat. It. Serie 6, 4C:1 (1985), 245–257 | MR | Zbl

[4] M. Gevre, “Sur la nature analitique des solutions des equations aux derivatives partielles”, Ann. Ec. Norm. Sup., Paris, 35 (1918), 129–190 | MR | Zbl

[5] S. G. Gindikin, L. R. Volevich, The method of Newton's polyhedron in the theory of PDE, Math. and its applications. Soviet series, Kluwer Academic Publishers, 1992 | MR

[6] L. Gȧrding, “Linear hyperbolic partial differential equations with constant coefficients”, Acta Math., 85 (1951), 1–62 | MR

[7] L. Hörmander, The analysis of linear partial differential operators, v. 2, Springer Verlag, 1983 | MR

[8] L. Hörmander, The analysis of linear partial differential operators, v. 1, Springer Verlag, 1983 | MR

[9] V. Ya. Ivri, “Well posedness in Gevrey class of the Cauchy problem for non strictly hyperbolic equation”, Math. Sb., 96 (1975), 390–413 (in Russian) | MR | Zbl

[10] V. Ya. Ivri, “Linear hyperbolic equations”, Partial differential equations – 4, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr., 33, VINITI, Moscow, 1988, 157–247 | MR

[11] G. G. Kazaryan, “On a family of hypoelliptic polynomials”, Izvestija Akad. Nauk Armjan. SSR, ser. matem., 1974, no. 9, 189–211 (in Russian) | MR | Zbl

[12] A. G. Khovanskii, “Newton polyhedra (algebra and geometry)”, Amer. Math. Soc. Transl. (2), 153 (1992), 1–15 | MR

[13] E. Larsson, “Generalized hyperbolicity”, Arkiv for Math., 7:2 (1966), 11–32 | MR

[14] V. N. Margaryan, H. G. Ghazaryan, “On fundamental solutions of a class of weak hyperbolic operators”, Eurasian Math. J., 9:2 (2018), 54–67 | MR

[15] V. N. Margaryan, G. H. Hakobyan, “On Gevrey type solutions of hypoelliptic equations”, Journal of Contemporary Math. Analysis, 31:2 (1996), 33–47 | MR | Zbl

[16] V. N. Margaryan, H. G. Ghazaryan, “On a class of weakly hyperbolic operators”, Journal of Contemporary Math. Analysis, 53:6 (2018), 53–69 | MR | Zbl

[17] W. Matsumoto, H. Yamahara, “On Cauchy Kowalevskaya theorem for system”, Proc. Japan Acad. ser. A, Math Sci., 67:6 (1991), 181–185 | MR | Zbl

[18] V. P. Mikhailov, “The behaviour of a class of polynomials at infinity”, Proc. Steklov Inst. Math., 91 (1967), 61–82 | MR

[19] L. Rodino, Linear partial differential operators in Gevrey spaces, Word Scientific, Singapoure, 1993 | MR | Zbl

[20] S. L. Svensson, “Necessary and sufficient conditions for the hyperbolicity of polynomials with hyperbolic principal parts”, Ark. Mat., 8 (1969), 145–162 | MR

[21] L. Zanghirati, “Iterati di operatori e regolarita Gevrey microlocale anisotropa”, Rend. Sem. Mat. Univ. Padova, 67 (1982), 85–104 | MR | Zbl