Geodesic
Summability of Formal Power-Series Solutions of Partial Differential Equations with Constant Coefficients
Contemporary Mathematics. Fundamental Directions, Proceedings of the International Conference on Differential and Functional-Differential Equations — Satellite of International Congress of Mathematicians ICM-2002 (Moscow, MAI, 11–17 August, 2002). Part 1, Tome 1 (2003), pp. 5-17 
W. Balser. Summability of Formal Power-Series Solutions of Partial Differential Equations with Constant Coefficients. Contemporary Mathematics. Fundamental Directions, Proceedings of the International Conference on Differential and Functional-Differential Equations — Satellite of International Congress of Mathematicians ICM-2002 (Moscow, MAI, 11–17 August, 2002). Part 1, Tome 1 (2003), pp. 5-17. http://geodesic.mathdoc.fr/item/CMFD_2003_1_a0/
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

We study Gevrey properties and summability of power series in two variables that are formal solutions of a Cauchy problem for general linear partial differential equations with constant coefficients. In doing so, we extend earlier results in two articles of Balser and Lutz, Miyake, and Schäfke for the complex heat equation, as well as in a paper of Balser and Miyake, who have investigated the same questions for a certain class of linear PDE with constant coefficients subject to some restrictive assumptions. Moreover, we also present an example of a PDE where the formal solution of the Cauchy problem is not $k$-summable for whatever value of $k$, but instead is multisummable with two levels under corresponding conditions upon the Cauchy data. That this can occur has not been observed up to now.

[1] Balser W., “Divergent solutions of the heat equation: on an article of Lutz, Miyake and Schäfke”, Pac. J. Math., 188 (1999), 53–63 | DOI | MR | Zbl

[2] Balser W., Formal power series and linear systems of meromorphic ordinary differential equations, Springer-Verlag, New York, 2000 | MR | Zbl

[3] Balser W., Miyake M., “Summability of formal solutions of certain partial differential equations”, Acta Sci. Math., 65 (1999), 543–551 | MR | Zbl

[4] Hibino M., “Divergence property of formal solutions for singular first order linear partial differential equations”, Publ. Res. Inst. Math. Sci., 35 (1999), 893–919 | DOI | MR | Zbl

[5] Lutz D. A., Miyake M., Schäfke R., “On the Borel summability of divergent solutions of the heat equation”, Nagoya Math. J., 154 (1999), 1–29 | MR | Zbl

[6] Miyake M., “Newton polygons and formal Gevrey indices in the Cauchy–Goursat–Fuchs type equations”, J. Math. Soc. Japan, 43 (1991), 305–330 | DOI | MR | Zbl

[7] Miyake M., “An operator $l=ai-d_t^jd_x^{-j-\alpha}-d_t^{-j}d_x^{j+\alpha}$ and its nature in Gevrey functions”, Tsukuba J. Math., 17 (1993), 83–98 | MR

[8] Miyake M., “Borel summability of divergent solutions of the Cauchy problem to non-Kovaleskian equations”, Partial differential equations and their applications, eds. Hua C., Rodino L., World Scientific Publ., Singapore, 1999, 225–239 | MR | Zbl

[9] Miyake M., Hashimoto Y., “Newton polygons and Gevrey indices for linear partial differential operators”, Nagoya Math. J., 128 (1992), 15–47 | MR | Zbl

[10] Miyake M., Ichinobe K., “On the Borel summability of divergent solutions of parabolic type equations and Barnes generalized hypergeometric functions”, RIMS Kokyuroku, 1158 (2000), 43–57 | MR | Zbl

[11] Miyake M., Yoshino M., “Toeplitz operators and an index theorem for differential operators on Gevrey spaces”, Funkc. Ekvacioj. Ser. Int., 38 (1995), 329–342 | MR | Zbl

[12] Ouchi S., Asymptotic expansion of singular solutions and the characteristic polygon of linear partial differential equations in the complex domain, Manuscript

[13] Ouchi S., “Formal solutions with Gevrey type estimates of nonlinear partial differential equations”, J. Math. Sci., Tokyo, 1 (1994), 205–237 | MR | Zbl

[14] Ouchi S., “Singular solutions with asymptotic expansion of linear partial differential equations in the complex domain”, RIMS Kokyuroku, 34 (1998), 291–311 | DOI | MR | Zbl

[15] Pliś M. E., Ziemian B., “Borel resummation of formal solutions to nonlinear Laplace equations in 2 variables”, Ann. Pol. Math., 67 (1997), 31–41 | MR | Zbl

[16] Ramis J.-P., “Les séries $k$-sommable et leurs applications”, Lect. Notes Phys., 126 (1980), 178–199 | DOI | MR