Summability in a monomial for some classes of singularly perturbed partial differential equations

• Autores: Sergio Alejandro Carrillo Torres
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 65, Nº 1, 2021, págs. 83-127
• Idioma: inglés
• DOI: 10.5565/publicacionsmatematiques.v65i1.383651
• Enlaces
  • Texto completo
• Resumen

  • The aim of this paper is to continue the study of asymptotic expansions and summability in a monomial in any number of variables, as introduced in [3, 15]. In particular, we characterize these expansions in terms of bounded derivatives and we develop Tauberian theorems for the summability processes involved. Furthermore,we develop and apply the Borel–Laplace analysis in this framework to prove the monomial summability of solutions of a specific class of singularly perturbed PDEs.

• Referencias bibliográficas
  • W. Balser, “Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations”, Universitext, Springer-Verlag, New York,...
  • W. Balser, Summability of power series in several variables, with applications to singular perturbation problems and partial differential...
  • M. Canalis-Durand, J. Mozo-Fernandez, and R. Sch ´ afke ¨ , Monomial summability and doubly singular differential equations, J. Differential...
  • M. Canalis-Durand, J. P. Ramis, R. Schafke, and Y. Sibuya ¨ , Gevrey solutions of singularly perturbed differential equations, J. Reine Angew....
  • S. A. Carrillo and J. Mozo-Fernandez ´ , Tauberian properties for monomial summability with applications to Pfaffian systems, J. Differential...
  • S. A. Carrillo and J. Mozo-Fernandez ´ , An extension of Borel–Laplace methods and monomial summability, J. Math. Anal. Appl. 457(1) (2018),...
  • S. A. Carrillo, J. Mozo-Fernandez, and R. Sch ´ afke ¨ , Tauberian theorems for k-summability with respect to an analytic germ, J. Math. Anal....
  • O. Costin, On Borel summation and Stokes phenomena for rank-1 nonlinear systems of ordinary differential equations, Duke Math. J. 93(2) (1998),...
  • O. Costin and S. Tanveer, Existence and uniqueness for a class of nonlinear higher-order partial differential equations in the complex plane,...
  • O. Costin and S. Tanveer, Nonlinear evolution PDEs in R+ × Cd: existence and uniqueness of solutions, asymptotic and Borel summability...
  • Y. Haraoka, Theorems of Sibuya–Malgrange type for Gevrey functions of several variables, Funkcial. Ekvac. 32(3) (1989), 365–388.
  • Z. Luo, H. Chen, and C. Zhang, On the summability of the formal solutions for some PDEs with irregular singularity, C. R. Math. Acad. Sci....
  • J. Martinet and J.-P. Ramis, Elementary acceleration and multisummability. I, Ann. Inst. H. Poincar´e Phys. Th´eor. 54(4) (1991), 331–401.
  • J. Mozo-Fernandez ´ , Entire functions polynomially bounded in several variables, Bull. Iranian Math. Soc. 46(4) (2020), 1117–1122. DOI: 10.1007/s41980-...
  • J. Mozo Fernandez and R. Sch ´ afke ¨ , Asymptotic expansions and summability with respect to an analytic germ, Publ. Mat. 63(1) (2019), 3–79....
  • J.-P. Ramis, Les s´eries k-sommables et leurs applications, in: “Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory” (Proc....
  • J.-P. Ramis and Y. Sibuya, Hukuhara domains and fundamental existence and uniqueness theorems for asymptotic solutions of Gevrey type, Asymptotic...
  • J. Sanz, Summability in a direction of formal power series in several variables, Asymptot. Anal. 29(2) (2002), 115–141.
  • B. V. Shabat, “Introduction to Complex Analysis. Part II. Functions of Several Variables”, Translated from the third (1985) Russian edition...