zbMATH — the first resource for mathematics

Differential equations on manifolds with singularities in classes of resurgent functions. (English) Zbl 0922.35035
The authors develop a new method of constructing asymptotic expansions for solutions of differential equations on manifolds with singularities near singular points of these manifolds. The main tool of this method is the theory of resurgent functions of power type in the multi-dimensional case. It is proved that the set of such functions forms an algebra with respect to the usual multiplication. The structures of the resurgent functions are given also. For example, if \(f\) is a resurgent function with simple singularities and its support consists, for given value of \(\rho\), of a single point, then it has an asymptotic expansion of the form \[ f(r)=e^{S(\ln r)}\sum^{\infty}_{k=0} a_k(\ln r), \] where \(\ln r=(\ln r_1,\ln r_2,\dots ,\ln r_n), r_k\in \mathbb{R} _+, k=1,\dots,n; S\) and \(a_k\) are some smooth complex-valued functions. This theory is applied to asymptotic expansions for the solutions of equations of the type \(Hu=f\), where \(H\) is a differential operator on the manifold \(M\) near the vertex \(V\). This operator has the form \[ H=t^{-m}\sum_{j=0}^m a_j(t) \left(t\frac{\partial}{\partial t}\right)^j,\quad 0\leq t\leq 1, \] where \(a_j(t), j=0,\dots,m\), are partial differential operators in local coordinates. Under some conditions on the structure of the manifold near \(V\) the authors prove the solvability of the above equation in the set of resurgent functions, if \(f\) is also a resurgent function. One-dimensional and two-dimensional cases are investigated in detail.

35C20 Asymptotic expansions of solutions to PDEs
58J05 Elliptic equations on manifolds, general theory
35B40 Asymptotic behavior of solutions to PDEs
35J70 Degenerate elliptic equations
35J25 Boundary value problems for second-order elliptic equations
Full Text: DOI
[1] Agmon, Properties of Solutions of Ordinary Differential Equations in Banach Space, Comm. Pure Appl. Math. 16 pp 121– (1963) · Zbl 0117.10001
[2] Kondrat’ev, Boundary Problems for Elliptic Equations in Domains with Conical or Angular Points, Trans. of Moscow Math. Soc. 16 (1967) pp 287–
[3] Trudy Mosk. Mat. Obshch. 15 pp 400– (1966)
[4] Kondrat’ev, Boundary-Value Problems for Partial Differential Equation in Non- Smooth Domains, Russian Math. Surveys 38 (2) pp 1– (1983)
[5] Schulze , B.-W. 1992
[6] Schulze, Conical Singularities and Asymptotics, Mathematics Topics 4, in: Pseudo-Differential Boundary Value Problems (1994)
[7] Melrose, The Atiyah - Patodi - Singer Index Theorem, Research Notes in Mathematics (1993) · Zbl 0796.58050
[8] Écalle, Les Fonctions Résurgentes, I, II, III (1981-1985)
[9] Ramis, Springer Lecture Notes in Physics No. 126, in: Les Series k-Sommables et Leurs Applications (1980) · Zbl 1251.32008
[10] Ramis , J.-P. 1991
[11] Martinet, Elementary Acceleration and Multisommability, Ann. Inst. H. Poincare 54 (1) pp 1– (1991)
[12] Malgrange, Introduction aux Travaux de J., Écalle, L’Enseignement Mathématique 31 pp 261– (1985)
[13] Malgrange, Travaux de Écalle et de Martinet-Ramis sur le Systémes Dinamiques., In: Séminaire Bourbaki Number 582 pp 1981– (1982)
[14] Malgrange , B. 1994
[15] Berry, Hyperasymptotics for Integrals with Saddles, Proc. R. Soc. Lond. pp 657– (1991) · Zbl 0764.30031
[16] Martinet, Les Derniers Manuscrits de Jean Martinet, Ann. Inst. Fourier, Grenoble 42 (31) (1992)
[17] Delabaere , E. Pham , F.
[18] Delabaere , E. Dillinger , H. Pham , F.
[19] Candelpergher , B. Nosmas , J. C. Pham , F. 1993
[20] Loday-Richard, Stokes Phenomenon, Multisummability and Differential Galois Groups Annales de l’Institur Fourier 44 (3) pp 92– (1994)
[21] Tougeron , J.-C. 1990
[22] Voros, The Return of Quartic Oscillator: The Complex WKB Method, Ann. Inst. H. Poincaré 39 (3) pp 211– (1983) · Zbl 0526.34046
[23] Sternin, On a Notion of Resurgent Function of Several Variables, Matematische Nachrichten 171 pp 283– (1995) · Zbl 0842.32006
[24] Sternin, Resurgent Analysis in Several Variables, I. General Theory. In: Partial Differential Equations. Models in Physics and Biology, Mathematical Research 82 (1994) pp 351– (1993) · Zbl 0831.35021
[25] Sternin, Borel - Laplace Transform and Asymptotic Theory. CRC-Press (1996)
[26] Sternin, Resurgent Analysis in Several Variables, II. Applications. In: Partial Differential Equations. Models in Physics and Biology, Mathematical Research 82 (1994) pp 378– (1993) · Zbl 0847.35019
[27] Schulze, Resurgent Analysis and Differential Equations with Singularities., In: Partial Differential Equations. Models in Physics and Biology, Mathematical Research 82 (1994) pp 401– (1993)
[28] Schulze, Resurgent Analysis in the Theory of Differential Equations with Singularities, Matematische Nachrichten 170 pp 1– (1994)
[29] Komatsu, Laplace Transform of Hyperfunctions., A New Foundation of Heaviside Calculus, J. Fac. Sc. Univ. Tokyo IA 34 pp 805– (1987) · Zbl 0644.44001
[30] Maslov , V. P. 1972
[31] Mishchenko, Lagrangian Manifolds and the Maslov Operator (1990)
[32] Sternin, Analytic Continuation of Solutions to Integral Equations and Localization of Singularities, Max - Planck -Institut für Mathematik 93 (1994)
[33] Sternin, Fourier - Maslov Transform in the Space of Multivalued Analytic Functions, Math. Notes 49 (5-6) pp 267– (1991) · Zbl 0755.32003
[34] Sternin, Differential Equations on Complex Manifolds (1994)
[35] Thom, Ensembles et Morphismes Stratifiés, Bul. A.M.S. 75 pp 240– (1969) · Zbl 0197.20502
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.