Resurgent analysis in the theory of differential equations with singularities.

*(English)*Zbl 0884.58094From the Introduction: “In Section 1 we introduce the notion of a resurgent function of power type and discuss different representations of these functions. Here we also describe an important subclass of the class of resurgent functions – the so-called resurgent functions with simple singularities and introduce the notion of Lagrangian uniformization.

Section 2 is aimed at solving the main resurgent equation (in the dual space) in the class of resurgent functions. Surely, resurgent equations are determined by the main (pseudo-)differential equation (problem) for which we search asymptotic expansions of solution. To make our considerations applicable to a sufficiently wide class of problems, we investigate a resurgent equation in a rather general situation – to a differential equation whose coefficients have power singularities on a manifold of arbitrary codimension. This class includes, in particular, classical problems on manifolds with singularities of the type of cone or edge, the class of degenerate equations, Sobolev problems and others. At the same time, the method of constructing and solving resurgent equations is quite universal, and the reader will be able to apply the method described in this section to other problems. The mentioned resurgent equation has the form of a sum of a (pseudo-)differential operator corresponding to the principal symbol (Hamiltonian) and translation operators (with different shifts) with coefficients also being \(\psi \text{DO}\)’s. A solution of such an equation is a resurgent function whose support consists of supports determined by the Hamiltonian together with lattices born by translation operators included in the resurgent equation.

In Section 3 we discuss some applications of the developed technique to problems on manifolds with singularities and to the Sobolev problem and present the computations for concrete examples of such problems”.

Section 2 is aimed at solving the main resurgent equation (in the dual space) in the class of resurgent functions. Surely, resurgent equations are determined by the main (pseudo-)differential equation (problem) for which we search asymptotic expansions of solution. To make our considerations applicable to a sufficiently wide class of problems, we investigate a resurgent equation in a rather general situation – to a differential equation whose coefficients have power singularities on a manifold of arbitrary codimension. This class includes, in particular, classical problems on manifolds with singularities of the type of cone or edge, the class of degenerate equations, Sobolev problems and others. At the same time, the method of constructing and solving resurgent equations is quite universal, and the reader will be able to apply the method described in this section to other problems. The mentioned resurgent equation has the form of a sum of a (pseudo-)differential operator corresponding to the principal symbol (Hamiltonian) and translation operators (with different shifts) with coefficients also being \(\psi \text{DO}\)’s. A solution of such an equation is a resurgent function whose support consists of supports determined by the Hamiltonian together with lattices born by translation operators included in the resurgent equation.

In Section 3 we discuss some applications of the developed technique to problems on manifolds with singularities and to the Sobolev problem and present the computations for concrete examples of such problems”.

##### MSC:

58J37 | Perturbations of PDEs on manifolds; asymptotics |

34A05 | Explicit solutions, first integrals of ordinary differential equations |

34E05 | Asymptotic expansions of solutions to ordinary differential equations |

35C20 | Asymptotic expansions of solutions to PDEs |

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\textit{B. W. Schulze} et al., Math. Nachr. 172, 261--281 (1995; Zbl 0884.58094)

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