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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.

MSC:
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
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