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Zbl 0873.35016
Boiti, Chiara; Nacinovich, Mauro
Extension of formal power series solutions. (English)
[J] Rend. Semin. Mat. Univ. Padova 96, 205-235 (1996). ISSN 0041-8994

In the classical Cauchy problem for a linear partial differential equation with initial data on a hypersurface, smooth initial data together with the equation allow to compute the Taylor series of a smooth solution at any given point of the hypersurface. This remark suggests further generalizations of the Cauchy problem, where the assumption that the initial data are given on a formally noncharacteristic initial manifold is dropped and we allow formal solutions (in the sense of Whitney) of the given system on any closed subset as initial data. The problem is then to find classical smooth solutions of the system, whose restrictions in the sense of Whitney are the given initial data.\par In this paper, we take up an extreme case of the generalized Cauchy problem, where the initial manifold reduces to a point. The initial data are then a (vector valued) formal power series $\varphi$ at a point $x_0\in\bbfR^n$, satisfying a system $P(x,D)\varphi=0$ at $x_0$ in the sense of formal power series, and we try to find a smooth function $u$, defined on a neighbourhood of $x_0$, having $\varphi$ as its Taylor series at $x_0$ and satisfying the system $P(x,D)u=0$ on a Schwartz-regular open domain $D\subset U$ with $x_0\in\overline D$. Most of the results will be obtained for systems of linear partial differential operators with constant coefficients.

MSC 2000:: *35E15 Initial value problems of PDE with constant coefficients
35C10 Series solutions of PDE

Keywords: Taylor series; initial manifold reduces to a point

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