Analytic solutions of P. D. E. 's (Q2756136)

This paper presents the proof of two results essentially due to Cattabriga and de Giorgi in 1971 about the solvability of partial differential equations with constant coefficients in the space of real analytic functions. The first theorem states that the operator \(\partial_{x_1}+ i\partial_{y_1}\) is not surjective on the space \(A(\mathbb{R}^3)\) of real analytic functions on \(\mathbb{R}^3_{x_1, y_1,x_2}\). The second theorem is an extension of the basic result of Cattabriga and de Giorgi that every linear partial differential operator \(P(D)\) with constant coefficients is surjective on \(A(\mathbb{R}^2)\). In the extension presented here the principal part of the operator \(P(D)\) on \(\mathbb{R}^N\) is assumed to decompose into the product of a hyperbolic and an elliptic factor.NEWLINENEWLINENEWLINEThe problem of the surjectivity of a linear partial differential operator \(P(D) \) with constant coefficients on the space of real analytic functions \(A(G)\) on an open subset \(G\) of \(\mathbb{R}^N\) has been investigated by several authors. \textit{L. C. Piccinini} [Boll. Unione Mat. Ital., IV. Ser. 7, 12-28 (1973; Zbl 0264.35003)] gave in 1973 the first example of a non-surjective operator on \(A(\mathbb{R}^3)\): proving a conjecture of Cattabriga and de Giorgi he showed that the heat equation is not surjective on \(A(\mathbb{R}^3)\). In the same year, \textit{L. Hörmander} [Invent. Math. 21, 151-183 (1973; Zbl 0282.35015)] provided a characterization of all surjective partial differential operators with constant coefficients which are surjective on \(A(G)\) for a convex open set \(G\) in terms of a Phragmén-Lindelöf condition on the complex variety of \(P\). A careful analysis of the Phragmén-Lindelöf conditions on analytic varieties and geometric characterizations of partial differential operators which are surjective on \(A(\mathbb{R}^3)\) and \(A(\mathbb{R}^4)\) has been recently accomplished by \textit{R. Braun}, \textit{R. Meise} and \textit{B. A. Taylor} in a series of articles; see e.g. [Math. Z. 232, 103-135 (1999; Zbl 0933.32047)] and other articles to be published soon. Hörmander's criterion is only valid for convex sets, since it uses Fourier analysis. \textit{T. Kawai} [J. Math. Soc. Japan 24, 481-517 (1972; Zbl 0234.35012)] proved that locally hyperbolic operators on special not necessarily convex bounded open subsets of \(\mathbb{R}^N\) are surjective. The assumption of boundedness was removed by \textit{A. Kaneko} [J. Fac. Sci. Tokio 32, 319-372 (1985; Zbl 0583.35013)]. \textit{M. Langenbruch} [Stud. Math. 140, 15-40 (2000; Zbl 0977.35035)] showed that if an operator \(P(D)\) is surjective on \(A(G)\), then \(P(D)\) admits shifted generalized elementary solutions which are real analytic on an arbitrary relatively compact subset of \(G\). This implies that any localization of the principal part of \(P\) is hyperbolic with respect to any normal vector to the boundary which is non-characteristic for the localization. Under additional assumptions, the principal part must be locally hyperbolic. In his preprint ``Characterization of surjective partial differential operators on spaces of real analytic functions'' form 2002, Langenbruch utilizes Palamodov's and Vogt's theory of the \(\text{Proj}^1\) functor to present a characterization of surjective partial differential operators on \(A(G)\) in the spirit of Kawai's work.