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Some Properties of Solutions to Weakly Hypoelliptic Equations

DOI: 10.1155/2013/526390

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Abstract:

A linear different operator is called weakly hypoelliptic if any local solution of is smooth. We allow for systems, that is, the coefficients may be matrices, not necessarily of square size. This is a huge class of important operators which coverall elliptic, overdetermined elliptic, subelliptic, and parabolic equations. We extend several classical theorems from complex analysis to solutions of any weakly hypoelliptic equation: the Montel theorem providing convergent subsequences, the Vitali theorem ensuring convergence of a given sequence, and Riemann's first removable singularity theorem. In the case of constant coefficients, we show that Liouville's theorem holds, any bounded solution must be constant, and any -solution must vanish. 1. Introduction Hypoelliptic partial differential equations form a huge class of linear PDEs many of which are very important in applications. This class contains all elliptic, overdetermined elliptic, subelliptic, and parabolic equations. Recall that a linear differential operator is called hypoelliptic if any solution to is smooth wherever is smooth. The study of hypoelliptic operators was initiated by H?rmander and others; see, for example, [1–4]. We generalize this class of operators even further by only demanding that any solution to be smooth. We call such operators weakly hypoelliptic. This is not to be confused with partially hypoelliptic operators as introduced by G?rding and Malgrange [5] nor with the almost hypoelliptic operators due to Elliott [6]. We show by example that the class of weakly hypoelliptic operators is strictly larger than that of hypoelliptic operators. The example of a weakly hypoelliptic but nonhypoelliptic operator that we give is defined on and is overdetermined elliptic on . It is of first order, and its principal symbol vanishes at . Thus, the class of weakly hypoelliptic operators allows for a certain degeneracy of the principal symbol on “small sets” and might be of interest for geometric applications. Holomorphic functions are the solutions to the Cauchy-Riemann equations which are elliptic in the case of one variable and overdetermined elliptic in the case of several variables. In any case, they are characterized as solutions to certain hypoelliptic PDEs. We show that the solutions to any weakly hypoelliptic equation share some of the nice properties of holomorphic functions which are familiar from classical complex analysis. Montel's theorem says that a locally bounded sequence of holomorphic functions subconverges to a holomorphic function. This does not hold for real analytic

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