### A link between global solvability and solvability over compacts for systems like : $(P({D}_{x},{D}_{y})u=f,\phantom{\rule{4pt}{0ex}}Qu=0)$

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We examine an elliptic equation in a domain Ω whose boundary ∂Ω is countably (m-1)-rectifiable. We also assume that ∂Ω satisfies a geometrical condition. We are interested in an overdetermined boundary value problem (examined by Serrin [Arch. Ration. Mech. Anal. 43 (1971)] for classical solutions on domains with smooth boundary). We show that existence of a solution of this problem implies that Ω is an m-dimensional Euclidean ball.

In this paper, we prove the existence of a global solution to an initial-boundary value problem for 1-D flows of the viscous heat-conducting radiative and reactive gases. The key point here is that the growth exponent of heat conductivity is allowed to be any nonnegative constant; in particular, constant heat conductivity is allowed.

BGG-operators form sequences of invariant differential operators and the first of these is overdetermined. Interesting equations in conformal geometry described by these operators are those for Einstein scales, conformal Killing forms and conformal Killing tensors. We present a deformation procedure of the tractor connection which yields an invariant prolongation of the first operator. The explicit calculation is presented in the case of conformal Killing forms.

Given a 3-dimensional vector field V with coordinates ${V}_{x}$, ${V}_{y}$ and ${V}_{z}$ that are homogeneous polynomials in the ring k[x,y,z], we give a necessary and sufficient condition for the existence of a Liouvillian first integral of V which is homogeneous of degree 0. This condition is the existence of some 1-forms with coordinates in the ring k[x,y,z] enjoying precise properties; in particular, they have to be integrable in the sense of Pfaff and orthogonal to the vector field V. Thus, our theorem links the existence...