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Dans cette note, nous prouvons l’existence de solutions indéfiniment différentiables d’un système de deux équations aux différences et appliquons la technique utilisée à l’étude des systèmes d’équations linéaires aux dérivées partielles.Dans chaque cas, on montre que les solutions sont les premières composantes des solutions d’un système matriciel que nous étudions.
In this paper are examined some classes of linear and non-linear
analytical systems of partial differential equations. Compatibility conditions are
found and if they are satisfied, the solutions are given as functional series in a
neighborhood of a given point (x = 0).
We study the solvability of equations associated with a complex vector field in
with or coefficients. We assume that is elliptic
everywhere except on a simple and closed curve . We assume that, on , is of infinite type and that vanishes to a constant order. The
equations considered are of the form , with satisfying compatibility
conditions. We prove, in particular, that when the order of vanishing of
is , the equation is solvable in the category but not in the category....
Using Clifford analysis in a multidimensional space some elliptic, hyperbolic and parabolic systems of partial differential equations are constructed, which are related to the well-known classical equations. To obtain parabolic systems Clifford algebra is modified and some corresponding differential operator is constructed. For systems obtained the boundary and initial value problems are solved.
This text is the act of a talk given november 18 2008 at the seminar PDE of Ecole Polytechnique. The text is not completely faithfull to the oral exposition for I have taken this opportunity to present the proofs of some results that are not easy to find in the literature. On the other hand, I have been less precise on the material for which I found good references. Most of the novelties presented here come from a joined work with Luigi Ambrosio.
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