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Pierre Bonneau

Publications and source records attributed to Pierre Bonneau.

5 recordsLinked to original sources
arXiv2023

Almost holomorphic curves in real analytic hypersurfaces

Using the theory of exterior differential systems, we study the existence of germ of pseudo-holomorphic disk in a real analytic hypersurface locally defined in a complex manifold equipped with J a real analytic almost complex structure. The integrable case in C n with J the multiplication by i has been intensively studied by several authors [DF], [DA1] and [DA2] for example. The non integrable case is drastically different essentially due to the following fact : in generic case, there is no J-invariant objects of dimension bigger than one. This simple observation leads to the non existence of some equivalents of Segree varieties or ideals of holomorphic functions which play a fundamental role in the complex case. Nevertheless in the almost complex case, we adopt the exterior differential system point of view of E.Cartan developed and clarified in [BCGGG].

math.CV↗
arXiv2018

Complex associated to some systems of PDE

In [WW1] and [WW2], the author constructed the complex associated to 1-regular functions. This complex is the equivalent of Dolbeault's complex for holomorphic functions if we replace the Cauchy-Riemann equations by the Cauchy-Fueter equations. In this paper, using the Cartan theory of linear Pfaffian system, we give a direct construction for the Cauchy-Fueter complex, at least in $\R^8$. Moreover, we give a sufficient condition in terms of Cartan's theory, to ensure that a complex associated to a linear PDE system with constant coefficients of order one, contains only operators of order one. In fact, the Cauchy-Fueter equation in $\R^8$ is an illuminating example for which this condition is not satisfied

math.AP↗
arXiv2012

A mi-chemin entre analyse complexe et superanalyse

In the framework of superanalysis we get a functions theory close to complex analysis, under a suitable condition (A) on the real superalgebras in consideration (this condition is a generalization of the classical relation 1 + i^2 = 0 in C). Under the condition (A), we get an integral representation formula for the superdifferentiable functions.We give a result of Hartogs type of separated superdifferentiability, a continuation theorem of Hartogs-Bochner type and a Liouville theorem for the superdifferentiable functions.

math.CV↗
arXiv2012

Entre analyse complexe et superanalyse

In the framework of superanalysis we get a functions theory close to complex analysis, under a suitable condition (A) on the real superalgebras in consideration. Under the condition (A), we get an integral representation formula for the superdifferentiable functions.We give a result of Hartogs type of separated superdifferentiability and a continuation theorem of Hartogs-Bochner type for the superdifferentiable functions.

math.CV↗