Éditeur(s) : HAL CCSD Résumé : To appear in TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS (16 pages). In the last two decades, many algebras of generalized functions have been constructed, particularly the so-called generalized Sobolev algebras. Our goal is to study the latter and some of their main properties. In this framework, we pose and solve a nonlinear degenerated Dirichlet problem with irregular data such as Dirac generalized functions. https://hal.archives-ouvertes.fr/hal-00004903
Éditeur(s) : HAL CCSDElsevier Résumé : International audience The aim of this paper is to prove that the framework of generalized functions of Sobolev type is more suitable to pose and solve some PDEs problems with very irregular data, than the one introduced by J.-F. Colombeau, when ${mathcal{C}}^{infty}$ estimates are not available or out of reach. In such type of algebras, one shows the existence and some qualitative properties of solutions for problems appearing in the mathematical modelisation of oil activities. Journal of Mathematical Analysis and applications
Éditeur(s) : HAL CCSD Résumé : We introduce a general context involving a presheaf A and a subpresheaf B of A. We show that all previously considered cases of local analysis of generalized functions (defined from duality or algebraic techniques) can be interpretated as the B-local analysis of sections of A. But the microlocal analysis of the sections of sheaves or presheaves under consideration is dissociated into a "frequential microlocal analysis " and into a "microlocal asymptotic analysis". The frequential microlocal analysis based on the Fourier transform leads to the study of propagation of singularities under only linear (including pseudodifferential) operators in the theories described here, but has been extended to some non linear cases in classical theories involving Sobolev techniques. The microlocal asymptotic analysis can inherit from the algebraic structure of B some good properties with respect to nonlinear operations. https://hal.archives-ouvertes.fr/hal-00190006
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