Caracterisations de champs de matrices, potentiels matrices et applications aux operateurs traces
Bassem Bahouli · HAL (Le Centre pour la Communication Scientifique Directe) · 2021
Many authors have used stress fields to solve the equilibrium equation of continuum me- chanics. Airy (1863) solved the two-dimensional case, Maxwell (1870) and Morera (1892) solved the three-dimensional case. The above solutions are special cases of those of Beltrami (1892). Gurtin gave an example of solutions that do not have Beltrami’s S = CurlCurlA representation. He showed that if the domain Ω is regular, then this representation is complete in the class of regular stress fields which are self-equilibrated.My thesis title is ”Characterizations of matrix fields, potential matrices and applications to trace operators”. In this work, we are interested by showing many characterizations ofvector fields, of matrix fields and especially by generalizing the result of Gurtin in the case when the open set and the stress fields are not regular.This thesis consists of five chapters. The first chapter presents the research problem ad- dressed in this thesis. It also presents the origin of the subject of research.In the second chapter, we study the operator . In particular, the existence of potential vectors in different functional frameworks.In Chapters 3 and 4, we will show some versions of Beltrami’s completeness and we deduce some Helmholtz decomopsitions for symmetric matrix fields.The last chapter is devoted to the study of the image of different trace operators of functions W 2,p (Ω), W 3,p (Ω) when Ω is a bounded open of R 2 with Lipschitz boundary. The essential ingredient is given by the Airy’s function or by the Beltrami representation.