Study of the Regularity of Solutions for a Elasticity System with Integrable Data with Respect to the Distance Function to the Boundary

In this article, we are interested in the existence, uniqueness and regularity of the solution of the linear elasticity system. More precisely, the quasi-static elasticity system. In the first part, we study the existence of a weak solution and the regularity in the space W01,p(Ω),∀p∈]1,+∞[\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^{1, p}_0(\Omega ),\ \forall p \in ]1, +\infty [$$\end{document} for a p-integrable source function. In the second part, the very weak solution is introduced which can be considered when the second member is a function with a very weak solution, for example, a locally integrable function. Such source functions lead to a lack of regularity for the solution in the fact that existence in classical spaces is no longer assured. So, to overcome this difficulty, the strategy consists in approaching it by another more regular problem “converging” towards the initial problem “in a direction to be specified”.