Mediterr. J. Math. (2018) 15:43
published online February 22, 2018
Springer International Publishing AG,
part of Springer Nature 2018
Study of the Regularity of Solutions for a
Elasticity System with Integrable Data with
Respect to the Distance Function to the
Nada Kassem El Berdan
Abstract. In this article, we are interested in the existence, uniqueness
and regularity of the solution of the linear elasticity system. More pre-
cisely, the quasi-static elasticity system. In the ﬁrst part, we study the
existence of a weak solution and the regularity in the space W
(Ω), ∀p ∈
]1, +∞[ 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 approach-
ing it by another more regular problem “converging” towards the initial
problem “in a direction to be speciﬁed”.
Mathematics Subject Classiﬁcation. 35J25, 35J60, 35850, 35B65, 35Q35.
Keywords. Linear elasticity system, very weak solution, regularity for
very weak solution, weighted spaces, duality method.
Scientists have for centuries attempted to write some models describing the
behavior of the material. More or less generally, accurate or robust, these
models are based on the representation of the deformation phenomena using
the vector ﬁelds and tensors. This describes, in particular, the deformation
of the object as well as the internal constraints (the internal forces involved
between portions) that it undergoes. The behavior laws then join the con-
straints with the resultant deformation.
The theory of linear elasticity lies within the framework of the descrip-
tion of slow deformable solids, and on the other hand it is imposed that the